Pseudo Master Curve Analysis of an Infinite Number of Parallel First-Order Reactions: Improved Distributed Activation Energy Model

Abstract

The so-called Distributed Activation Energy Model (DAEM) has been used extensively, mainly to analyze pyrolysis reactions of solid reactants. The model expresses many parallel first-order reactions using the distributions of activation energy f(E) and frequency factor k₀(E). Miura and Maki presented a method to estimate both f(E) and k₀(E) in the DAEM in 1998. This model has been used successfully by many researchers. In this paper more general basic equations are derived for describing an infinite number of parallel first-order reactions by extending the basic equations for the finite number of parallel first-order reactions. Revisiting the Miura–Maki method based on the general basic equations, a graphical analysis method that may be called “Pseudo Master Curve Analysis” is presented. The method not only supplements the Miura–Maki method but gives the underlying concept of the Miura–Maki method clearly. It is also shown that the graphical method can be applicable to analyze single reactions and the experimental data obtained using isothermal reaction techniques. Next, a method that improves the estimation accuracy of k₀(E) is presented. Practical examples analyzing several experimental data are also given to show the usefulness and validity of the Miura–Maki method and the graphical method. Through the examination, it is proposed that the DAEM should be renamed, for example, as the Distributed Rate Constant Model (DRCM).

1. Introduction

The so-called distributed activation energy model (DAEM), originally developed by Vand,¹ has been widely used to analyze complex reactions such as pyrolysis of fossil fuels, thermal regeneration reaction of activated carbon, etc.²⁻¹² The model assumes that many irreversible parallel first-order reactions that have different rate parameters occur simultaneously. This model relates the unreacted fraction of solid reactant 1 – x against time t by

1

where f(E) is a normalized function of E, and k₀ and E are, respectively, frequency factor and activation energy of the first-order reaction of the fraction f(E)dE.

When this equation was derived by Vand, k₀ was assumed to be the same for all reactions a priori. The author calls this assumption a “constant k₀ assumption” in this paper. Then most researchers followed the “constant k₀ assumption” when using the DAEM. In addition, f(E) had been approximated by the Gaussian distribution to facilitate kinetic analysis. This approximation enabled representation of a set of an infinite number of parallel first-order reactions using only three parameters, k₀ and two parameters defining the Gauss function f(E). The constant k₀ value and thus defined f(E), however, are interrelated.^13,14 The work of Lakshmanan et al.,¹⁵ for example, clearly showed that a number of sets of k₀ and f(E) describe the experimental data within the limits of experimental accuracy. To overcome this weakness, Miura¹⁶ and Miura and Maki¹⁷ presented a simple method to estimate both f(E) and k₀(E) from three sets of experimental data obtained at different constant heating rates without assuming functional forms for f(E) and k₀(E). This analysis method has been widely and successfully used to analyze complex reactions including pyrolysis of heavy carbonaceous resources.¹⁸⁻³⁵

Although many factors and elements in addition to the kinetic model must be taken into account for practical application, this paper is limited to just kinetic analysis. It is first shown that eq 1 is not a suitable basic equation representing a set of an infinite number of parallel first-order reactions. More general basic equations are derived by natural extension of a finite number of parallel first-order reactions to the infinite number of parallel first-order reactions. The Miura–Maki method is revisited to clarify the underlying concept and the validity of the method using the derived general equations. Next, a general graphical analysis method that may be called “Pseudo Master Curve Analysis” is presented. The method exactly follows the Miura–Maki method, estimates E and k₀ simultaneously, and gives the basic concept of the analysis method properly. It is also shown that the graphical method can be applicable to analyze single reactions and the experimental data obtained using the isothermal reaction technique. Next, a method that improves the estimation accuracy of k₀(E) is presented to overcome the weakness of the Miura–Maki method. Practical examples analyzing several experimental data are also given to show the usefulness and validity of the proposed graphical method. Through these examinations the significance of the proposed graphical method is shown.

2. Theory

The first task of this paper is to derive general basic equations representing a set of an infinite number of parallel first-order reactions. Since it is done by natural extension of the number of reactions from one to finite and finite to infinite, basic equations are shown in the order of single first-order reaction, N parallel first-order reactions, and an infinite number of parallel first-order reactions.

2.1. General Basic Equations

2.1.1. Single Reaction

We assume that the changing rate of the conversion of solid reactant x, dx/dt, is given by

2

where k₀ and E are, respectively, frequency factor and activation energy. This equation is formally integrated to give

3-a

or

3-b

where τ is a dimensionless time. The kinetic analysis is to estimate k₀ and E by fitting experimentally obtained x vs t and/or dx/dt vs t relationships with theoretical ones. The simplest way to obtain x vs t and/or dx/dt vs t relationships may be to measure the weight change of the solid reactant at a constant temperature (isothermal reaction) or under a programmed temperature change (Temperature-Programmed Reaction with a constant heating rate a; TPR) in a fixed gas atmosphere using a sensitive thermobalance. Then eq 3-a and 3-b are applied to the isothermal reaction and the TPR for the kinetic analysis.

For isothermal reaction, the dimensionless time τ is rewritten as

4

For TPR, T is related to time t by

5

Inserting this relationship to eq 3-a and integrating it gives

6

where the initial temperature T₀ was set equal to 0 because T₀ can be chosen so that the reaction rate at T = T₀ can be regarded as zero. Rearranging eq 6 with the substitution of u = E/RT gives

7

where p(u) is called the “p-function” in the field of thermal analysis. Unfortunately, p(u) cannot be integrated analytically. Employing the following approximate equation for p(u)³⁶

8

eq 7 is reduced to

9

The term RT²/aE can be regarded as a kind of time for the TPR.

Equation 3-a or 3-b combined with eq 4 gives the basic equation for isothermal reaction, and eq 3-a or 3-b combined with eq 9 gives the basic equation for TPR of integral form.

2.1.2. N Parallel First-Order Reactions

N independent parallel first-order reactions are defined as the sum of N individual first-order reactions whose reaction rates are defined similarly as in eq 2. The reaction rate of the i-th reaction is written as

10

where x* represents the initial reduced amount of reactant, Δx^*_i is the initial reduced amount of the i-th reactant, is the conversion of the i-th reactant, and k_0i and E_i are, respectively, the frequency factor and activation energy of the i-th reaction. Mathematically, Δx^*_i and are defined so as to satisfy

11

Equation 10 is integrated to give

12-a

or

12-b

with

13-a

13-b

The overall conversion rate dx/dt and the overall unreacted fraction 1 – x can be written as

14

15

2.1.3. Infinite Number of Parallel First-Order Reactions (N → ∞)

Since this model is an extension of N parallel first-order reactions to parallel first-order reactions with N → ∞, the reaction rate of the infinitesimal amount of reduced reactant is written by increasing the number of reactions N to infinity. The rate equation corresponding to eq 10 is written as follows:

16

where x_dx*(x*) is the conversion of the reaction of the infinitesimal fraction dx* for x* = x*, and k₀ (x*) and E₀ (x*) are, respectively, the frequency factor and activation energy of the reaction of dx*. This shows that Δx^*_i and Δx^*_i, , k_0i, and E_i in eq 10 are replaced, respectively, by dx*, x_dx* (x*), k₀ (x*), and E₀ (x*) in eq 16. Mathematically, dx* and x_dx*(x*) are defined similar to eq 11 as follows:

17

Equation 16 is integrated to give

18-a

or

18-b

with

19-a

19-b

where the dimensionless time τ_dx*(x*) is regarded as a general time for the reaction of the fraction dx* for x* = x*.

The overall conversion rate dx/dt and the overall unreacted fraction 1 – x can can be formulated by increasing the number of reactions, N, from N to ∞ in eqs 14 and 15 to give

20

21

Equations 20 and 21 are derived as a natural extension of eqs 15 and 16. Equation 21 for TPR, for example, gives a 1 – x value at a specified a and T. To obtain the 1 – x value, 1 – x_dx* (x*) vs x* relationships calculated for covering 0 to 1 of x* using E (x*) vs x* and k₀ (x*) vs x* relationships at the specified a and T must be integrated over 0 to 1 of x*. It is stressed here that k₀ (x*) and E (x*) are the values for the reaction of dx* for x* = x*. More importantly, note that eqs 20 and 21 are applicable for a single reaction and N parallel reactions also. In this sense, eqs 20 and 21 are general basic equations for parallel first-order reactions. It is shown later how eq 21 works using assumed E (x*) vs x* and k₀ (x*) vs x* relationships.

