A new diagnostic when determining the activation energy by the advanced isoconversional method

Accepted Manuscript Title: A New Diagnostic when Determining the Activation Energy by the Advanced Isoconversional Method Author: James S. Campbell John R. Grace C. Jim Lim David W. Mochulski PII: DOI: Reference:

S0040-6031(16)30113-7 http://dx.doi.org/doi:10.1016/j.tca.2016.05.004 TCA 77510

To appear in:

Thermochimica Acta

Received date: Revised date: Accepted date:

8-4-2016 3-5-2016 7-5-2016

Please cite this article as: James S.Campbell, John R.Grace, C.Jim Lim, David W.Mochulski, A New Diagnostic when Determining the Activation Energy by the Advanced Isoconversional Method, Thermochimica Acta http://dx.doi.org/10.1016/j.tca.2016.05.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A New Diagnostic when Determining the Activation Energy by the Advanced Isoconversional Method James S. Campbell, John R. Grace, C. Jim Lim and David W. Mochulski Department of Chemical and Biological Engineering, University of British Columbia, 2360 East Mall, Vancouver, Canada, V6T 1Z3. Corresponding author: James S. Campbell, +16044429441, [email protected]

Highlights   

An Optimization Indicator is introduced which identifies and diagnoses possible error in TGA data. The results suggest that limestone activation energies 200 kJ.mol‐1 for calcination in nitrogen are questionable. Strong activation energy vs. conversion trends for limestone calcination are likely due to errors.

Abstract The Advanced Isoconversional (AIC) method involves minimization of a function which uses conversion data at a minimum of three heating rates to determine an activation energy ( conversion ( ). Reactions that can be described by a single model give an while reactions which are described by more than one model give an conversion curves due to error can lead to

vs.

) at different values of

which is independent of ,

which varies with . Shifts in the

trends which falsely indicate, or mask, true multi-

model kinetics. It was found that the minimum value obtained during optimization of what we call the ‘Optimization Indicator’ ( ) can indicate whether trends in

vs.

are likely to be error-derived in the

case of single-model reactions or artifacts in the case of multi-model reactions.

and

for the

calcination of CaCO3 are used to demonstrate the experimental application of this new diagnostic.

Keywords Nonisothermal kinetics Advanced Isoconversional method Optimization Indicator

1. Introduction The kinetic equation for a single-mechanism, solid-state reaction at a particular heating rate ( ) describes the rate of reaction (

/

) as a function of two variables: temperature ( ) and conversion ( ): (1) 1



This temperature dependence is referred to as the Arrhenius equation, with the activation energy, while the conversion dependence,

factor and

being the pre-exponential

, is usually described by one

of 17 solid-state reaction models (Khawam and Flanagan, 2006). Knowledge of

and

,

,

commonly referred to as the kinetic triplet, allows prediction of the rate at a given temperature and conversion. This information has applications across a wide range of solid-solid and gas-solid reactions. Gas-solid reactions, such as biomass gasification, are often complex, with a single activation energy insufficient to describe the kinetics. In such cases, isoconversional methods can be employed to determine

at multiple values of . For example, a reaction described by a single mechanism should which is independent of

produce an produce

, while a reaction described by more than one model should

which varies with . In the case of multi-model reactions, different values of

can be chosen

as initial guesses, leaving the other two values of the kinetic triplet(s) to be determined by other methods. All that is required to estimate

by an isoconversional method is conversion data at three or more

heating rates. Several isoconversional methods are available for

determination. Among the earliest were the

Friedman (Friedman, 1964), Flynn-Wall-Ozawa (Ozawa, 1965; Flynn and Wall, 1966) and KissingerAkahira-Sunrose (Akahira et al., 1971) methods. The more recent Advanced Isoconversional (AIC) method of Vyazovkin (1997; 2001) is thought (Berzins and Actins, 2012) to be the most precise and reliable of these methods. This method produces values for

when a function (defined below in Eq. 13)

is minimized. Mochulski (2014) noticed that the value of this minimization function indicates the ability of the method to reconcile the experimental data using a consistent

