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Ind. Eng. Chem. Res. 2003, 42, 3642-3646
RESEARCH NOTES Dehydration Kinetics of Ulexite by Thermogravimetric Data Using the Coats-Redfern and Genetic Algorithm Method Hu 1 seyin Okur* and C ¸ igˇ dem Eymir Department of Chemical Engineering Faculty, Atatu¨ rk University, 25240 Erzurum, Turkey
There are various methods for determining the reaction parameters using thermal analysis data. In recent years, genetic algorithms have been considered as a new method for the determination of kinetic parameters. In the present study, calcination kinetics of ulexite was investigated by thermogravimetry (TG) using the Coats-Redfern and genetic algorithm methods. The parameters of reaction (activation energy, frequency factor, and order of reaction) and their relative standard deviations were calculated by the Coats-Redfern and genetic algorithm methods. Also, the results obtained from both methods were compared. For the second region of the dehydration process, the Coats-Redfern method gave meaningless results although the genetic algorithm method gave acceptable results. As a result, it can be said that the Coats-Redfern method would not be used for reactions having low activation energies. 1. Introduction The boron reserves commercially recoverable are mostly in the form of hydrated boron minerals, and it is reported that 54% of the world’s known boron reserves are in Turkey.1,2 The extra weight of water and other impurities in raw materials increase transportation and energy costs for anhydrous borate production. Therefore, the demand for raw hydrated borates has declined in recent years. This has forced the producers to produce the dehydrated borates and boron products.3 To predict and understand an industrially important dehydration process, its kinetics must be known. In recent years, different methods have been applied to obtain kinetic parameters of reactions from experimental nonisothermal thermogravimetric data. In these methods, the mass change of material heated at a constant heating rate is recorded continuously, and these thermogravimetric data are then interpreted with the use of different approaches. The most known methods are the Suzuki method,4 the Coats-Redfern method,5 Doyle’s approximation, and McCarty’s approximation.6 When boron minerals are heated for dehydration, internal thermal reactions occur. The mineral first loses its water of crystallization, followed by either production of amorphous material or recrystallization into new phases. Colemanite decrepitates as a result of the sudden release of confined water vapor within micropores during this thermal treatment, while ulexite does not decrepitate. Instead, it only exfoliates as a result of gradual water vapor removal, and the structure becomes amorphous with numerous microcracks and interstices.3 The thermal decomposition of various boron compounds and ulexite has been studied by different * To whom correspondence should be addressed. Tel.: +90(442)23-4582. E-mail:
[email protected].
investigators using different methods. They used usually the Coats-Redfern and Suzuki methods to calculate the kinetic parameters from thermogravimetry (TG) data.2,7,8 S¸ ahin et al.9 utilized the methods of CoatsRedfern and genetic algorithm (GA) to obtain the kinetic parameters of dehydration of ammonium pentaborate. 2. Theoretical Background 2.1. GA. During the last 30 years, there has been a growing interest in problem-solving systems based on principles of evolution and hereditary: such systems maintain a population of potential solutions, they have some selection process based on fitness of individuals, and some “genetic” operators. GAs are among such techniques; they are stochastic algorithms whose search methods model some natural phenomena: genetic inheritance and Darwinian strife for survival.10 Their popularity can be attributed to their freedom from dependence on functional derivatives and to their incorporation of these characteristics:11 (i) GAs are parallel-search procedures that can be implemented on parallel-processing machines for massively speeding up their operations. (ii) GAs are applicable to both continuous and discrete (combinatorial) optimization problems. (iii) GAs are stochastic and less likely to get trapped in local minima, which inevitably are present in any practical optimization application. (iv) GAs’ flexibility facilitates both structure and parameter identification in complex models such as neural networks and fuzzy inference systems. A GA (as any evolution program) for a particular problem must have the following five components: (1) a genetic representation for potential solutions to the problem; (2) a way to create an initial population of potential solutions; (3) an evaluation function that plays the role of the environment, rating solutions in terms
10.1021/ie020929n CCC: $25.00 © 2003 American Chemical Society Published on Web 06/25/2003
Ind. Eng. Chem. Res., Vol. 42, No. 15, 2003 3643
Figure 1. Producing the next generation in GAs.