Equations 20 and 21 do not include the distribution function f(E) which is involved in the basic equation for DAEM as given by eq 1. Equation 1 can be obtained when we set

22

in eq 21. This means that eq 1 is a modified form of the original basic eq 21. When Vand developed eq 1, he assumed the same k₀ for all reactions a priori. Under the constant k₀ assumption only the E (x*) vs x* relationship defines reactions. In other words, the difference of reactions among different dx* fractions is given by the difference of only E, which might naturally lead to writing dx* as given by eq 22. Since every dx* fraction has different k₀ and E in general, however, both E (x*) vs x* and k₀ (x*) vs x* relationships are required to express the x vs t or x vs T relationship. This shows that the kinetic analysis for parallel first-order reactions with N → ∞ is to estimate the E (x*) vs x* and k₀ (x*) vs x* relationships from experimental x vs t or x vs T data. Once the E (x*) vs x* and k₀ (x*) vs x* relationships are estimated, dx/dt vs t and x vs t are calculated for any heating profiles using eqs 20 and 21. This discussion clarifies that the DAEM is not an appropriate terminology to represent the infinite number of parallel first-order reactions. It should be renamed, for example, as the Distributed Rate Constant Model (DRCM).

3. Methods of Kinetic Analysis

3.1. Generation of x vs T Relationships Using Assumed E (x*) vs x* and k₀ (x*) vs x* Relationships

In 1995 Miura¹⁶ presented a method to estimate both E(x*) vs x* and k₀(x*) vs x* relationships from experimental x vs T data obtained using the TPR technique with three or more heating rates. This method was named the Differential method later. In 1998 Miura and Maki¹⁷ presented a simpler and more accurate analysis method than the Differential method. It was named the Integral method. In this paper, a graphical analysis method that is equivalent to and/or supplements the Integral method is proposed. Then the Integral method is re-examined here to clearly show the underlying concept of the method. To do so, simulated x vs T data are generated for single reactions (N = 1) and parallel first-order reactions with N → ∞ by assuming E(x*) vs x* and k₀(x*) vs x* relationships as shown, respectively, in panels (a) and (b) of Figure 1. For the single reaction, E and k₀ are assumed, respectively, as 250 kJ mol s^–1 and 2 × 10¹⁶ s^–1. They are represented by the horizontal solid lines when shown as the E(x*) vs x* and k₀(x*) vs x* relationships. The relationships for N → ∞ are replotted as f(E) vs E and k₀ vs E relationships in Panels (c) and (d) of Figure 1 for reference. Since most of the works using the original DAEM assumed that f(E) is given by a Gaussian distribution, the assumed E(x*) vs x* relationship for N → ∞ is given so that f(E) is represented by

23

with E̅ = 250 kJ mol^–1 and σ_E = 30 kJ mol^–1. In addition, the assumed k₀ (x*) vs x* relationship is also given so that g(ln k₀) vs ln k₀ will be represented by the Gaussian distribution defined as

24

with k̅₀ = 2 × 10¹⁶ s^–1 and σ_lnk₀ = 2.4. With the assumed f(E) and g(ln k₀), the ln k₀ vs E relationship is given by the linear line as shown in Panel (d) of Figure 1. The k₀ vs E relationship in addition to the f(E) vs E relationship defines the parallel first-order reactions with N → ∞ when we stick to eq 1 as the basic equation.

Figure 1.

E (x*) vs x* and k₀ (x*) vs x* relationships assumed and those estimated.

Panels (a) and (b) of Figure 2 show the x vs T relationships generated for a = 2, 5, and 10 K min^–1, respectively, for the single reaction and the reactions with N → ∞. For the single reaction, eq 3-b was simply applied with eq 9 to generate the x vs T relationships. For the reactions with N → ∞, eq 21 is used with eq 19-b. However, eq 21 is applicable for both a single reaction and the reactions with N → ∞ as stated above. Then it was examined how eq 21 works to generate x vs T relationships for both the single reaction and the reactions with N → ∞. To obtain a 1 – x value for a specified a and T using eq 21, 1 – x_dx*(x*) vs x* relationships calculated for covering 0 to 1 of x* using E(x*) vs x* and k₀(x*) vs x* relationships must be integrated over 0 to 1 of x* as stated above. Panels (a) and (b) of Figure 3 show the 1 – x_dx*(x*) vs x* relationships constructed for four T values at a = 2 K min^–1, respectively, for the single reaction and for the reactions with N → ∞. The 1 – x value at T = 675 K, for example, is given by the hatched area in each Panel. The x vs T relationships given in Panel (b) of Figure 2 were generated by integrating the 1 – x_dx*(x*) vs x* relationships constructed at many T values for a = 2, 5, and 10 K min^–1.

Figure 2.

x vs T relationships generated using assumed E (x*) vs x* and k₀ (x*) vs x* relationships.

Figure 3.

1 – x_dx* (x*) vs x* relationships for calculating the x vs T relationship at a = 2 K min^–1.

3.2. Integral Method and Graphical Analysis for Single Reaction

We will analyze the x vs T relationships given in Panel (a) of Figure 2 by assuming that the reaction will be represented by a first-order reaction. The target of the analysis is to estimate k₀ and E. There are several established methods for this purpose. Here the so-called Kissinger–Akahira–Sunose method,³⁷⁻³⁹ which is applicable to any single reaction, is used to estimate E. The graphical method presented by Ozawa,⁴⁰ which is called “Master Curve Analysis”, is used to estimate k₀. This is because the two methods give a clue about developing a graphical method for the parallel first-order reactions with N → ∞.

The Kissinger–Akahira–Sunose method utilizes the Arrhenius type plot to estimate E. When it is applied to the first-order reaction, the governing equation is represented as

25

This is obtained by rearranging eq 3-a in combination with eq 9. This equation shows that the plot of ln(a/T²) vs 1/T at the same x value for different heating rates of a₁, a₂, a₃, ··· gives a straight line. The activation energy can be obtained from the slope of the linear line. What must be stressed here is that the method can be applicable only when parallel linear lines are obtained for different x values because eq 25 is derived for a single reaction by assuming a constant E value. Nonparallel plots for different x values indicate that the assumption is not valid and that the target reaction is not a single reaction. Applying the method to analyze the x vs T relationships given in Panel (a) of Figure 2 gives exactly parallel linear lines for different x values and gives E = 250 kJ mol^–1 from the slope. The k₀ value is obtained exactly as k₀ = 2 × 10¹⁶ s^–1 using one point of the parallel lines, in other words using a set of a, T, and x values with the estimated E value.

When x vs t relationships obtained experimentally with three or more temperatures under isothermal conditions are used for the kinetic analysis, eq 3-a combined with eq 4 is rearranged after taking the natural logarithm to give

26

This equation shows that the Arrhenius type plots of ln(1/t) vs 1/T for the same x value at different temperatures gives parallel straight lines. The plot also allows us to estimate E from the slope and k₀ from the intercept of one ln(1/t) vs 1/T plot.

Ozawa’s graphical method was developed to find suitable reaction models and k₀ using the so-called master curves, x vs τ relationships, constructed for various reaction models including first-order reaction. Since the Kissinger–Akahira–Sunose method estimates both E and k₀ when the target reaction is known as first-order reaction, Ozawa’s graphical method is not necessary in principle. Here it is shown how Ozawa’s method works for first-order reaction to estimate k₀ using the known E because the graphical method for parallel first-order reactions with N → ∞ is developed using a similar concept. This method directly utilizes the x vs τ relationship. Figure 4 shows the x vs τ relationship given by eq 3-b with the solid line. What is stressed here is that the abscissa τ is taken as the logarithm scale. The shape of the curve does not change by changing the abscissa to βτ (β: a positive constant) as long as the abscissa is taken as a logarithm scale. The dimensionless time τ is set equal to kt (= k₀e^–E/RT·t) for isothermal reaction and to k₀e^–E/RT·RT²/aE for TPR as shown above. We will call Figure 4 the mother diagram here. The mother diagram is utilized to find k₀ for both isothermal reaction and TPR by following the procedure given below:

• (i)
  Convert t or T to the time θ defined by

  27-a

  27-b
• (ii)Plot x against θ by taking the abscissa same logarithm scale as the mother diagram for different temperatures or different heating rates. Let us call the collection of the x–θ relationships the “x–θ diagram” by using the same notation used by Criado.⁴¹ When experimental data exactly follow the first-order reaction and the E value is correctly estimated, the x–θ relationships for different temperatures or different heating rates converge to a single line.
• (iii)The x–θ diagram can be exactly superimposed on the mother diagram when experimental data exactly follow the first-order reaction. Comparing the abscissas of the mother diagram and the x–θ diagram at the best superimposition, k₀ can be obtained as the proportional constant.