. In this study, this value is termed the

Optimization Indicator ( ). Thermogravimetric analysis (TGA) is a common method for obtaining conversion data from reactions which result in change of sample mass upon heating. Data should be gathered according to recommended practices (Vyazovkin et al., 2014) which are, typically, heating rates below 10°C.min-1 and sample masses below 10 mg, although the best experimental conditions depend on the type of sample under investigation. For a decomposing solid, such as CaCO3, large masses, high heating rates, low purge gas flow rate and/or a poorly distributed sample (too thick) can result in thermal gradients, selfcooling and mass transfer effects. Improper minimization of heat and mass transfer effects can create variation in

with

which can falsely indicate, or mask true, multi-step kinetics. In this study we

investigate a new diagnostic, the Optimization Indicator, for indication of data which are potentially unsuitable for determination of the kinetic parameters. In addition, we show that in some instances, the Optimization Indicator could aid in selection of

values as initial guesses for multi-model fitting. To our

knowledge, this study is the first application of the Optimization Indicator to kinetic analysis.

2

2. Theory To put the findings of this paper in their proper context, we must first lay out the mathematical framework of the AIC method (Vyazovkin, 1997; Vyazovkin, 2000). The heating rate ( ) in Eq. 1 can be removed when the temperature is expressed as a function of time, . Then, separation of variables and integration gives: (2) where

is equal to

⁄

→

. Integrating over small time intervals

, Eq. 2 becomes:

(3) There is no fully analytical solution for Eq. 3, which means that

must be determined numerically. The

integral expression, for each , can be written more concisely as: ′

,

(4)

where: ′

,

The integral form of the reaction model,

(5) , is assumed to be constant at equivalent conversions at any

heating rate. Thus for two heating rates: (6) where subscripts

and

denote heating rates 1 and 2, respectively. This is the founding assumption of

any integral isoconversional method and shows why two or more runs at different heating rates are required. For two heating rates: ,

,

(7)

so that: ,

,

(8)

Dividing both sides by one another leads to: ,

(9)

, ,

(10)

, 3

Summing Eqs. 9 and 10 and minimizing, we obtain: ,

,

,

,

This minimum value is given the symbol

2

(11)

and termed the Optimization Indicator. The minimum of the

sum of any positive variable and its reciprocal is always 2. For three heating rates: ,

,

,

,

,

,

,

,

,

,

,

,

(12)

6 If

heating rates are applied, the expression can be generalized to: ′

,

′

,

1

(13)

Perfect optimization would give the exact integer corresponding to the permutations of experiments. For two rates,

is equal to 2, for three rates,

is 6, while for four rates,

is 12 and so on.

Simplifying Eq. 12 to: , , ,

where

, ,

(14)

,

,

and

. Plotting the function for the case

where

1 and 0.8

1.2 gives a 3-dimensional graph (Fig. 1). The extra dimension is removed by

setting

1. It is seen that a minimum value occurs at 6, whenever

must always be equal to, or greater than, For example, for 3-rate optimization, optimization,

.

1 , deviating further from the integer as

increases.

will be greater than, but very close to 6, while for 6-rate

will be greater than, but deviate much further from 30, because the optimizer is trying to

reconcile data at many more rates. Mochulski (2014) noted that

could provide an indication of the data

quality, or, more specifically, the reconcilability of the data at different heating rates. Highly incongruent data give values of

which deviate greatly from their whole integer minima. Two-rate optimization was a

special case which always gave the value of exactly 2 due to exact definition of the system (vs. over definition). This can be seen as further confirmation that

should be obtained using at least three

heating rates, since Arrhenius type plots require more than two points to reveal possible nonlinearity that can be an important source of kinetic information (Vyazovkin et al., 2014).

4

3. Results 3.1 Simulated case study Four columns of data - time ( ), ‘furnace’ temperature ( ), ‘sample’ temperature ( ), and conversion ( ) were constructed at three constant heating rates for a first-order reaction. The chosen time values were 0 to 5000 s in steps of 1 s. Conversion values were constructed by numerical integration of Eq. 1 for a single-model reaction with

=200 kJ.mol-1, =1e18 s-1 and

with an intercept of 400 K.

was constructed at three constant heating rates (2, 4 and 8 K.min-1) with an

intercept of 400 K. .

, i.e.

was equivalent to

was higher than

=F1 at heating rates of 2, 4 and 8 K.min-1

for the data at 2 and 4 K.min-1. For the 8 K.min-1 data,

by an amount proportional to the instantaneous reaction rate ( .

in order to simulate a self-heating effect. Similarly,

/ )

for simulation of a self-cooling effect.