of their “fitness”; (4) genetic operators that alter the composition of children; (5) values for various parameters that the GA uses (population size, probabilities of applying genetic operators, etc.). GAs require the natural parameter set of the problem to be coded as a finite-length string (analogous to chromosomes in biological systems) containing characters, features, or detectors (analogous to genes), taken from some finite-length alphabet. Usually, the binary alphabet that consists of only 0 and 1 is taken. Each feature takes on different values and may be located at different positions. The total package of strings is called a structure or population (or genotype in biological systems). Therefore, the entire population of individuals is created in a computer as a binary string.12 The first step after creating a generation is to calculate the fitness value of each member in the population. For a maximization problem, the fitness value fi of the ith member is usually the objective function evaluated at this member (or point). After evaluation, a new population is created. The selection operation determines which parents participate in producing offspring for the next generation, and it is analogous to survival of the fittest in natural selection. Usually members are selected for mating with a selection probability proportional to their fitness values. This selection only gives a general description of the basics of GAs; detailed implementations vary considerably. For instance, we may choose a policy of always keeping a certain number of best members when each new population is generated; this principle is usually called elitism. Then this population is evolved with the use of some genetic operation, such as crossover and mutation. Crossover. To exploit the potential of the current gene pool, we use crossover operators to generate new chromosomes that we hope will retain good features from the previous generation. Crossover is usually applied to select pairs of parents with a probability equal to a given crossover rate. One-point crossover is the most basic crossover operator, where a crossover point on the genetic code is selected at random and two parent chromosomes are interchanged at this point. The effect of crossover is similar to that of mating in the natural evolutionary process, in which parents pass segments of their own chromosomes on to their children. Therefore, some children are able to outperform their parents if they get “good” genes or genetic traits from both parents. One example of one-point crossover between two parent individuals can be given as follows:
Mutation. Crossover exploits current gene potentials, but if the population does not contain all of the encoded information needed to solve a particular problem, no amount of gene mixing can produce a satisfactory solution. For this reason a mutation operator capable of spontaneously generating new chromosomes is included. The most common way of implementing mutation is to flip a bit with a probability equal to a very low given mutation rate. A mutation operator can prevent any single bit from converging to a value throughout the entire population and, more important, it can prevent the population from converging and stagnating at any local optima. The mutation rate is usually kept low so good chromosomes obtained from crossover are not lost. If the mutation rate is high (above 0.1), the GA performance will approach that of a primitive random search.11 An example of mutation can be given as follows:
Figure 1 is a simple schematic diagram illustrating how to produce the next generation from the current one. 2.2. Coats-Redfern Method. In this work, the decomposition reaction of ulexite can be described as thermal decomposition of a solid substance. This type of reaction can be written as
Asolid f Bsolid + Cgas
(1)
The decomposition rate of a solid material, Asolid, can be expressed as
dx/dt ) k(1 - x)n
(2)
where x is the conversation fraction, t is time, k is the rate constant, and n is the reaction order. If the heating rate is expressed as q ) dT/dt and the rate constant k ) k0 exp(-E/RT), eq 2 can be written as
-E dx k0 ) (1 - x)n exp dT q RT
( )
(3)
where k0 is the frequency factor, T is the absolute temperature, E is the activation energy, and R is the universal gas constant. Integration of eq 3 with n * 1 and the boundary conditions 0 f x for the conversation fraction and T0 f T for the temperature gives the following expression:
1 - (1 - x)1-n k0 -E ) exp dT 1-n q RT
∫
( )
(4)
The right-hand side of this equation has no exact
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Ind. Eng. Chem. Res., Vol. 42, No. 15, 2003
Table 1. Chemical Composition of Ulexite component
wt %
component
wt %
CaO B2O3 Na2O
13.66 42.43 7.24
H2O impurity
35.43 1.24
integral, but the following equation can be obtained when the right-hand side of the equation is expanded into an asymptotic series and the higher-order terms are ignored: 2