Figure 4.

Superimposition of the x–θ diagram on the x – τ mother diagram for the single first-order reaction and the x vs τ_dx* (x^*_Select) diagram for the parallel reactions with N → ∞.

The x – θ diagrams constructed by the following procedure (ii) with the estimated E = 250 kJ mol^–1 for a = 2, 5, and 10 K min^–1 converged to a single curve. It was superimposed on the x – τ diagram by taking θ as the top axis in Figure 4. Reading that θ = 1 × 10^–16 s at τ = 2, k₀ is estimated to be 2 × 10¹⁶ s^–1. This graphical method allows us to select the k₀ value by carefully viewing whole experimental data. Since it is critical to collect reliable and reproducible experimental data for practical application of any analysis methods, the graphical method is powerful not only to estimate k₀ but also to examine the accuracy of experimental data.

3.3. Integral Method for Parallel First-Order Reactions with N → ∞

3.3.1. New Finding That the General Basic Equation Provided

To extend the idea of the graphical method for a single reaction to the parallel first-order reactions with N → ∞, the performance of eq 21 was examined at nine x levels of 0.1, 0.2, ···, and 0.9. This can be done by reading T values corresponding to the selected x levels from Panel (b) of Figure 2, as exemplified as T_a2, T_a5, and T_a10 at x = 0.3, and by inserting them into eq 18-b with eq 19-b. Figure 5 shows thus calculated 1 – x_dx*(x*) vs x* relationships for a = 2, 5, and 10 K min^–1 at the nine x levels. The 1 – x_dx*(x*) values spread over a wide range of x* at all the x levels, indicating that many reactions are occurring at any x levels. The 1 – x_dx*(x*) vs x* relationships are dependent on the heating rate, but we can clearly see that the 1 – x_dx*(x*) vs x* relationships for a = 2, 5, and 10 K min^–1 almost exactly come together at only one point marked by the key ▽ at each x level. We name here the x* value at the point of the key ▽, x^*_Select, as exemplified for x = 0.3. Namely, the x^*_Select value for x = 0.3 is 0.308. The 1 – x_dx*(x*) value at x* = x^*_Select, 1 – x_dx* (x^*_Select), is written using eq 18-b as

28

where τ_dx*(x^*_Select) is the value of τ_dx*(x*) for x* = x^*_Select. The significant and important finding here is that eq 28 holds independent of heating rates, at least for a = 2, 5, and 10 K min^–1, at x^*_Select that is uniquely related to each x.

Figure 5.

1 – x_dx*(x*) vs x* and 1 – x_dx* (x^*_Select) vs x^*_Select relationships at several x levels for a = 2, 5, and 10 K min^–1.

To show this finding more clearly, a diagram showing the x vs τ_dx*(x^*_Select) relationship was constructed. This could be done by estimating x^*_Select values for x covering 0 to 1. For this purpose, 1 – x_dx*(x*) vs x* relationships were constructed at 99 x levels of 0.01, 0.02, ···, and 0.99 for a = 2, 5, and 10 K min^–1, and the x^*_Select value corresponding to each x level was estimated. Figure 6 shows thus estimated x^*_Select vs x relationships at 99 x levels covering 0.01 to 0.99 including the points marked by the key ▽ at the nine x levels in Figure 5. Using the x^*_Select vs x relationships, the x vs τ_dx*(x^*_Select) relationship was constructed at 99 x levels as plotted in Figure 4 with the x vs τ relationship for the single reaction. This can be done because τ_dx*(x^*_Select) is reduced to τ for the single reaction if E(x^*_Select) and k₀(x^*_Select) are the same for all reactions. The x vs τ_dx* (x^*_Select) relationship is regarded as the mother diagram for the assumed parallel first-order reactions with N → ∞. The x vs τ_dx*(x^*_Select) diagram gives a key to develop a graphical analysis method for the parallel first-order reactions with N → ∞, although it can be constructed only when E(x)* vs x* and k₀(x*) vs x* relationships are given.

Figure 6.

True or assumed x^*_Select vs x relationships for the parallel reactions with N → ∞.

Exactly the same diagram is also constructed under isothermal conditions using the τ_dx*(x^*_Select) that is obtained by setting x* = x^*_Select in eq 19-a.

Then the question is how we utilize eq 28 and/or the underlying concept of the mother diagram to estimate E(x*) vs x* and k₀(x*) vs x* relationships from experimentally obtained x vs T relationships.

3.3.2. Miura–Maki Method

Miura and Maki gave an answer to the question above in 1998.¹⁷ Here the essence of the Miura–Maki method is shown based on the finding above. Let us assume that the temperatures giving an x for more than three heating rates of a₁, a₂, a₃, and ··· are, respectively, T₁, T₂, T₃, and ···. Equation 28 is applied at the x level, and the following relationships hold at the x^*_Select corresponding to the x level.

29

These relationships are used to estimate the E(x*) vs x* and k₀(x*) vs x* relationships as follows:

• 1Equation 29 shows that E(x^*_Select) can be obtained from any two sets of (a_i, T_i) from more than three sets of (a_i, T_i) in principle. For better estimation of E(x^*_Select), it is of course recommended to use more than three sets of (a_i, T_i).
• 2Next, k₀(x^*_Select) is estimated using a set of (a_i, T_i) with the estimated E(x^*_Select) when the value of 1 – x_dx*(x^*_Select) is given.
• 3Repeating the above procedure at many x levels, E(x^*_Select) and k₀(x^*_Select) are obtained against x.
• 4If x^*_Select is related to x, E(x^*_Select) vs x and k₀(x^*_Select) vs x relationships are converted to E(x*) vs x* and k₀(x*) vs x* relationships.

The above procedure shows that two approximations or assumptions are needed to obtain E(x*) vs x* and k₀(x*) vs x* relationships: the first approximation is for the values of 1 – x_dx*(x^*_Select) at all x levels, and the second one is the relationship of x^*_Select and x.

In 1994 Miura¹⁶ presented a method to estimate f(E) and k₀(E) involved in the basic equation, eq 1. The method is based on the approximation that only one reaction, the reaction of only dx*(x^*_Select) using the notation of this paper, is occurring at a selected x. This approximated a priori that the x^*_Select is equal to the corresponding x, that is, x^*_Select ≅ x. The approximation, x^*_Select ≅ x, is judged to be well acceptable, because the x^*_Select values are rather close to the corresponding x values as Figure 6 clearly shows. The E(x^*_Select) value corresponding to each x was obtained by using the methodology that Miura and Maki named the Differential method later. To estimate the corresponding k₀(x^*_Select) from the E(x^*_Select) estimated, the approximation, 1 – x_dx*(x^*_Select) ≅ 0.58, was employed for all reactions. We will call the k₀(x^*_Select) using the approximation “First estimate of k₀(x^*_Select)” and will represent it as k_0,Est1(x^*_Select). Then k_0,Est1(x^*_Select) is obtained from eq 28 as

30

In 1998 Miura and Maki¹⁷ found that eq 28 with the approximation 1 – x_dx*(x^*_Select) ≅ 0.58 for all reactions, or eq 30, can be rearranged to give

31

This equation allowed us to estimate E(x^*_Select) from the slope and k_0,Est1(x^*_Select) from the intercept by performing the Arrhenius type plot of ln(a/T²) vs 1/T with three or more heating rates at each x level without knowing the x^*_Select value. If we say this concretely for x = 0.3 using Panel (b) of Figure 2 and Figure 5, the plot of ln(a/T²) vs 1/T performed for 1/T_a2, 1/ T_a5, and 1/T_a10 gives an E(x^*_Select) value of x* (= x^*_Select) = 0.308. Performing the Arrhenius type plots of ln(a/T²) vs 1/T at the same x levels covering 0 to 1, we can estimate both E(x^*_Select) vs x and k_0,Est1(x^*_Select) vs x relationships. This can be done by using the x vs T relationships measured with three or more heating rates. With the approximation of x^*_Select ≅ x for all x values the E(x^*_Select) vs x and k_0,Est1(x^*_Select) vs x relationships are replaced by the E(x*) vs x* and k_0,Est1(x*) vs x* relationships. This is the interpretation of the Miura–Maki method based on the basic equation, eq 21. The Miura–Maki method was named the Integral method and was found to be a more accurate analysis method than the Differential method.

