Ind. E n g . Chem. Res. 1987,26, 1037-1040
imental results. On the other hand, Roddy et al. (1971) found the exponent to be unity. That this reaction mechanism could not fully explain the [H+] dependency of rates is evident from an earlier study by the same group [USAEC, 19651 where it has been shown that the exponent on [H+] is 1 for [HA] < 0.1 M but changes to 2 for [HA] > 1.0 M. Thus, it seems that the exponent should better be left as an adjustable parameter in the rate expressions in order to bring agreement with the experimental values. Conclusions The liquid-liquid jet reactor offers an interesting device to study the kinetics of extraction which are controlled by chemical reactions at the interface. It has the advantage of the possibility of precise determination of the interfacial area. The results in this study with such a device showed the mechanism to be similar to that in the Lewis cell and not to the one in dispersion mixing, though the latter yields an average contact time of the same order of magnitude as in the jet reactor. Further, the [H+] ion dependency is much weaker here than that reported for the Lewis cell or dispersion mixing. It is suggested that the exponent on [H+] is kept as an adjustable parameter in the rate expressions to bring agreement with experimental results. Nomenclature A = interfacial area, m2 a = specific interfacial area A / V, m-l C = concentration, mol/L C1, Cz = constants in eq 6 and 9 D = diameter of the mixing chamber, mm d = drop diameter, mm d32 = Sauter mean drop diameter, mm [Fe3+],[H+], [HA], etc. = concentration of the respective species, mol/L h = height of the jet, mm HA, HDEHP = bis(2-ethylhexyl) phosphate klm, k l d , k , = rate constants for lm, Id, 2s steps, respectively M = molarity of HDEHP, mol/L N = rotation of the rotor blades, revolutions/g Id, lm,2s = rate-controlling steps in the reaction mechanism R = chamber diameter/jet diameter
1037
r = rate of reaction, mol/(m2 s-l) rlm,rld, rb = individual rates t = time, s ui = interfacial velocity of jet, mm/s a = average jet velocity, mm/s V = volume of the jet phase, m3 Greek S y m b o l s = 4c~/ ( w + 4112) y = (Ra- 4Ri + 3 + 4 In R ) / ( R 4In R - R4 + 2R2 - In R - 1) v, = collision frequency of drops, 8-l $= volume fraction of the dispersed phase p1 = chamber-phase viscosity, kg/ms pz = jet-phase viscosity, kg/ms p, = density of the continuous phase, kg/m3 u = interfacial tension, mN/m Registry No. HDEHP,298-07-7;Fe, 7439-89-6.
B
Literature Cited England, D. C.; Berg, J. C. AZChE J. 1971,17, 113. Freeman, R. W.; Tavlarides, L. L. Chem. Eng. Sci. 1980,35, 559. Freeman, R.W.;Tavlarides, L. L. Chem. Eng. Sci. 1982,37, 1547. Garner, F. H.; Mina, P. Trans. Faraday SOC.1959,55,1627. Howarth, W.J. Chem. Eng. Sci. 1964,19,33. Kimura, S.;Miyanchi, T. Chem. Eng. Sci. 1966,21, 1057. Kleknik, Yu. B.; Navrotskaya, V. A. Russ. J. Znorg. Chem. 1967,12, 1648. Kletenik, Yu. B.; Navrotskaya, V. A. Russ. J. Znorg. Chem. 1969,14, 997. Kuester, J. L.; Mize, J. H. Optimization Techniques with FORTRAN, McGraw-Hill: New York, 1973;p 240. Lewis, J. B. Chem. Eng. Sci. 1954,3, 248. Madden, A. J.; Dammerell, G. L. AIChE J. 1962,8,233. Marquardt, D. W. J . Appl. Math. 1963,11, 431. Peppard, D.F.; Ferraro, J. R.; Mason, G. W. J. Inorg. Nucl. Chem. 1958,7,231. Roddy, J. W.; Coleman, C. F.; Arai, S. J. Znorg. Nucl. Chem. 1971, 33,1099. Sprow, P. B. Chem. Eng. Sci. 1967,22,435. USAEC Report ORNL-3830, 1965,p 217. Vogel, A. I. A Text Book of Quantitative Inorganic Analysis; ELBS and Longman: London, 1969.