The instantaneous reaction rate was obtained by taking the first derivative (forward differencing) of the conversion with respect to time (see Fig. 2). Introduction of a simulated self-heating effect shifted the peak to higher temperature and gave a right-shifted skew to the rate curve while introduction of a simulated self-cooling effect shifted the peak to lower temperature and gave a left-shifted skew to the rate curve. Finally,

and

were determined for

from 0.04 to 0.96 in steps of 0.02 by the Advanced

Isoconversional method using the three columns of simulated data ( , rates producing three results: vs.

vs.

for no perturbation,

vs.

, and

) for the three heating

for the self-heating perturbation and

for the self-cooling perturbation (see Figs. 3 and 4). It must be mentioned that these are merely

approximations for the self-heating and cooling effects. A more detailed simulation of these phenomena was performed by Lyon et al., 2012, where the error in the measured maximum reaction rate was related to the sample mass, heat of reaction, heating rate and a thermokinetic parameter which was a function of the activation energy, thermal conductivity, heat capacity and density. Background noise is always present to some degree since vibrations can cause motion of the TGA balance and variations in purge gas flow rate can create motion of the TGA pan. These effects generate random noise in the measured mass, which is reflected in noise in the calculated conversion. Thus conversion values, at all three rates (not just 8 K.min-1 as was done for the self-heating/cooling), were constructed according to the same kinetic parameters as before but with the addition of a random perturbation of conversion no greater than +0.0005 or less than -0.0005. In all cases, derivative (forward differencing) is shown in Fig. 2.

and

. The first

were then determined as outlined in Fig. 3

and 4. The self-heating perturbation resulted in underestimation of the value for underestimation of 14% at

, reaching a maximum

=0.62 while the self-cooling perturbation led to overestimation for

,

reaching a maximum overestimation of 17% at =0.62 (Fig. 3). The slightly larger error for self-cooling is due to the asymmetry in the exponential function in Eq. 1. The curvature of the

vs.

trend is to be

expected, given that the rate values (see Fig. 2) deviated most from their true position at =0.62, i.e. at 5

the rate maximum. Underestimation usually results when rate curves are shifted apart in the temperature dimension, while overestimation results when curves are shifted together (Khawam and Flanagan, 2005). For example, in the case of the self-heating effect, the 8 K.min-1 rate data are shifted to higher temperatures and thus away from the rate data at 2 and 4 K.min-1, data leading to underestimation. The simulated ‘noisy’ data resulted in

with between ±5% error and no trend with respect to , as would be very close to its

expected with randomly perturbed conversion data. The unperturbed data gave accurate value. Fig. 4 shows the corresponding results for

. The self-heating perturbation resulted in

maximum of 6.0156 at =0.62 while the self-cooling perturbation resulted in of 6.0295 at =0.62. Therefore whenever ‘noisy’ data resulted in

was nearest 6,

which reached a

which reached a maximum

was nearest the true value. The simulated

from 6.000015-6.006685 and showed no trend with

.

for the unperturbed

data is very close to, but always slightly greater than 6 (6.00000000000236-6.00000043348988) which suggests that, the lower the error in the conversion data, the closer the value of the self-cooling perturbation is greater than for the self-heating perturbation. was 17% overestimated while

for the self-

for the self-heating effect was just 0.26% greater than 6.

and the deviation of

Therefore the relation between error in

for

for the self-cooling effect

for the self-heating effect was underestimated by 14%, yet

cooling effect was 0.5% greater than 6 while more sensitive than

is to 6. Note that,

from 6, is not linear. In addition,

is

to the direction of the shift in the rate curve.

As previously mentioned, trends of

with

are often interpreted as requiring multiple models to account

for the kinetics. A researcher, without any other information on the reaction, might wrongly conclude that the trends in

were due to multi-step kinetics rather than the presence of a self-heating or cooling effect.

Observation of trends in

vs.