Here, the author must make a correction to the statement made to develop the Miura–Maki method. Miura and Maki assumed that “only a reaction having E(x^*_Select) and k₀(x^*_Select) occurs at x (x^*_Select ≅ x)” to develop eq 31. However, the statement is not acceptable, as Figure 5 clearly shows. Then the statement must be replaced by the above finding: “the conversions x_dx(x^*_Select) are the same for different a values at x^*_Select (≅ x)”. This correction changes neither the analysis procedure nor the estimated E(x*) vs x* and k_0,Est1(x*) vs x* relationships. Then, the procedure presented by Miura and Maki is written using the correction as follows:

• (1)Measure x vs T relationships with three or more heating rates.
• (2)Calculate the values of a/T² at selected x values from the x vs T relationships obtained for the different heating rates.
• (3)Plot ln(a/T²) vs 1/T at the selected x values.
• (4)Determine the E(x^*_Select) and k_0,Est1(x^*_Select) values at the selected same x levels by following eq 31 to obtain E(x^*_Select) vs x and k_0,Est1(x^*_Select) vs x relationships.
• (5)The E(x^*_Select) vs x and k_0,Est1(x^*_Select) vs x relationships are converted to E(x*) vs x* and k_0,Est1(x*) vs x* relationships using the approximation x^*_Select ≅ x.

Here it is worth noting again that the E(x^*_Select) vs x and k_0,Est1(x*) vs x relationships are obtained without knowing x^*_Select values. The approximation x^*_Select ≅ x is needed to relate x to x* through x^*_Select. Needless to say, it is not always necessary to perform the plots given by procedure (3). The procedure is added to show that the plot is the same as that for single reactions. Both E(x^*_Select) and k_0,Est1(x^*_Select) are estimated by just applying eq 31 to more than three sets of (a_i, T_i). The plot is, however, informative to know the soundness of experimental data or to judge the applicability of models.

It is possible to derive a similar analysis method for isothermal reactions. Equation 18-a with eq 19-a is written for x* = x^*_Select as follows:

32

Reorganizing this equation using the approximation, 1 – x_dx*(x^*_Select) ≅ 0.58, gives

33

Then the plots ln(1/t) vs 1/T at selected x (≅ x^*_Select) values for experimental x vs t data obtained with three or more temperatures give both E(x*) vs x* and k_0,Est1(x^*_Select) vs x* relationships. However, the application of the analysis method for isothermal reactions may be limited because it is rather difficult to obtain reliable x vs t data with three or more temperatures for slow reactions, for example.

3.3.3. Analysis of the Parallel First-Order Reactions with N → ∞ Using the Miura–Maki Method

The Miura–Maki method thus corrected was applied to analyze the reactions with N → ∞. Figure 7 shows the plots of ln(a/T²) vs 1/T following procedures (2) and (3) for the x vs T relationships given for N → ∞ in Panel (b) of Figure 2. The plots are made at 99 levels of x ranging from 0.01 to 0.99, but lines are drawn at 9 x levels of 0.1, 0.2, ···, and 0.9 to avoid the cram of lines. The plots are not parallel to each other, and their slopes clearly increase with the increase of x. Following the procedures (4) and (5), E(x*) vs x* and k_0,Est1(x*) vs x* relationships estimated at 99 levels of x are shown by the key △, respectively, in Panels (a) and (b) of Figure 1. The estimated E(x*) vs x* relationship is close to the E(x*) vs x* relationship assumed, indicating the validity of the proposed method. However, the k_0,Est1(x*) values are slightly smaller at lower x* levels and slightly larger at higher x* levels than the assumed k₀(x*) vs x* relationship. This comes from the accuracy of the approximation made to estimate k_0,Est1(x^*_Select), 1 – x_dx*(x^*_Select) ≅ 0.58 for all reactions. In section 4, a method is developed to make the k_0,Est1(x*) vs x* relationship closer to the true k₀(x*) vs x* relationship. It is, however, surprising that the plots of ln(a/T²) vs 1/T at the selected x values, which are the same as for the analysis of a single reaction, give E(x*) vs x* and k_0,Est1(x*) vs x* relationships for the parallel first-order reactions with N → ∞. The estimated E(x*) vs x* and k_0,Est1(x*) vs x* relationships converted to f(E) vs E, and k_0,Est1 vs E relationships are also shown in Panels (c) and (d) of Figure 1 for reference.

Figure 7.

Arrhenius type plot proposed by Miura and Maki to estimate E(x*) vs x* and k_0,Est1(x*) vs x* relationships.

3.3.4. Analysis of Single Reactions Using the Miura–Maki Method

The Miura–Maki method was developed for parallel first-order reactions with N → ∞. It is, however, informative to know how the method analyzes a single reaction. To do so, the x vs T relationships for the single first-order reaction given in Panel (a) of Figure 2 were analyzed by following the above procedure. The E(x*) vs x* and k_0,Est1(x*) vs x* relationships estimated from the ln(a/T²) vs 1/T plots made at 99 levels of x ranging from 0.01 to 0.99 are shown by the diamond, respectively, in Panels (a) and (b) of Figure 1. Since the Arrhenius like plots estimating E values at the same x levels are exactly the same as the analysis method of the single reaction, the method gave exact E values at any levels of x* for the single reaction. On the other hand, the estimated k_0,Est1(x*) values are highly dependent on the x* levels as shown in Panel (b). The k_0,Est1(x*) values are the same as the assumed k₀ value only at x* = 0.42. This is because the approximation 1 – x_dx* (x^*_Select) ≅ 0.58 for all reactions is exactly valid only at x = 1 – 0.58 = 0.42. Since we do not know how many reactions are involved in the reaction system of concern before analyzing experimental data, the trend of the k_0,Est1(x*) vs x* relationship shown in Panel (b) of Figure 1 becomes a criterion to judge if the reaction system contains many reactions. If a similar trend of the k_0,Est1(x*) vs x* relationship as for the single reaction is observed, it suggests that the reaction of concern is a single reaction. If some x* range where a constant E(x*) and decreasing k_0,Est1(x*) with increasing x* appeared, the x range corresponding to the x* range should be treated as a single reaction with E, and the k₀ selected from the middle of the x* range which gives the same E(x*).

The discussion presented above clearly shows that the Miura–Maki method is valid and powerful to analyze the parallel first-order reactions with N → ∞ for estimating E(x*) vs x* and k_0,Est1(x*) vs x* relationships. The result for the single reaction shows that a constant E(x*) and swinging k_0,Est1(x*) vs x* relationship can be a criterion to guess the number of reactions involved in the target reaction.

3.4. Graphical Analysis Method Presented

3.4.1. Concept of the Graphical Method

The Miura–Maki method presented one of the ways to estimate E(x*) vs x* and k_0,Est1(x*) vs x* relationships based on eq 21. The method was valid, as exemplified above. The analysis procedure is judged to be straightforward and simple enough. If we dare to say the weakness of the analysis procedure, we cannot easily determine the meaning of each procedure. The plots of ln(a/T²) vs 1/T, which are the core procedure of the analysis method, for example, neither give the overall image of the analysis method nor the degree of correctness of the procedure. Then the possibility of an analysis method that uses eq 21 and gives more insight into each procedure was examined. The extension of the graphical method for a single reaction to that of the parallel first-order reactions with N → ∞ was examined as a candidate.