Received f o r review August 12, 1985 Revised manuscript received April 7, 1986 Accepted September 17, 1986
Obtaining the Kinetic Parameters from Thermogravimetry Using a Modified Coats and Redfern Technique P. E r n e s t o Fischer,* Chon Shin Jou,and S h y a m S . Gokalgandhi Department of Chemical Engineering, Texas T e c h University, Lubbock, Texas 79409
T h e accuracy of the Coats and Redfern (1964) technique (CR) t o retrieve kinetic parameters has been tested by using numerical integration applied to a first-order irreversible reaction. The numerical integration simulated thermogravimetric analysis data of a first-order irreversible reaction. A wide range of activation energies and frequency factors was used and covers the range of values reported in the literature. Predicted values are in the average within 1%and 21% of the values used in the numerical solution. To improve the accuracy of the values predicted by CR technique, correlations are presented which correct the activation energy within 0.15% of its assumed value and the frequency factor within 1% accuracy. Different types of thermoanalytical methods such as thermogravimetry (TGA), thermovolumetry,or differential thermal analysis have been applied to retrieve kinetic data from pyrolysis reactions. In TGA, for example, a solid or
liquid sample is heated with a constant-temperature ramp, recording the weight loss due to the decomposition reaction at different time/temperature levels. The retrieval of kinetic parameters from the weight loss vs. temperature
0888-5885/87/2626-1037$01.50/00 1987 American Chemical Society
1038 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987
data can be done by using one of several techniques, such as those of Freeman and Carroll (1958), Horowitz and Metzger (1963), Coats and Redfern (1964), MacCallum and Tanner (1970),Gorbachev (1975), Reich and Stivala (1980), and modifications thereof. Of these techniques, the one developed by Coats and Redfern is the simplest to use. Unfortunately, there is no simple theoretical way to predict the accuracy of the kinetic parameters obtained with this method. Published comparisonsof the CR technique with the others have been done with experimental TGA data (House 1981; Nutall et al., 1983); however, since the true kinetic parameters or reaction mechanisms are not known, the results are unreliable. There have been limited attempts to test this equation with theoretical data, but the ranges studied cover only a very narrow range of activation energies and frequency factors (Flynn and Wall 1966; Ravindran et al., 1974). In this paper, the differential equations which depict the pyrolysis reactions are solved numerically by using typical values of activation energy, frequency factors, and heating ramps as reported in the extensive TGA literature. The resulting solution yields conversion-temperature profiles for each set of selected values. These profiles are treated by the Coats and Redfern technique to obtain values of the activation energy and frequency factors. These values are then compared to those selected previously, and an error criterion is established. Coats and Redfern Technique. For the decomposition reaction of component A to a gaseous component B,
The mathematical description of the weight loss during heating can be represented by using an Arrhenius-type reaction equation:
If this reaction is carried in a TGA, the sample will be pyrolyzed with a constant heating ramp, P, defined as p = -d T (3) dt
Table I. Representative Values Used in the Numerical Integration of Equation 12 io-3~ io 20 30 40 50 60 k 0 / P 1 X lo2 1 X lo3 1 X lo4 1 X lo5 1 X IO6 1 X lo7
termine the reaction order. It is then very important to obtain a good estimate for the reaction order; otherwise, considerable error can result (Flynn and Wall, 1966). We define A _ -1-a
(7)
A0
where a is defined as the fraction of reactant A converted. After integrating the left-hand side of eq 6, it can be written as
Unfortunately, the right-hand side of this equation has no simple analytical solution. Coats and Redfern (1964) developed a simplified mathematical technique to obtain an approximation for this equation in the form In
-(ln [l - a]) =In
T2
[ %[ y ] ] 1-
-E/RT
(9)
This expression can be further simplified to
This is the equation of a line. To obtain the kinetic parameters, a linear regression of this expression is required, and from the values of the slope and intercept, the values of the activation energy and frequency factor can be estimated. Numerical Determination of Conversion Data and Kinetic Parameters. For a first-order reaction, the decomposition rate based on the unconverted fraction can be written as
By use of this parameter in eq 2 to. remove the time dependency for this equation, with the initial conditions (4) To integrate this equation, two boundary conditions are needed: T=T, A=Ao
T=T
A = A
(5)
Then the integral will be
For simplicity, a reaction order of 1 (i.e., n = 1)is used to integrate eq 6. This simplification will not diminish the generality of the CR approximation,because this technique applies only to the right-hand side of eq 6. In practice, the determination of kinetic parameters for a reaction requires a three-parameter search: reaction order, activation energy, and frequency factor. The Coats and Redfern approximation is usually used to find the last two, and some other optimization technique is needed to de-
The numerical integration of this equation can be easily performed with great accuracy by using Gear’s (1971) method. The solution of eq 11 requires the knowledge of three parameters (i.e., activation energy, frequency factor, and heating ramp). In this equation, two of them are combined as a ratio: the frequency factor and heating ramp. This will allow the reduction of the search space from a three-dimensionalproblem to only two dimensions. The values used for the integration are reported in Table I, and they span the typical values reported in the literature. From the numerical solution of eq 11, values of temperature and fraction unconverted can be obtained. These values then can be expressed as shown in eq 10. Table I1 shows the results obtained for one particular set of conditions. Similar data sets can be obtained for any combination of parameters in Table I. Some parameter combinations cover temperature ranges not commonly found in practice, but these results have been included in this work to produce a more general model.