(as in Fig. 3), which are echoed by trends in

vs. , (as in Fig. 4), might

imply that one or more of the curves is subject to error unrelated to the mechanism. Writing code to produce

as an output of the optimization process, in addition to

, is not difficult and could prevent

researchers from drawing false conclusions regarding multi-step mechanics, or at the least, indicate whether the data ought to be repeated using lower heating rates and/or smaller masses (within reason, such that reliable and fast measurements can be made) to see if these trends disappear. From this simulated case study, it can be supposed that ‘high quality’ experimental data ought to produce which is close to 6 and randomly distributed. However, solid-state reactions are rarely, if ever, singlestep. In order to investigate a multi-model reaction, three different reactions (A, B and C), each composed of two equally weighted overlapping first order (F1) processes, were constructed by numerical integration using the kinetic triplets summarized in Table 1 at heating rates of 5, 10 and 20°C.min-1. The rate data at 20 K.min-1 for these three reactions is shown in Fig. 5. The only parameter that was changed was the activation energy for one of the reactions (highlighted in bold in Table 1). The % contribution of each reaction to the total conversion is 50%.

and

were determined as previously described. 6



Fig. 6 shows kJ.mol-1

at

vs.

for this simulated multi-step process. For simulation B,

=0.04, increasing to nearly 135

approximately 130

kJ.mol-1

at

is approximately 120

=0.68, then decreasing and stabilizing at

above a conversion of about 0.8. The

vs.

plot has the potential to give

for determination of the other two components of the kinetic triplet. However,

good initial guesses of the false peak at

kJ.mol-1

=0.6-0.8 is greater than the prescribed

have intuitively expected an

vs.

for either mechanism. Instead, we might

curve which begins at 120 kJ.mol-1 and gradually increases to ~130

kJ.mol-1. Indeed, simulation A, where the overlap of the two reactions is greater, showed much less of an artifactual hump, while simulation C, where the two reaction rate peaks (see Fig. 5) were well separated, had a very sharp peak in the conclusions from

vs.

vs.

plot. Great care must therefore be exercised in drawing mechanistic

trends and in choosing the values of

this example, the lowest and highest

for initial estimates in model fitting. In

values provided the best estimates for

In Fig. 6 the artifactual peaks in the

vs.

.

curves do not reflect the kinetic triplets used in their

construction, at least, not under any intuitive interpretation. A number of false conclusions might have been drawn from the

results. One might have proposed that simulations B and C, for example, were

tri-mechanistic, with

values derived from the first and last few values of conversion as well as at the vs.

peak. However, the misleading peaks in the

curve between

=0.5-0.8 corresponded exactly to

the point of inflection between the two peak maxima of the rate data. In addition, the

values are most

unstable and deviate furthest from 6 (see Fig. 7) between the same conversion values at which the

vs.

plot gave the false hump. For simulation A, the artifactual hump in

is much smaller than in B and C. A researcher might have

concluded from the rate data, that the process obeyed a predominantly single mechanism and that the variation in

was due to error. However, the

erratic behaviour around the hump in the

vs.

vs.

plot for simulation A (see Fig. 7) shows similar

plot as simulations B and C. This could indicate to the

researcher the existence of a second ‘hidden’ mechanism which is not obvious from the rate or activation energy data. The results indicate that

should be chosen such that its value corresponds to values of

is least overlap of the reactions, which is, not coincidentally, also where values

where there

values tend to be

closest to the whole integer which in this example was 6. Therefore with the extra information provided by , one can adopt more accurate choices for

before attempting model fitting using non-linear least

squares for example. Poor estimates could lead to incorrect determination of the remainder of the kinetic triplet. Values of

where

is closest to 6 ought to be the first choice as estimates for model fitting. This

is merely one simulated example, and it remains to be seen whether

is useful for experimental multi-

model cases. It is likely that other errors will be large enough to mask model-related deviation of the use of

should in no way replace the checks and procedures as recommended in Vyazovkin et al.,

2014, but merely supplement them.

7

. Thus

3.2 Experimental case study Limestone (

) calcination is one of the most intensely studied reactions in solid-state kinetics,

making it a good starting point for testing the Optimization Indicator.