The essence of the Master Curve Analysis for a single reaction was to estimate E first, to convert experimental x vs T or x vs t relationships to x vs θ relationships with the estimated E and to superimpose the x vs θ diagram on the mother diagram showing the x vs τ relationship (Figure 4) to find k₀ as shown above. On the other hand, the graphical method proposed in this paper uses a theoretical x–θ diagram as given in Figure 8 where the abscissa τ of the x–τ diagram shown for the single reactions in Figure 4 is converted to θ by assigning several k₀ values of 1 × 10¹³, 1 × 10¹⁴, ..., and 1 × 10²⁰ s^–1, for instance. The curve for every k₀ value is identical and shifts to 10^–1 times smaller θ with the 10 times increase of k₀. This diagram allows us to estimate E and k₀ simultaneously from the x vs T data obtained with three or more heating rates for a single reaction. The idea is to convert the x vs T relationships to x vs θ relationships by arbitrarily assigning several E values. The x vs θ relationships with true E will converge on one of the theoretical x vs θ relationships with true k₀. This analysis method may be theoretically the same as the Master Curve Analysis for the single reaction shown above, but the proposed graphical method does not require the estimation of E beforehand. In this sense the method may be called “Pseudo Master Curve Analysis”. To exemplify how this method works, the x vs T relationships given for the single reaction in Panel (a) of Figure 2 were just converted to x vs θ relationships for ten values of E = 200, 210, ···, and 290 kJ mol^–1, and they are plotted on the theoretical x–θ diagram as shown in Figure 9. Three curves for a = 2, 5, and 10 K min^–1 converge to a single curve only at E = 250 kJ mol^–1 as expected, and the shapes of the curves are identical with the theoretical x–θ curve. To show this, the theoretical x vs θ relationship with k₀ = 2 × 10¹⁶ s^–1 is included in Figure 9. Then the frequency factor k₀ in addition to E are obtained. In principle, the frequency factor k₀ for a single reaction can be obtained with one set of known a, T, and x. If we set x = 0.42 for a single first-order reaction in parallel to the approximation made to develop the Miura–Maki method, the following relationship holds between k₀ and θ from eq 3-a:

34

Figure 8.

Theoretical x–θ diagram proposed for single first-order reactions.

Figure 9.

Plots of the x vs θ relationships constructed from the x vs T relationships on the theoretical x–θ diagram for the single first-order reaction.

Scaling the top axis to satisfy eq 34 as given as “k_0,Est1 for 1st-order reaction” in Figures 8 and 9, we can estimate k₀ by reading the top axis corresponding to x = 0.42. This is exemplified by the red broken lines and the key ▲ in Figure 9: k₀ is estimated as 2 × 10¹⁶ s^–1 by just reading the top axis corresponding to x = 0.42 on the x–θ relationship for E = 250 kJ mol^–1. This examination shows how the graphical method works for the analysis of a single reaction. Since the graphical method was developed for analyzing the parallel first-order reactions with N → ∞, this example was shown to give the underlying concept of the graphical method proposed.

3.4.2. Analysis of the Parallel First-Order Reactions with N → ∞ Using the Graphical Method

Then the proposed graphical method was applied to analyze the parallel first-order reactions with N → ∞, for which the method was proposed. To convert the x vs T relationships for N → ∞ shown in Panel (b) of Figure 2 to x vs θ relationships, the E values estimated at x = 0.1, 0.2, ···, 0.9 by the Miura–Maki method were employed. The E values are given by the ordinate values of the points marked by the key ▽ in Panel (a) of Figure 1, and they are equal to true E(x^*_Select) values at the x^*_Select levels corresponding to x = 0.1, 0.2, ···, and 0.9 as discussed above. The E values were chosen to examine the consistency of the graphical method and Miura–Maki method. The x–θ relationships thus constructed are plotted on the theoretical x–θ diagram (Figure 8) as shown in Figure 10. The three x–θ curves for a = 2, 5, and 10 K min^–1 for each E value spread over more than 6 spans of θ, indicating that the span is much larger than that of a single first-order reaction, and hence the reaction consists of many reactions. The curves shift to smaller θ values with the increase of E value assigned and clearly converge or come close together only at one point marked by the red dot. Every marked point gives the corresponding x from the ordinate and k_0,Est1(x^*_Select) from the top axis. The nine x values are almost exactly equal to 0.1, 0.2, ···, and 0.9 for the nine E values employed. The k_0,Est1(x*) vs x* relationships shown in Panel (b) of Figure 1 that were estimated at 99 x levels are plotted as x vs k_0,Est1(x^*_Select) relationships by the key △ in Figure 10. The x vs k_0,Est1(x^*_Select) relationships exactly pass the nine points marked by the key ▽, showing the consistency of the graphical method and the Miura–Maki method. Since we know both E(x*) vs x* and k₀(x*) vs x* relationships for this reaction, we can construct “x vs θ_dx*(x^*_Select) assumed” from the x vs τ_dx*(x^*_Select) relationship shown in Figure 4 by defining θ_dx*(x^*_Select) as follows:

35

Figure 10.

Plots of the x vs θ relationships constructed from the x vs T relationships on the theoretical x–θ diagram for the parallel reactions with N → ∞.

The “x vs θ_dx*(x^*_Select) assumed” relationship exactly coincides with the x vs k_0,Est1(x^*_Select) relationship marked by the key △ in Figure 10. The nine points marked by the key ▽ in Figure 4 correspond to the points marked by the same key ▽ in Figure 10.

Using the graphical method we can get both E(x^*_Select) vs x and k_0,Est1(x^*_Select) vs x relationships equivalent to the E(x*) vs x* and k_0,Est1(x*) vs x* relationships estimated by the Miura–Maki method by constructing the x–θ curves for wide range of E values from the x vs T relationships. The analysis procedure is more straightforward and gives a clear image of the meaning of the analysis method. The proposed graphical method and the Miura–Maki method may be used supplementarily for practical purposes, as will be shown below.

4. Improvement of the “Miura–Maki Method”

The final evaluation of the validity of the analysis method will be done by examining if the estimated E(x*) vs x* and k_0,Est1(x*) vs x* relationships can reproduce the data used for the analysis. Additional testing may be to examine if the estimated relationships can be used to predict x vs T relationships at different heating profiles. Figure 11 compares the assumed x vs T relationships for a = 0.1, 2, 5, 10, 100, and 1000 K min^–1 and the corresponding x vs T relationships calculated using the estimated E(x*) vs x* and k_0,Est1(x*) vs x* relationships shown, respectively, in Panels (a) and (b) of Figure 1. The assumed x vs T relationships for a = 2, 5, and 10 K min^–1 were used for the rate analysis, and those for a = 0.1, 100, and 1000 K min^–1 were used to test the validity of the E(x*) vs x* and k_0,Est1(x*) vs x* relationships for prediction purposes. The calculated x vs T relationships shown by the broken lines fitted fairly well not only the x vs T relationships used for the rate analysis but those relationships used for prediction purpose. In this sense, the E(x*) vs x* and the k_0,Est1(x*) vs x* relationships obtained are accurate enough for practical application. The degree of discrepancy between the estimated and assumed x vs T relationships shown in the figures may be much smaller than the experimental error and/or data handling artifact.

Figure 11.

Comparison of assumed and estimated x vs T relationships for the parallel reactions with N → ∞.

4.1. Can We Approach the True k₀(x*) vs x* Relationship

The above discussion showed that both the Miura–Maki method and the proposed graphical method are valid to estimate E(x*) vs x* and k₀(x*) vs x* relationships rather accurately. The estimated E(x*) vs x* relationship was actually so close to the assumed one, but the estimated k_0,Est1(x*) vs x^*_B relationship was still a little bit different from the assumed relationship as shown in Panels (a) and (b) of Figure 1. The estimated k_0,Est1(x*) slightly overestimated true k₀(x*) at smaller x* values and slightly underestimated true k₀(x*) at larger x* values. This discrepancy caused smaller and larger x values of the estimated ones at lower and higher temperatures than those assumed as shown in Figure 11. These examinations suggest that the k_0,Est1(x*) values will approach the true ones by decreasing them at lower x* levels and by increasing them at higher x* levels so that the estimated x vs T relationships will show better agreements with the assumed x vs T relationships. This is expected to be done if we can find the way to approach the assumed (true) x_dx*(x^*_Select) values from the estimated E(x*) vs x* and k_0,Est1(x*) vs x* relationships. The possibility is examined next.