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 1039 Table 11. Results from the Integration of Equation 12 and Calculated Parameters for the Coats and Redfern
2
-11
-12
~
-16.5721 -16.0233 -15.5031 -15.0093 -14.5398 -14.0929 -13.6671 -13.2608 -12.8728 -12.5019 -12.1469
0.9667 0.9400 0.8961 0.8277 0.7276 0.5933 0.4325 0.2677 0.1309 0.0460 0.0102
730 750 770 790 810 830 850 870 890 910 930
nParameters: E = 30000, ko/B = 1 X
1.370 1.333 1.299 1.266 1.235 1.205 1.176 1.149 1.124 1.099 1.075
lo6.
-13
-15 4
-16
-17
Table 111. Linear Regression Results for Using the Coats and Redfern Model of the Data Given in Table 11" intercept 4.005 10"k 0.8241 slope -15.021 10-3~ 29.85 R2 1.0000
~
20
30
40
50
60
knlB 1 x 102 1 x 103 1 x 104 1 x 105 1 x 106 1 x 107 1 x 102 1 x 103 I x 104 1 x 105 1 x 106 1 x 107 1 x 102 1 x 103
1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x
104 105 106
107 102
103 104 105 106 10' 102 103 104 105 106
107 102 1 x 103 1 x 104 1 x 105 1 x 106 1 x 107
10-3~ 9.84 9.88 9.91 9.93 9.94 9.96 19.70 19.78 19.84 19.87 19.90 19.92 29.58 29.69 29.77 29.82 29.85 29.88 39.46 39.61 39.70 39.76 39.81 39.84 49.35 49.52 49.63 49.71 49.76 49.81 59.24 59.44 59.57 59.66 59.72 59.77
ko/B 7.056 X 10' 7.416 X lo2 7.723 X lo3 7.953 x 104 8.192 X lo6 8.324 X lo6 7.144 X 10' 7.519 X lo2 7.786 X lo3 8.031 X lo' 8.218 X lo6 8.343 X lo6 7.222 X 10' 7.568 X lo2 7.842 X lo3 8.046 X lo' 8.241 X lo5 8.371 X lo6 7.270 X 10' 7.606 X lo2 7.861 X lo3 8.072 X lo' 8.245 X lo5 8.397 X lo6 7.307 X 10' 7.640 X lo2 7.883 X lo3 8.091 X lo4 8.258 X lo5 8.409 X lo6 7.338 X 10' 7.651 X lo2 7.916 X lo3 8.109 X lo4 8.278 X lo5 8.413 X lo6
The linear regression results are given in Table I11 and plotted in Figure 1. Similarly, the same procedure has been followed for all combinations of parameters in Table I, and the results are given in Table IV. From these numerical results, it is evident that the Coats and Redfern technique will produce a straight line from conversion vs. temperature data for all the data sets. A statistid analysis of the regression for the model (eq 10) for these data sets
t
Numerical Solution
Figure 1. Comparison between the numerical data (E = 30000, k / B = 1 X lo6) and its Coats and Redfern regression. 1.7
1
1.5 1.6
1.4
Table IV. Comparison between the Selected Set of Parameters from Table I and Predicted Values Obtained from the CR Techniaue from Simulated TGA Data used in numerical simulation predicted by CR 10-3~ 10
1 ooo./r
- Coats and Redfern
1.3 1.2 1.1
-
-
1 -
0.9 0.8 0.7
-
-
-
1 E4
1 E5
-
0.5 0.6
0.4
n r _.L
1 E6
-
1 E7
,
10
30 ACTIVATION ENERGY
50
, Ksol/mola
Figure 2. Percentage error committed calculating the activation energy using the Coats and Redfern equation, for different values of activation energy and k / p .
