Ind. Eng. Chem. Res. 2001, 40, 1465-1470
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Calcination Kinetics of Ammonium Pentaborate Using the Coats-Redfern and Genetic Algorithm Method by Thermal Analysis O 2 mer S¸ ahin,*,† Mustafa O 2 zdemir,† Mehmet Aslanogˇ lu,† and U 2 . Gu 1 rbu 1 z Beker‡ Department of Chemistry, Harran University, S¸ .URFA, Turkey, and Department of Chemical Engineering, Istanbul Technical University, 80626 Maslak-Istanbul, Turkey
The calcination kinetics of ammonium pentaborate was investigated by thermogravimetry (TG) using the Coats-Redfern and genetic algorithm methods. Various methods for determining the reaction parameters using thermal analysis data were improved. Genetic algorithms can be considered as a new method for the determination of kinetic parameters by TG. In this study, the genetic algorithm procedure is applied to the determination of the coefficients of reaction (activation energy, frequency factor, and order of reaction) and their relative standard deviations. The Coats-Redfern method was applied to the same TG data to calculate the reaction parameters. The results obtained from both methods were also compared. Introduction More than 200 compounds of boron that are available in nature take different names, such as tincal, colemanite, ulexite, and kernite, depending on the ratio of boron to sodium, calcium, magnesium, etc. Boron and its compounds have a wide field of application in industry. Particularly, boric oxide is a well-known substance, and its potential rich chemistry has recently been discovered. This interesting chemical activity is a result of the high viscosity of its molten form. Boric oxide is mainly utilized in the production of many special glass compositions, enamels and alloys; in the preparation of fluxes; and as a catalyst in organic reactions.1 The production methods of boric oxide can be categorized into four main groups: (1) From boric acid: Boric oxide can be produced in molten form at 832 K by dehydration of boric acid or under vacuum at a maximum temperature of 423 K.2-6 (2) From metal borate: Production of boric oxide from metal borate gives a product having purity of 93-94%. More details on these methods can be found elsewhere.7-9 (3) From the ester of boric acid: This method consists of the esterification of boric acid with ethanol.10 (4) From ammonium pentaborate (APB): Ammonium pentaborate octahydrate, (NH4)2O‚5B2O3‚8H2O (APB), particles can be transformed into boric oxide in a fluidized bed reactor11 according to the following stoichiometric expression: heat
(NH4)2O‚5B2O3‚8H2O 98 5B2O3+2NH3+9H2O The kinetic parameters of calcination of solid materials have been obtained through various theoretical methods using thermogravimetric (TG) devices, differential thermal analysis (DTA), and differential scanning calorimetery (DSC). These theoretical methods have been improved for different devices. Using DTA data, Kissinger12 developed a method for determining the kinetic parameters of magnesite, calcite, brucite, kaolinite, and halloysite. Using thermal analysis derivative curves, * Corresponding author. Fax: +90 414 315 1998. E-mail: osahin@ harran.edu.tr. † Harran University. ‡ Istanbul Technical University.