Figure 12 schematically shows the assumed (true or experimentally obtained) x vs T relationship by the thick solid line in the vicinity of an x ≅ x^*_Select point at a certain heating rate of a in which dx* is expanded to 1. To make the discussion simple, let us assume that the estimated E(x*) vs x* is the same as the assumed (true) one. The red thin solid line is the assumed (true) x_dx*(x^*_Select) vs T relationship. This can be calculated by using the assumed (true) E(x^*_Select) and k₀(x^*_Select) values. Since the rate analysis is to estimate E(x*) vs x* and k₀(x*) vs x* relationships from the true x vs T relationships, the thin red solid line cannot be drawn before analysis. We define here the temperature of a specified x on the true x vs T relationship as T_True (denoted by the key ◆). Knowing E(x^*_Select) and T_True, k_0,Est1(x^*_Select) was estimated to satisfy eq 28 as

36

The red thin dashed line in Figure 12 shows the x_dx*(x^*_Select) vs T relationship calculated with E(x^*_Select) and k_0,Est1(x^*_Select). Equation 36 is satisfied at the point marked by the key × on the relationship. On the other hand, the true x_dx*(x^*_Select) is given by the point marked by the key △ on the assumed (true) x_dx*(x^*_Select) vs T relationship and expressed with the assumed E(x^*_Select) and k₀(x^*_Select) as

37

The x vs T relationship calculated using the estimated E(x*) vs x* and k_0,Est1(x*) vs x* relationships covering the full range of x is shown by the thick broken line. The temperature corresponding to the specified x on the x vs T relationship is defined as T_Est1 (indicated by ▲), which is probably slightly different from T_True. Then the estimated x_dx*(x^*_Select) value with the estimated E(x^*_Select) and k_0,Est1(x^*_Select) at T = T_Est1 (indicated by Δ) is calculated by

38

Equating the estimated x_dx*(x^*_Select) (Δ) and the true x_dx*(x^*_Select) (◇), namely, equating eq 38 and eq 37, the true k₀(x^*_Select) can be expressed as follows

39

Since all quantities on the right-hand side of eq 39 are known, true k₀(x^*_Select) can be obtained. However, the estimated E(x^*_Select) is not exactly the true one, as shown above. Then, we call the k₀(x^*_Select) calculated by eq 39k_0,Rev(x^*_Select), which is a revised value of k_0,Est1(x^*_Select), and rewrite eq 39 as follows:

Figure 12.

Schematic diagram showing the scheme to estimate the k_0,Rev(x*) vs x* relationship.

40

Then eq 40 gives k_0,Rev(x^*_Select) at the specified x from the known quantities. Repeating this procedure over the entire range of x, we can obtain the k_0,Rev(x*) vs x* relationship.

It is possible to develop the equation to obtain k_0,Rev(x^*_Select) from k_0,Est1(x^*_Select) for isothermal reactions. Equations 37 and 38 are written for isothermal reactions for a constant temperature T and a selected x as follows:

41

42

where t_True and t_Est1 are, respectively, times which give the selected x value on the true x vs t relationship and the x vs t relationship calculated using the estimated E(x*) vs x* and k_0,Est1(x*) vs x* relationships. Equating eq 41 and eq 42, the true k₀(x^*_Select) can be expressed as follows:

43

Since the estimated E(x^*_Select) is not exactly the true one as discussed above, the k₀(x^*_Select) calculated by eq 43 is replaced by k_0,Rev(x^*_Select), which means the revised value of k_0,Est1(x^*_B), to give

44

This equation is utilized to obtain the k_0,Rev (x*) vs x* relationship for isothermal reactions.

4.2. Test of the Validity of the k_0,Rev (x*) vs x* Relationship

Since only one set of assumed (true or experimental) and estimated x vs T relationships is required to estimate the k_0,Rev(x*) vs x* relationship, the x vs T relationship assumed and that estimated using k_0,Est1(x*) vs x* for a = 5 K min^–1, which are given, respectively, by the key △ and by the red broken line in Figure 11, were used here. The k_0,Rev(x*) vs x* relationship estimated is included in Panel (b) of Figure 1 by the key ▲. They are also shown as the k_0,Rev vs E relationships in Panel (d) of Figure 1. In both figures the k_0,Rev(x*) vs x* relationship approached more closely to the assumed k₀(x*) vs x* relationship than the k_0,Est1(x*) vs x* relationship did. The k_0,Rev vs E relationship looks so close to the assumed k₀ vs E relationship.

The x vs T relationships calculated by using the E(x*) vs x* and k_0,Rev(x*) vs x* relationships are shown by the solid lines for a = 0.1, 2, 5, 10, 100, and 1000 K min^–1 in Figure 11. The calculated x vs T relationships fitted almost exactly not only the x vs T relationships for a = 2, 5, and 10 K/min, used for the rate analysis, but also those assumed for a = 0.1, 100, and 1000 K min^–1, used for prediction purposes. These examinations showed that the Miura–Maki method and/or the graphical method with the revised k_0,Rev(x*) vs x* relationship can almost exactly estimate both E(x*) vs x* and k₀(x*) vs x* relationships, and the estimated relationships can be used for not only reproducing the x vs T relationships but also for predicting x vs T relationships at significantly different heating conditions.

As a conclusion, both the Miura–Maki method and the graphical method proposed are valid to estimate E(x*) vs x* and k_0,Est1(x*) vs x* relationships from the x vs T relationships obtained experimentally with three or more heating rates. The k_0,Est1(x*) vs x* relationship can be upgraded to the k_0,Rev(x*) vs x* relationship that is much closer to the true k₀(x*) vs x* relationship if necessary.

If the reasoning leading to eq 40 is correct, the k_0,Rev(x*) vs x* relationship should exactly coincide with the assumed k₀(x*) vs x* relationship when the estimated E(x*) vs x* relationship is the same as the assumed (true) E(x*) vs x* relationship. This is examined in the Supporting Information for the k_0,Rev(x*) vs x* relationship.

5. Results and Discussion

5.1. Direct Application of the Proposed Analysis Method to Practical Pyrolysis

It is known that coal is a macromolecule consisting of chains of polyaromatic nuclei and many different types of functional groups attached to them. When the coal consisting of such a structure is pyrolyzed, many different reactions that have different rate parameters take place, yielding various products. Then the pyrolysis reaction is expected to be modeled by the parallel first-order reactions with N → ∞. Then the proposed analysis methods were applied to analyze the pyrolysis reaction of two coals: an Australian brown coal, Morwell, and a US bituminous coal, Pittsburgh No. 8 (PITT). The x vs T relationships were measured for the two coals using a sensitive thermobalance (Shimadzu TG50A) at three heating rates of a = 5, 10, and 20 K min^–1. They are shown in Panels (a) and (b) of Figure 13 by the keys. The x vs T relationships were analyzed by the following procedure.

• 1.The most important and essential task of kinetic analysis is to acquire reliable x vs T relationships covering a wide range of x, but it is not easy in general. For Morwell, the x vs T relationships for x < 0.05 are so close that the relationships for 0.05 < x < 0.95 were analyzed at 91 levels of x in every 0.01 of x. For PITT, the x vs T relationships for a = 10 and 20 K/min intersect at x ≅ 0.02, and those for a = 5 and 10 K/min intersected at x ≅ 0.87, although particular care was taken to acquire the data. Then only the x vs T relationships ranging from x = 0.05 to 0.8 were analyzed.
• 2.Following the revised Miura–Maki method, the Arrhenius like plots of ln(a/T²) vs 1/T were performed in every 0.01 of x covering x = 0.05 to 0.95 for Morwell and covering x = 0.05 to 0.8 for PITT as shown, respectively, in Panels (a) and (b) of Figure 14. The lines connecting three points are drawn in every 0.05 of x to avoid the overcrowding of lines. From the slope and the intercept of each plot, both E(x^*_Select) and k_0,Est1(x^*_Select) values corresponding to the x^*_Select value can be obtained by following eq 31. They are shown as the E(x*) vs x* and k_0,Est1 (x*) vs x* relationships using the approximation, x^*_Select ≅ x, in Panels (a) and (b) of Figure 15 for Morwell and in Panels (a) and (b) of Figure 16 for PITT. Since some errors or artifacts are involved for obtaining experimental data and/or analyzing data, it is rare to obtain exactly linear lines passing three points. Fairly good linear lines passing three points could be drawn for Morwell, but it was rather hard to draw linear lines for PITT. This is why the maximum and minimum values for each averaged E(x*) and k_0,Est1 (x*) are shown with error bars for PITT. For Morwell, the estimated E(x*) increased monotonously from ∼150 kJ mol^–1 to ∼500 kJ mol^–1 with the increase of x*, indicating that the pyrolysis reaction of Morwel consists of many reactions. The k_0,Est1 (x*) also increased monotonously in parallel with E(x*). For PITT, the E(x*) values estimated are almost the same as ∼235 kJ mol^–1 for ∼0.15 < x < ∼0.5, and the x* dependency of the corresponding k_0,Est1 (x*) values is very close to that of the single reaction shown in Panel (b) of Figure 1. This suggests that the range ∼0.15 < x < ∼0.5 will be represented by a single reaction with E = ∼235 kJ mol^–1 and k₀ = ∼7 × 10¹⁴ s^–1. The ranges x < ∼0.15 and x > ∼0.5 are judged to consist of many reactions.
• 3.The graphical analysis was performed to supplement the above analysis. The experimentally obtained x vs T relationships shown in Figure 14 were just converted to x vs θ relationships by assigning 14 E values of 160, 180, 200, ···, 400, and 420 kJ mol^–1 for Morwell and 12 E values of 190, 200, 210, ···, 290, and 300 kJ mol^–1 for PITT. In addition, x–θ diagrams for E = 235 kJ mol^–1 were also constructed for PITT by referring to the above analysis. They are plotted on the theoretical x–θ diagram as shown, respectively, in Figure 17 for Morwell and in Figure 18 for PITT. Three x–θ curves for a = 5, 10, and 20 K min^–1 spread over more than 8 spans of θ for Morwell, indicating that the span is much larger than that of a single first-order reaction, and hence the pyrolysis reaction of Morwell consists of many reactions. The curves shift to smaller θ values with the increase of E value assigned and clearly converge or come close together only at one point on every set of curves marked by the key ▽. Every marked point gives the E(x^*_Select) value and the corresponding x from the ordinate and k_0,Est1(x^*_Select) from the top axis. Thus, estimated E(x^*_Select) and k_0,Est1(x^*_Select) values are plotted by the same key ▽ against x* using the approximation, x^*_Select ≅ x, respectively, in Panels (a) and (b) of Figure 15. All the marked points are on the E(x*) vs x* and k_0,Est1(x*) vs x* relationships estimated by the Miura–Maki method as expected. In contrast, the E(x*) vs x* and k_0,Est1(x*) vs x* relationships in Panel (b) of Figure 16 are plotted as the x vs k_0,Est1(x^*_Select) relationship by the red dot in Figure 17. The 14 points marked by the red dot are all on the relationship. Thus, it was shown that the proposed graphical method is equivalent to the Miura–Maki method and that its analysis procedure is more straightforward and gives a clear image of the meaning of the analysis method.