27
-
26
-
25
-
24 23
-
10
1 E3
30 ACTIVATION ENERGY
50
, Kcol/mole
Figure 3. Percentage error committed calculating k / O using the Coats and Redfern equation, for different values of activation energy and kfB.
has given values of R2 of 1.0000 (all significant figures). F-tests also confirm the predictions of a linear correlation for this technique. Even though these are high correlation coefficients from the CR technique, the values of the kinetic parameters retrieved differ from the input values. The activation energy values are within 1.0% of the true value (Figure 2), and the frequency factor values are consistently smaller with an average error of 21 % (Figure 3). Close inspection of Figures 2 and 3 indicates the possibility of a correlation
1040 Ind. Eng. Chem. Res., Vol. 26, No. 5 , 1987 Table V. Coefficients for Eauations 12a and 12b iii
1
2
3
4
0.555 183 0.031323
-0.012 98 0.99909
0.97449 -0.002 1565
0.000642 36 0.000065379
~
a
0.06 0.05
8
0.04
accuracy of the kinetic parameters retrieved has been improved markedly.
0.03 0.02
z
g K
1 2
0.01
o
-0.01
$ -0.02 W
k? k!
-0.03 -0.04
-0.05 -0.06
10
30
50
ACTIVATION ENERGY , K c a l / m o l e
Figure 4. Percentage error committed calculating the activation energy using eq 12b with the values obtained from the Coats and Redfern equation, for different values of activation energy and k / @ .
%
’
v 1 ‘1
‘3 L
0.5 0.4 0.3 0.2 0.1
O
(L
-0.1
0
-0.2 -0.3
Conclusions This paper demonstrates that the Coats and Redfern technique can correlate TGA data and that the kinetic parameters obtained with it can be used to obtain higher accuracy values by using eq 12a and 12b. The activation energy can be computed then with a maximum error of 0.1% and the frequency factor with a maximum error at 1% , an order of magnitude increase in accuracy. Nomenclature A = reactant concentration, concentration units A. = initial concentration, concentration units B = product concentration, concentration units c1 = parameters defined in eq 12a and 12b E = activation energy, kcal/gmol k0 = Arrhenius frequency factor R = ideal gas constant, 1.9872 kcal/(gmol K) T = temperature of the reacting system, K To = initial temperature, K t = time Greek S y m b o l s cy
= unconverted fraction of reactant A
= heating ramp, K/time A = heat input to the reaction system -0.9
Subscripts th = theoretical values
CR = Coats and Redfern values
Literature Cited Coats, A. W.; Redfern, J. P. Nature (London) 1964,201, 68. Flynn, J. H.; Wall, L. A. J.Res. Natl. Bur. Stand., Sect. A 1966, 70A, 487.
Freeman, E. S.; Carroll, B J . Phys. Chem. 1958, 62, 394. Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall: Englewood, NJ, 1971; pp 158-166
Gorbachev, V. M. J. Therm. Anal. 1975, 8, 349. Horowitz, H. H.; Metzger, G. Anal. Chem. 1963, 35, 1464. House, J. E. Thermochim. Acta 1981,47, 379. McCallum, J. R.; Tanner, J. Nature (London) 1970,225, 1127. Nutall, H. E.; Guo, T. M.; Schrader, S.; Thakur, D. S. ACS Symp. Ser. 1983,230, 269. Ravindran, P. V.; Radhakrishnan, T. P.; Sundaram, A. K. Therm. Anal. 1974, 1 , 85. Reich, L.; Stivala, S. S. Thermochim. Acta 1980, 36, 103.
The values of the coefficients are given in Table V. The percent error committed when using these expressions is shown in Figures 4 and 5, and i t is apparent that the
Receiued for review June 26, 1984 Reuised manuscript received November 14, 1986 Accepted February 25, 1987