Ozawa13 determined the kinetic parameters of various polymer materials. Salvador and Calvo14 and Mu and Perlmutter15 utilized the method of Coats and Redfern16 to obtain the kinetic parameters of dehydration of zinc acetate dihydrate and carbonates, carboxylates, oxalates, acetates, formates, and hydroxides. Although many studies have been done on the decomposition of APB,1,11 we have not found any studies on the kinetics of APB by nonisothermal analysis. We also have not found any applications of genetic algorithms (GAs) for the determination of the kinetic parameters by thermal analysis. Therefore, the main aim of this work is to determine the kinetic parameters of thermal decomposition of APB using the CoatsRedfern16 and GA methods. The parameters obtained by both methods are also compared, and the results are discussed. Theoretical Background Various methods have been advanced in the literature for the calculation of the activation energy and kinetic parameters of decomposition reactions from an analysis of the thermogravimetric (TG) curve. In this work, the decomposition reaction of APB can be described as the thermal decomposition of a solid substance. This type of reaction can be formulated as
Asolid f Bsolid + Cgas The rate of decomposition of this type reaction is expressed as
dx E ) k0(1 - x)n exp dt RT
(
)
(1)
Equation 1 is generally used for comparison of prior nucleation and diffusion models.17 This equation reduces to the Erofeev form of Mampel’s unimolecular law for n ) 1; it conforms to three-dimensional and two-dimensional shrinking core models for n ) 1/3 and 1/2, respectively; and the equation follows Avrami’s nucleation law (constant density of nuclei and one-dimensional nucleus growth) for n ) 0.18 For these reasons, the data from each decomposition step were tested
10.1021/ie000690f CCC: $20.00 © 2001 American Chemical Society Published on Web 02/21/2001
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Ind. Eng. Chem. Res., Vol. 40, No. 6, 2001
empirically by fitting to eq 1 integrated to the general expression. Equation 1 can be integrated under the conditions n *1 and T ) T0 + βt. After integration, the general expression for the decomposition rate of APB becomes
[
]
( )(
2
)
k0RT 1 2RT 1 -E -1 ) 1exp n - 1 (1 - x)n-1 βE RT E (2) In this study, the relative standard deviation was calculated by the following route. Equation 2 can be subdivided into two parts in the case of a known reaction order n.
part 1 Y1 )
[
]
1 1 -1 n - 1 (1 - x)n-1
(3)
This side of eq 2 can be calculated easily under the assumption of a constant n value.
k0RT2 E 2RT 1part 2 Y2 ) exp βE RT E
(
)(
)
(4)
On the other hand, this part of eq 2 can be regarded as the regression analysis side. Thus, the values of the relative standard deviation (σ) can be calculated as follows:
σ)
x( ∑
)
Y1 - Y2 Y1 N-1
2
(5)
Equation 2 can be linearized under the assumption of 2RT/E , 1 for application of the Coats-Redfern method;16 the term (1 - 2RT/E) is then equal to 1. Under these conditions, the last form of eq 2 can be rewritten as
]} ( )
{ [
ln
k0R E 1 1 1 - 1 2 ) ln (6) n-1 n - 1 (1 - x) βE RT T
From a plot of ln{1/(n -1)[1/(1 - x)n-1 - 1]1/T2} vs 1/T utilizing the TG data, the kinetic constants can be obtained by the Coats and Redfern16 method. In the case of n ) 1, the last integrated form of eq 1 is
[
ln -
] ( )
ln(1 - x) T
2
) ln
k0R E βE RT
(7)
From a plot of ln[-ln(1 - x)/T2] vs 1/T, the kinetic constants of the thermal decomposition of APB can be obtained by the Coats-Redfern method. The same TG data were used for the application of the GA method through eq 2. The advantage of the GA method is that the constants of eq 2 (n, k0, and E) can be found without any assumptions or linearization. However, as shown by Wolf,19 the increasing number of unknown constants results in a low efficiency for searches by the GA method. For instance, constant values of A ) k0R/βE in eq 2 were calculated for each constant value of n and E obtained randomly in the GA program as follows:
x)1
A)
1
∑ x)0n - 1 T)Ti
[
1
(1 - x)n-1
( )( E
T2 exp ∑ RT T)T 0
-1
1-
]
)
(8)
2RT E
After calculation of each A value, which depends on n and E, all of the GA’s operations were applied to the values of A. The efficiency of the GA method was increased by this kind of calculation. For this reason, after an explanation of the basic GA methodology, both the Coats-Redfern and the GA methods have been applied to determine the constants of the kinetic parameters for the thermal decomposition of APB. Genetic Algorithm (GA) Special problems related to the fitting of kinetic parameters on the basis of a model without the assumption of any values for E, n, and A are emphasized, and the strategy applied in this study is illustrated. The operation steps of genetic algorithm can be characterized by the following features:20,21 (1) definition of solution space and the initial distribution of individual solution (2) an evaluation function (selection mode) (3) a schema for encoding the solution (4) a mode of crossover (5) a mode of probability of mutation The initial population of individuals is usually generated randomly for certain boundary conditions. The boundaries used in this study are summarized as follows:
initial solution space ) generated randomly 1 < E/R < 3 × 105 0.01 < n