  The x–θ curves for a = 5, 10, and 20 K min^–1 for PITT show different behavior as compared with those for Morwell as Figure 18 shows. They spread over more than 6 spans of θ, indicating that the span is much larger than that of a single first-order reaction, and hence the pyrolysis reaction of PITT also consists of many reactions. However, the curves for ∼0.15 < x < ∼0.5 are rather steep, and their span is well in the range of a single first-order reaction. The x–θ curves constructed in every 10 kJ mol^–1 of difference of E are judged to converge or come close together only at one point on every set of curves marked by the key ▽. However, there existed a big jump of x value from 230 to 240 kJ mol^–1, indicating that the E values for the reactions occurring at ∼0.15 < x < ∼0.5 are within 230 to 240 kJ mol^–1. The x–θ curves for E = 235 kJ mol^–1 included by referring to the estimated E(x*) vs x* relationship shown in Panel (a) of Figure 16 almost coincided for ∼0.15 < x < ∼0.5 on which five points are marked by the key ▽. Every marked point gives the E(x^*_Select) value and corresponding x from the ordinate and k_0,Est1(x^*_Select) from the top axis. Thus, estimated E(x^*_Select) and k_0,Est1(x^*_Select) values are plotted by the same key ▽ against x* using the approximation, x^*_Select ≅ x, respectively, in Panels (a) and (b) of Figure 16. All the marked points are on the E(x*) vs x* and k_0,Est1(x*) vs x* relationships estimated by the Miura–Maki method. In contrast, the E(x*) vs x* and k_0,Est1(x*) vs x* relationships in Panel (b) of Figure 16 are plotted as the x vs k_0,Est1(x^*_Select) relationship by the key △ in Figure 18. The 19 points marked by the key ▽ are all on the relationship. Thus, it was shown again that the proposed graphical method is equivalent to the Miura–Maki method. In addition, the graphical analysis suggested more clearly that the x range, ∼0.15 < x < ∼0.5, will be represented by a single first-order reaction with E = ∼235 kJ mol^–1.

• 4.The validity of the E(x*) vs x* and k_0,Est1(x*) vs x* relationships estimated was examined by comparing the experimental x vs T relationships with those calculated using the estimated E(x*) vs x* and the k_0,Est1 (x*) vs x* relationships in Panels (a) and (b) of Figure 13. For Morwell, the calculated x vs T relationships shown by the broken lines almost exactly coincided with the experimental relationships. On the other hand, the calculated x vs T relationships slightly overestimated the experimental x vs T relationships at smaller x region and underestimated them at larger x region for PITT. This is probably because the approximation, x^*_Select ≅ x, becomes more accurate as the number of reactions increases.
• 5.The k_0,Est1(x*) vs x* relationships estimated were revised by following the procedure given in the previous section using eq 40 and the experimental and calculated x vs T relationships for a = 10 K/min for both Morwell and PITT, although this procedure may be unnecessary for Morwell. They are shown by the key ▲ as k_0,Rev(x*) vs x* relationships, respectively, in Panel (b) of Figure 15 and in Panel (b) of Figure 16. As expected, the k_0,Rev(x*) vs x* relationship for Morwell is rather close to the k_0,Est1(x*) vs x* relationship. For PITT, the k_0,Rev(x*) vs x* relationship is slightly different from the k_0,Est1(x*) vs x* relationship.
• 6.The x vs T relationships calculated with the estimated E(x*) vs x* and k_0,Rev(x*) vs x* relationships are shown by the solid lines for Morwell and PITT, respectively, in Panels (a) and (b) of Figure 13. They fitted the experimental x vs T relationships almost exactly for both Morwell and PITT. For Morwell, however, the broken lines and solid lines almost coincided, showing that the k_0,Est1(x*) vs x* relationship is good enough compared to the estimated k₀ (x*) vs x* relationship.

Figure 13.

Comparison of the experimental x vs T relationships with those calculated using the estimated E(x*) vs x* and the k₀(x*) vs x* relationships.

Figure 14.

Arrhenius-type plot proposed by Miura and Maki to estimate E(x*) vs x* and k_0,Est1(x*) vs x* relationships.

Figure 15.

E(x*) vs x* and k₀(x*) vs x* relationships estimated for Morwell.

Figure 16.

E(x*) vs x* and k₀(x*) vs x* relationships estimated for PITT.

Figure 17.

Plots of the x vs θ relationships constructed from the experimental x vs T relationships on the theoretical x–θ diagram for Morwell.

Figure 18.

Plots of the x vs θ relationships constructed from the experimental x vs T relationships on the theoretical x–θ diagram for PITT.

5.2. Application of Proposed Analysis Method with Some Modification

The analysis described above directly followed the analysis procedure proposed. It is judged that the E(x*) vs x* and the k₀ (x*) vs x* relationships are estimated accurately for Morwell. For PITT, some modification is necessary, although the x vs T relationships are almost exactly reproduced with the estimated E(x*) vs x* and the k_0,Rev(x*) vs x* relationships. The estimated E(x*) vs x* and the k_0,Rev(x*) vs x* relationships for the range of ∼0.15 < x < ∼0.5 suggest that the range will be represented by a single first-order reaction with E = ∼235 kJ mol^–1 and k₀ = ∼7.0 × 10¹⁴ s^–1, as stated above. Then the range 0.14 ≤ x ≤ 0.49 was assumed to be represented by a single reaction by eye observation and was analyzed separately by the following procedure.

• 1.To extract the x vs T relationships for the x range where the single first-order reaction is dominating from the whole x vs T relationships. The x vs T relationships were divided into three ranges. Range I: 0.05 ≤ x ≤ 0.14. Range II: 0.14 ≤ x ≤ 0.49. Range III: 0.49 ≤ x ≤ 0.80, as shown in Figure 19.

  Since the experimental x vs T relationships were almost exactly reproduced by the E(x*) vs x* and the k_0,Rev(x*) vs x* relationships as shown in Panel (b) of Figure 13, the x vs T relationships for Range I and Range III were calculated, respectively, using the E(x*) vs x* and the k_0,Rev (x*) vs x* relationships for 0.05 ≤ x* ≤ 0.14 and 0.49≤ x* ≤ 0.80. They are shown by the thin broken lines in Figure 19. The x vs T relationships for Range II were extracted by subtracting the calculated x vs T relationships for Ranges I and III from the experimental x vs T relationships, and they are shown by the keys ▲, ▲, and ▲ in Figure 19.