1 - (1 - x)1-n k0RT 2RT -E 1exp ) 1-n qE E RT
(
) ( )
(5)
Equation 5 can be linearized under the assumption of 2RT/E , 1 for application of the Coats-Redfern method;5 the term 1 - 2RT/E is then equal to 1. Under these conditions, the last form of eq 5 can be written as
[
]
( )
k0R 1 - (1 - x)1-n 1 E ) exp 2 1-n qE RT T
(for n * 1) (6)
For n ) 1, the following equation can be obtained from eq 3 with the same assumption:
-ln(1 - x) T2
)
k0R -E exp qE RT
( )
(for n ) 1)
(7)
A plot of ln[-ln(1 - x)/T 2] vs 1/T (for n ) 1) or ln{[1 (1 - x)1-n]/[(1 - n)T2]} vs 1/T (for n * 1) gives a straight line of slope -E/R. The frequency factor can be calculated from the intercept of this straight line. 3. Experimental Section The ulexite used in the present work was obtained from the deposits around Eskisehir-Kirka in Turkey. The composition of the sample was determined by analytical analysis (Table 1). Four different particle sizes (-1180 + 850, -850 + 600, -600 + 425, and -300 + 212 µm) of the mineral were used in the thermogravimetric experiments. The heating rate was 10 °C/min for these particle sizes. Besides, they were tested at heating rates of 5 and 20 °C/min for -600 + 425 µm particle size. A Shimadzu model TG -50 system was employed for the differential TG (DTG) and TG measurements. For TG, a given amount of sample was put into a platinum crucible, and its mass loss due to dehydration was recorded at a constant heating rate under a N2 gas flow rate of 30 mL/s. 4. Results and Discussion The TG and DTG curves of ulexite mineral are given in Figure 2, as a sample, for a particle size fraction of -1180 + 850 µm. The TG and DTG analyses showed that thermal decomposition of ulexite began at 65-70 °C for all particle size fractions. For all particle size fractions, two basic endothermic peaks were observed; one of them at 150-175 °C and the other, larger than the first one, at 190-200 °C. The rate of the dehydration process was very high in the interval 100-214 °C; after this region the process continued at a very low rate up to 600-650 °C. Later, almost no mass loss was observed. The dehydration process takes places in two steps because of there are two TG peaks and two DTG regions with different nominal mass loss rates. The application
Figure 2. TG and DTG curves for ulexite.
of the Coats-Redfern method confirmed the DTG results; i.e., the process occurred in two steps (Figure 2). From the TG experiments and the plot of the CoatsRedfern method, it was observed that the second process started at the conversion fraction of approximately 0.71-0.76 for all particle size fractions used, but the beginning temperature of the second step increased from 211 to 215 °C with increasing particle size. The dehydration reaction of ulexite is given as
Na2O‚2CaO‚5B2O3‚16H2O f Na2O‚2CaO‚5B2O3‚nH2O + (16 - n)‚H2O where n is the number of moles of water remaining after dehydration. The conversation fractions were calculated as the ratio of the mass loss at a given temperature to the total mass loss at the end of the process. In the Coats-Redfern method, to determine the reaction order of each dehydration step of the process, the values 0, 1/3, 1/2, 2/3, 3/4, 1, 4/3, 5/2, 2, 5/2, and 3 were tested. The reaction order was determined as the value giving the best fit to the plot of ln[f(x)/T 2] vs 1/T. The relative standard deviation was calculated by the following route. Equation 5 can be subdivided into two parts in the case of a known reaction order n.
Y1 ) f(x)
first part
(8)
This side of eq 5 can be calculated easily under the assumption of a constant n value.
second part
k0RT 2 2RT -E 1exp Y2 ) qE E RT
(
) ( )
(9)
On the other hand, this part of eq 5 can be regarded as the regression analysis side. Thus, the values of the relative standard deviation (δ) can be calculated as follows:
δ)
x
(
∑
)
Y1 - Y2 Y1 N-1
2
(10)
From the slope of the straight line and the equation (6 or 7), the activation energies and frequency factors were calculated and given in Table 2 for the first region. In the second region, the slope (-E/RT) of the straight line was obtained as positive values (as seen in Figure 3) for some reaction orders having a good correlation
Ind. Eng. Chem. Res., Vol. 42, No. 15, 2003 3645 Table 2. Activation Energies and Frequency Factors from the Coats-Redfern Method for Region I particle size (µm)
heating rate (°C/min)
r2
reaction order n
k0 (s-1)
E (cal/mol)
standard deviation
-1180 + 850 -850 + 600 -600 + 425 -300 + 212 -600 + 425 -600 + 425
10 10 10 10 5 20
0.993 574 223 0.997 959 078 0.997 145 122 0.996 241 355 0.993 986 219 0.998 355 514
1 1 1 1 1 1
270.453 782.868 404.626 183.091 41.924 954.296
10 707.138 747 11 674.607 336 11 001.140 576 10 346.702 492 9 513.554 998 11 487.956 734
0.310 352 0.309 950 0.309 852 0.310 537 0.310 487 0.313 104
Table 3. Activation Energies and Frequency Factors from GA Method for Region I particle size (µm)
heating rate (°C/min)
run no.