(calcite polymorph) was

purchased from Sigma Aldrich (CAS No. 471-34-1, Product No. 310034, Japanese Limestone, Lot. MKBV0812V, particle size ≤30 μm, purity >98%). Decomposition data were gathered in a TA Instruments Q-500 Series TGA (weighing precision +/-0.01%) from 25 to 990°C at constant heating rates of 2, 4 and 8°C.min-1 with initial masses of ~7 mg, and also at rates of 5, 10 and 20°C.min-1 at masses of ~11 mg. General guidelines based on empirical data (Vyazovkin et al., 2014) and analytical analysis (Lyon et al., 2012) suggest that the sample mass should be less than 10 mg and the heating rate less than 10°C.min-1 for quantitative nonisothermal analysis. Therefore the former tests are within the recommended best practices while the latter are not. A dry, high-purity, N2 purge gas was used with cross-flow of 60 ml.min-1 and down-flow of 40 ml.min-1. Each sample was thinly and evenly spread on the bottom of a platinum pan. Buoyancy effects were accounted for by subtracting an inert sample background run at each heating rate. Data within the temperature range of mass loss were selected and transformed to conversions between 0 and 1 according to: ° °

where

°

is the mass at 500°C,

°

(15)

°

is the mass at 800°C and

the mass at temperature ( ).

Conversion data, with their corresponding time and temperature values, were used to determine the activation energy (

) and the Optimization Indicator ( ).

at

< 0.04 and

> 0.96 are omitted,

because those estimates have been found to be sensitive to minor errors in baseline determination (Vyazovkin et al., 2011). Fig. 8 shows the rate data for calcination of 8

°C.min-1

between 500°C and 800°C at heating rates of 2, 4 and

(mass ~7mg) and 5, 10 and 20 °C.min-1 (mass ~11mg). As the heating rate increases, the

temperature corresponding to the maximum rate and temperature of the reaction completion both shift to higher temperatures. This is an obvious result of increased heating rate allowing less time for the calcination to progress. and

were determined for the two systems, with the results plotted in Fig. 9. For the system with

smaller initial mass and lower heating rate, 1

to 225 kJ.mol-1 while

increased nearly linearly with increasing

from 205 kJ.mol-

had values randomly distributed between 6.0008 and 6.04200 (increasing very

slightly with increasing conversion). For the system with larger mass and higher heating rate, decreased approximately linearly from 180 kJ.mol-1 to 130 kJ.mol-1, while 6.04 at

=0.04 to 6.22 at

=0.84, then decreased to 6.12 at

recommended practices gave a pattern of

increased nearly linearly from

=0.96. Thus the system which satisfied

which better resembled the simulated case where only

random error was present, while the system operating outside the recommended conditions gave a statistically significant trend in

with . Thus the magnitudes and trends in the values of 8



indicate that

the system at lower heating rate and smaller initial mass provided values of underlying kinetics. It is suspected (L’Vov, 2002) that many reported

more representative of the

results, for the decomposition of

alkaline-earth carbonates, obtained by Arrhenius methods, are 15–30% underestimated, with this being attributed, among other factors, to the self-cooling effect. The higher values for

for the ‘best practices’

data support this finding. In addition, the findings presented here have implications for studies (e.g. Sanders and Gallagher, 2002; Rodriguez-Navarro et al., 2009) which suggest that strong trends in calcination result from multi-model kinetics. The results, i.e. the much steeper slope of the ‘poor quality’ data, as well as the significant trend in

, suggest that much stricter experimental limitations would be

needed and repetitions performed before multi-model kinetics could be confirmed.

4. Conclusion The Advanced Isoconversional method uses extent of conversion at three or more heating rates, producing values for activation energy (

) when a function (Eq. 13) is minimized. The value of this

minimized function, termed here the Optimization Indicator ( ), indicates the ability of the method to reconcile the experimental data using a consistent

. In a simulated first order reaction, lack of

agreement between the sample and furnace temperature, due to a self-heating or self-cooling effect created a trend in

vs.

similar to the trend in

vs. . Thus

could potentially be used as an indicator

for data containing non-kinetic influences related to experimental variables such as mass and heat transfer. Indeed

and

showed trends with increasing

for experimental CaCO3 calcination data

gathered according to poor practice, and much less so for data gathered according to recommended practices.

was also used in a simulated multi-model reaction to indicate where variation of

was artifactual. The use of

with

as an indicator for the reconcilability and quality of the gathered data is a

novel addition to

determination by the Advanced Isoconversional method. This study provides some

examples of how

can be used, but it remains to be seen if it has the potential to be applied consistently

to solid-state reactions. There may also be opportunity to improve upon simple averaging of experimental results, using

to eliminate suspect data. To the best of our knowledge this is the first application of

the Optimization Indicator to experimental data, and it is hoped that its application might be extended in the future.