• 2.The extracted x vs T relationships for Range II were analyzed using the graphical method proposed to estimate the E and k₀ values of the range accurately. The x vs T relationships for Range II were converted to x vs θ relationships and are plotted on the theoretical x–θ diagram as shown in Figure 20 as was done in Figure 18. The x vs θ relationships for Ranges I and III are also shown by the broken lines for reference. The x vs k_0,Est1(x^*_Select) relationship shown in Figure 18 is also replotted in the figure for reference. Since the x vs θ relationships for a = 5, 10, and 20 K min^–1 are expected to coincide at around E = ∼235 kJ mol^–1 for Range II from the above discussion, the x vs θ relationships were constructed in every 0.1 kJ mol^–1 between 235 and 236 kJ mol^–1 to find the E value that shows the best agreement of the three x vs θ relationships. It was judged that the best agreement is obtained at E = 235.5 kJ mol^–1 as shown in the figure. The k₀ value is obtained as 7.0 × 10¹⁴ s^–1 from the top axis corresponding to x = 0.287 (= 0.14 + 0.42 × 0.35). The theoretical x vs θ relationship for k₀ = 7.0 × 10¹⁴ s^–1, drawn by the blue solid line to adjust the span of Range II, showed good agreement with the x vs θ relationships for E = 235.5 kJ mol^–1. It also showed fairly good agreement with the x vs k_0,Est1(x^*_Select) relationship for ∼0.2 < x < ∼0.4. This suggests that the k₀ value of Range II may be estimated from the top axis corresponding to x = 0.287 of the x vs k_0,Est1(x^*_Select) relationship also. Thus, the E and k₀ values of Range II were estimated to be, respectively, 235.5 kJ mol^–1 and 7.0 × 10¹⁴ s^–1.
• 3.Now the E(x*) vs x* and k₀(x*) vs x* relationships covering 0.05 ≤ x* ≤ 0.80 are finally estimated. They are represented as the E_Mod(x*) vs x* and k_0,Mod(x*) vs x* relationships and are shown by the blue broken lines in Panels (a) and (b) of Figure 16. The x vs T relationships calculated using the relationships are shown by the solid lines in Figure 19 for both Range II and for 0.05 ≤ x ≤ 0.80. They fitted the corresponding experimental x vs T relationships almost exactly.

Figure 19.

Comparison of the experimental x vs T relationships with those calculated using the estimated E(x*) vs x* and _0,Rev(x*) vs x* relationships and the E_Mod(x*) vs x* and k_0,Mod(x*) vs x* relationships for PITT.

Figure 20.

Plots of the x vs θ relationships constructed from the extracted x vs T relationships for Range II on the theoretical x–θ diagrams for PITT.

The E_Mod (x*) vs x* and the k_0,Mod (x*) vs x* relationships thus estimated are judged to be closer to the true E (x*) vs x* and k₀ (x*) vs x* relationship than the E (x*) vs x* and the k_0,Rev(x*) vs x* relationships are for PITT. There may be a better strategy to extract and to estimate the reaction occurring in the range of ∼0.15 < x < ∼0.5, but the author believes that the analysis above is good enough, judging from the accuracy of experimental data and artifact of analysis procedure. Strictly speaking, the range of ∼0.72 < x < ∼0.85 of the x vs T relationships for Morwell is judged to be represented by a single reaction with E = ∼350 kJ mol^–1 and k₀ = ∼3 × 10¹⁹ s^–1 as Figures 15 and 17 show. However, further analysis of the x range was not performed using the same reasoning. This discussion will answer the comment raised by Fiori et al.³² on the Miura–Maki method.

The practical application of the proposed method showed that the Miura–Maki method and the graphical method are valid to analyze a complex reaction through a straightforward procedure and that the two methods used supplementally give a clear image of the underlying concept and the procedure of the analysis methods.

6. Conclusions

• 1.General basic equations were derived for describing an infinite number of parallel first-order reactions by extending the basic equations for the finite number of parallel first-order reactions. Based on the basic equations it was clearly shown that the target of the kinetic analysis is to estimate the frequency factors and activation energies of the infinite number of reactions. This discussion clarified that the DAEM is not an appropriate terminology to represent the infinite number of parallel first-order reactions. It should be renamed, for example, as the Distributed Rate Constant Model (DRCM).
• 2.The Miura–Maki method, which enabled us to estimate both f(E) and k₀(E) in the DAEM from more than three sets of experimental data obtained using the temperature-programmed reaction technique (TPR), was reexamined to be consistent with the general basic equations derived. Through the reexamination, a graphical analysis method that may be called “Pseudo Master Curve Analysis” was presented. The method not only supplements the Miura–Maki method but also gives the underlying concept of the Miura–Maki method clearly. The graphical method enabled us to estimate both E and k₀ simultaneously for all reactions without performing the Arrhenius type plots. It was also shown that the graphical method can be applicable to analyze a single reaction and the experimental data obtained using the isothermal reaction technique.
• 3.The Miura–Maki method almost exactly estimated the E values of an infinite number of parallel first-order reactions, but the estimated k₀ values were a little bit different from the true ones. To overcome the weakness, a method that improves the estimation accuracy of k₀ values was presented.
• 4.Practical application of the graphical method with the aid of the Miura–Maki method to analyze pyrolysis reactions of two kinds of coal showed that it is powerful to overlook the target reaction, to evaluate the accuracy of experimental data, and to understand the reasoning of the analysis method. Thus, the validity and usefulness of the graphical method with the aid of the Miura–Maki method were clarified.

Glossary

Abbreviations

a

constant heating rate of solid reactant for TPR, K s^–1, K min^–1

dx*

reduced infinitesimal amount of reactant of parallel reactions with N → ∞, dimensionless

E

activation energy, J mol^–1, kJ mol^–1

E (x*)

E value for the reaction of dx* at x* = x*, J mol^–1, kJ mol^–1

E_Mod(x*)

final E (x*) value estimated for PITT coal, J mol^–1, kJ mol^–1

E̅

mean E value of the Gaussian distribution defined by eq 23, J mol^–1

f(E)

distribution function of activation energy, mol J^–1

g(ln k₀)

distribution function of ln k₀, dimensionless

k₀

frequency factor, s^–1

k₀(x*)

k₀ value for the reaction of dx* at x* = x*, s^–1

k_0,Est1(x*)

first estimate of k₀(x*) by the Miura–Maki method or the Graphical method, s^–1

k_0,Mod(x*)

final k₀(x*) value estimated for PITT coal, s^–1

k_0,Rev(x*)

k₀(x*) revised from k_0,Est1(x*) using eq 40, s^–1

Ink̅₀

mean Ink₀ value of the Gaussian distribution defined by eq 24, dimensionless

N

total number of parallel reactions

p(u)

p function defined by eq 8, dimensionless

T

temperature, K

T_Est1

temperature at a selected x on x vs T relationship calculated with estimated E and k_0,Est1, K

T_True

temperature at a selected x on true or experimentally obtained x vs T relationship, K

T₀

temperature at which heating is started for TPR, K

t

time, s, min

t_Est1

time at a selected x on x vs t relationship calculated with estimated E and k_0,Est1, s, min

t_True

time at a selected x on true or experimentally obtained x vs t relationship, s, min

u

E/RT, dimensionless

x

overall conversion of solid reactant, dimensionless

x_dx*(x*)

conversion of dx* for x* = x*, dimensionless

x_dx*(x^*_Select)

conversion of dx* for x* = x^*_Select, dimensionless

conversion of Δx^*_i, dimensionless

x*

initial reduced amount of solid reactant, dimensionless

x^*_Select

x* value corresponding to x at which eq 28 holds independent of a, dimensionless

Δx^*_i

reduced amount of reactant of the i-th reaction, Σ (Δx^*_i) = 1, dimensionless

θ

time defined by eq 27-a or 27-b, (e^–E/RT·RT²/aE for TPR), s, min

θ_dx*(x^*_Select)

time defined by eq 35, s

σ_E

standard deviation of the Gaussian distribution defined by eq 23, J mol s^–1

σ_lnk0

standard deviation of the Gaussian distribution defined by eq 24, dimensionless

τ

dimensionless time defined by eq 4 or eq 9, (≅ k₀e^–E/RTT·RT²/aE for TPR), dimensionless

τ_dx* (x*)

general time for the reaction of fraction dx* for x* = x*defined by eq 19-a or 19-b, dimensionless

τ_dx*(x^*_Select)

τ_dx* (x*) value for x* = x^*_Select, defined by eq 28, dimensionless

general time for the reaction of fraction Δx^*_i defined by eq 13-a or 13-b, dimensionless

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.4c05744.

• Details on the k_0,Rev(x*) vs x* relationship (PDF)

The author declares no competing financial interest.