reaction order n
k0 (s-1)
E (cal/mol)
standard deviation
-1180 + 850 -850 + 600 -600 + 425 -300 + 212 -600 + 425 -600 + 425
10 10 10 10 5 20
5 2 2 5 3 1
1.157 810 1.092 095 1.014 698 0.926 349 0.847 492 1.062 889
3960.806 224 1459.706 479 613.582 047 219.529 746 42.350 118 1625.979 645
12 897.963 154 12 025.422 707 11 187.165 999 10 349.226 127 9 385.319 118 11 769.046 026
0.062 669 0.045 966 0.055 394 0.062 886 0.074 251 0.043 514
Table 4. Activation Energies and Frequency Factors from the GA Method for Region II particle size (µm)
heating rate (°C/min)
run no.
reaction order n
k0 (s-1)
E (cal/mol)
standard deviation
-1180 + 850 -850 + 600 -600 + 425 -300 + 212 -600 + 425 -600 + 425
10 10 10 10 5 20
2 2 4 1 3 2
0.499 965 0.473 718 0.462 052 0.496 077 0.512 604 0.470 801
0.000 415 0.000 379 0.000 368 0.000 408 0.000 216 0.000 747
55.143 024 33.343 399 22.859 554 48.647 193 58.638 975 34.535 227
0.014 104 0.013 030 0.010 013 0.013 252 0.013 712 0.010 217
Figure 3. Coats-Redfern treatment for ulexite dehydration.
coefficient. This was meaningless because the activation energy value should not be less than zero. The same TG data were used for the GA method through eq 5. The advantage of the GA method is that n, k0, and E can be found without any assumption or linearization. A GA program was prepared in Matlab 6.0. As shown by Wolf and Moros,13 the increasing number of unknown constants results in a low efficiency for searches by the GA method. For instance, constant values of A ) k0/q in eq 4 were calculated for each constant value of n and E obtained randomly in the GA program as follows: x)11
∑
A)
x)0 T)Ti
- (1 - x)1-n 1-n
( ) -E
∑ ∫exp RT
T)T0
(11)
A total of 100 values of n and E were generated randomly using boundary conditions. The boundary conditions were selected as 0.0001 < n < 3 and 0.0001 < E < 16 000. After calculation of each A value, which depends on n and E, all of the GAs’ operations were applied to a function as 1/(1 + δ) and tried to reach its maximum value. The efficiency of the GA method was increased by this calculation method. During the study of the program, it was observed that the program reached minimum standard derivation at 60-70 iterations. Therefore, the number of iteration steps for the reaction stages was selected as 100. It is well-known that GA is a stochastic method. Thus, although the results for the kinetic parameters (E, k0, and n) obtained at the end of each run of the GA program are very close to each other, they are not the same. To observe this feature of the GA method, five runs each with 100 iteration steps were performed at all heating rates and particle sizes. At the beginning of a GA program run by eq 5, the kinetic parameter data were near the results obtained by the Coats-Redfern method for the first region. However, the activation energy values were very low, and negative values of the frequency factors (k0) were obtained for the second region. This explains why we are receiving negative activation energy values by the Coats-Redfern method. It is obvious that the assumption of 2RT/E , 1 for application of the Coats-Redfern method is valid for great E values. However, it was not sufficient to use eq 5 in this situation. Higher-order terms in the asymptotic series must also be taken into consideration. Instead of this, in the studies the righthand side of eq 4 was calculated by numeric integration. Also, Y2 was calculated for the relative standard deviation as follows:
dT
Y2 )
k0 q
dT ∫0Texp(-E RT )
(12)
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Ind. Eng. Chem. Res., Vol. 42, No. 15, 2003
The results, having the lowest standard deviation of five runs, obtained with GA of the dehydration process are given in Tables 3 and 4. It can be seen from Tables 2 and 3 that there are some differences between the values of the kinetic parameters calculated by the Coats-Redfern and GA methods for the first region. The values of activation energies, k0, and reaction order are close to each other in the first region, for certain a particle size. However, the standard deviation values are better in the GA method.