Acknowledgements The authors are grateful to Carbon Management Canada and to the National Sciences and Engineering Research Council of Canada for financial support of this research. We would also like to acknowledge the helpful input from the referees of this paper.

References

9

AKAHIRA, T. and SUNOSE, T., 1971. Method of determining activation deterioration constant of electrical insulating materials. Res.Rep.Chiba Inst.Technol.(Sci.Technol.), 16, pp. 22-31. BĒRZIŅŠ, A. and ACTIŅŠ, A., 2012. Evaluation of Kinetic Parameter Calculation Methods for Non-Isothermal Experiments in Case of Varying Activation Energy in Solid-State Transformations/Neizotermisko Eksperimentu Kinētisko Parametru Noteikšanas Metožu Izvērtēšana Mainīgas Aktivācijas Enerģijas Gadījumā Cietfāžu Pārvērtībām. Latvian Journal of Chemistry, 51(3), pp. 209-227. FLYNN, J.H. and WALL, L.A., 1966. A quick, direct method for the determination of activation energy from thermogravimetric data. Journal of Polymer Science Part B: Polymer Letters, 4(5), pp. 323-328. KHAWAM, A. and FLANAGAN, D.R., 2005. Role of isoconversional methods in varying activation energies of solidstate kinetics: II. Nonisothermal kinetic studies. Thermochimica Acta, 436(1), pp. 101-112. KHAWAM, A. and FLANAGAN, D.R., 2006. Solid-state kinetic models: basics and mathematical fundamentals. The Journal of Physical Chemistry B, 110(35), pp. 17315-17328. L’VOV, B.V., 2002. Mechanism and kinetics of thermal decomposition of carbonates. Thermochimica acta, 386(1), pp. 1-16. LYON, R.E., SAFRONAVA, N., SENESE, J. and STOLIAROV, S.I., 2012. Thermokinetic model of sample response in nonisothermal analysis. Thermochimica Acta, 545, pp. 82-89. MOCHULSKI, D.M., 2014. Multiple reaction solid state kinetic parameter determination and its application to woody biomass, Master’s Thesis, University of British Colombia OZAWA, T., 1965. A new method of analyzing thermogravimetric data. Bulletin of the Chemical Society of Japan, 38(11), pp. 1881-1886. RODRIGUEZ-NAVARRO, C., RUIZ-AGUDO, E., LUQUE, A., RODRIGUEZ-NAVARRO, A.B. and ORTEGAHUERTAS, M., 2009. Thermal decomposition of calcite: Mechanisms of formation and textural evolution of CaO nanocrystals. American Mineralogist, 94(4), pp. 578-593. SANDERS, J.P. and GALLAGHER, P.K., 2002. Kinetic analyses using simultaneous TG/DSC measurements: Part I: decomposition of calcium carbonate in argon. Thermochimica acta, 388(1), pp. 115-128. VYAZOVKIN, S., 1997. Advanced isoconversional method. Journal of Thermal Analysis, 49(3), pp. 1493-1499. VYAZOVKIN, S., 2001. Modification of the integral isoconversional method to account for variation in the activation energy. Journal of Computational Chemistry, 22(2), pp. 178-183. VYAZOVKIN, S., BURNHAM, A.K., CRIADO, J.M., PÉREZ-MAQUEDA, L.A., POPESCU, C. and SBIRRAZZUOLI, N., 2011. ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data. Thermochimica Acta, 520(1), pp. 1-19. VYAZOVKIN, S., CHRISSAFIS, K., DI LORENZO, M.L., KOGA, N., PIJOLAT, M., RODUIT, B., SBIRRAZZUOLI, N. and SUÑOL, J.J., 2014. ICTAC Kinetics Committee recommendations for collecting experimental thermal analysis data for kinetic computations. Thermochimica Acta, 590, pp. 1-23.

10

6.25

6.00 6.05 6.10 6.15 6.20 6.25

f(x,y,z) z=1

6.20

6.15

6.10 6.05

y

6.00 1.15

1.10

1.05

1.00

x Fig. 1 Plot of the function

0.95

0.90

0.85

, ,

0.80

for 0.8

11

1.20 1.15 1.10 1.05 1.00 0.95 0.90 0.85 0.80

x, y

1.2 and z

1

Unperturbed Randomly perturbed Self-heating perturbed Self-cooling perturbed

0.005

0.003

-1

d/dt (s )

0.004

0.002

0.001

0.000

-0.001 480

500

520

540

Ts (K) Fig. 2 Rate (

/ ) vs. Ts (‘sample’ temperature) for a reaction with

=F1 at a heating rate of 8 K.min-1.

12

=200 kJ.mol-1, =1e18 s-1 and

240 230

-1

Ea (kJ.mol )

220 210 200 190 180 No error Self-heating Self-cooling Random noise

170 160 0.0

0.2

0.4

0.6

0.8

1.0

 Fig. 3

vs.

determined by the Advanced Isoconversional method for a simulated first order reaction

with no error, rate proportional errors (self-heating and cooling) and random error.

13

No error Self-heating Self-cooling Random noise

6.030 6.025



6.020 6.015 6.010 6.005 6.000 0.0

0.2

0.4

0.6

0.8

1.0

 Fig. 4

vs.

determined by the Advanced Isoconversional method for a simulated first order reaction

with no error, rate proportional errors (self-heating and cooling) and random error.

14

0.006

A 0.005

B

‐1

d/dt (s )

0.004

0.003

C

0.002

0.001

0.000 400

450

500

550

600

650

Temperature (K) Fig. 5 Simulated rate vs. temperature data for three different reactions, composed of two overlapping parallel first order processes at 20 K.min-1 with kinetic triplets provided in Table 1. The only parameter that differed was the activation energy of the second reaction.

15

145

140 C

Ea (kJ.mol‐1)

135

B 130 A 125

120

115 0.0

Fig. 6

vs.

0.2

0.4

0.6

for three simulated reactions (A, B and C) with kinetic triplets given in Table 1

16

0.8

1.0

6.00000020 A



6.00000015

6.00000010

6.00000005

6.00000000 0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

6.0000012 B 6.0000010



6.0000008

6.0000006

6.0000004

6.0000002

6.0000000 0.0



17

6.000010 C

6.000008



6.000006

6.000004

6.000002

6.000000 0.0

0.2

0.4

0.6

0.8

1.0

 Fig. 7

vs.

for three simulated reactions (A, B and C) with kinetic triplets given in Table 1

0.005

0.0030

20oC.min‐1 8oC.min‐1

0.0025

0.004

d/dt (s‐1)

d/dt (s‐1)

0.0020 0.0015 4oC.min‐1

0.0010 o

2 C.min

0.003

10oC.min‐1

0.002 5oC.min‐1

‐1

0.001

0.0005 0.0000

0.000 500

550

600

650

700

750

800

500

o

550

600

650

700

750

800

o

Temperature ( C)

Temperature ( C)

Fig. 8 Rate vs. temperature for CaCO3 at (left) 2, 4 and 8°C.min-1 (initial mass of ~7 mg) and (right) 5, 10, and 20°C.min-1 at (initial mass of ~11 mg)

18

6.5

200

6.4

150

6.3 Ea (2, 4, 8oC.min‐1)



‐1

Ea (kJ.mol )

250

Ea (5, 10, 20oC.min‐1)

100

6.2

(2, 4, 8oC.min‐1) (5, 10, 20oC.min‐1)

50

6.1

0

6.0 0.0

0.2

0.4

0.6

0.8

1.0

 Fig. 9

and

vs.

for CaCO3 calcination at 2, 4 and 8°C.min-1 at masses of ~7 mg and 5, 10, and

20°C.min-1 at masses of ~11 mg

19

Table 1 Kinetic triplets and % contribution to total conversion for three simulated, equally-weighted parallel dual mechanism reactions where F1 = 1 Code ↓

(s-1)

(kJ.mol-1)

Reaction →

1

2

1

2

1

2

1

2

A

1e10

1e10

120

125

F1

F1

50

50

B

1e10

1e10

120

130

F1

F1

50

50

C

1e10

1e10

120

135

F1

F1

50

50

20

% of total