5. Conclusion A method for the estimation of the thermal decomposition rate constant of ulexite is proposed, and its utility in describing the random and systematic variations in the kinetic parameters obtained by the GA method is explained. It is shown that the kinetic parameters obtained by using the GA method are better than those obtained by the Coats-Redfern method. Also, we have obtained the kinetic parameter values that were not obtained by the Coats-Redfern method. The results show that the assumption of 2RT/E , 1 can be used as a first approximation for the reactions having high E values, even if this assumption does not give complete accuracy. To use the Coats-Redfern method for reactions having low activation energy is not suitable. Thermal decomposition kinetic parameters of ulexite can be obtained sensitively through changes in the GA operations, such as the kind of crossover and probability of mutation. Also this method is applicable for other processes such as adsorption, heat transfer, and mass transfer.
Literature Cited (1) Davies, T. W.; C¸ olak, S.; Hooper, R. M. Boric Acid Production by the Calcination and Leaching of Powdered Colemanite. Powder Technol. 1991, 65, 433. (2) Tunc¸ , M.; Ers¸ ahan, H.; Yapıcı, S.; C¸ olak, S. Dehydration Kinetics of Ulexite from Thermogravimetric Data. J. Therm. Anal. 1997, 48, 403. (3) S¸ ener, S.; O ¨ zbayogˇlu, G.; Demirci, S¸ . Changes in the Structure of Ulexite on Heating. Thermochim. Acta 2000, 362, 107. (4) Suzuki, M.; Misic, D. M.; Koyama, O.; Kawazoe, K. Study of Thermal Regeneration of Spent Activated Carbons: Thermogravimetric Measurement of Various single Component Organics Loaded on Activated Carbons. Chem. Eng. Sci. 1978, 33, 271. (5) Coats, A. W.; Redfern, J. P. Kinetic Parameters from Thermogravimetric Data. Nature 1964, 201, 68. (6) Duswalt, A. The Practice of Obtaining Kinetic Data by Differential Scanning Calorimetry. Thermochim. Acta 1974, 8, 57. (7) Stoch, L.; Waclawska, I. Thermal Decomposition of Ulexite. J. Therm. Anal. 1990, 36, 2045. (8) Gu¨lensoy, H. Tu¨ rkiyedeki Bor Mineralleri ile Bunların Dehidrolanmaları, C¸ o¨ zu¨ nu¨ rlu¨ kleri ve Kat Cisim Reaksiyonları Hakkında; S¸ irketi Mu¨rettibiye Basımevi: Istanbul, Turkey, 1961. (9) S¸ ahin, O ¨ .; O ¨ zdemir, M.; Aslanogˇlu, M.; Beker, U ¨ . G. Calcination Kinetics of Ammonium Pentaborate Using the CoatsRedfern and Genetic Algorithm Methods by Thermal Analysis. Ind. Eng. Chem. Res. 2001, 40, 1465. (10) Michalewicz, Z. Genetic Algorithms + Data Structures ) Evolution Programs, 2nd ed.; Springer: New York, 1992. (11) Jang, J. S.; Sun, C. T. Mizutani, E. Neuro-Fuzzy and Soft Computing; Prentice Hall: Upper Saddle River, NJ, 1997. (12) Goldberg, D. E. Genetic Algorithms in Search, Optimization, and Machine Learning; Addison-Wesley: Reading, MA, 1997. (13) Wolf, D.; Moros, R. Estimating Rate Constants of Heterogeneous Catalytic Reactions Without Supposion of Rate Determining Surface Stepssan Application of a Genetic Algorithm. Chem. Eng. Sci. 1997, 52, 1189.
Received for review November 20, 2002 Revised manuscript received May 30, 2003 Accepted June 2, 2003 IE020929N
