Method for Determining the Activation Energy Distribution Function of

Article pubs.acs.org/JPCB

Method for Determining the Activation Energy Distribution Function of Complex Reactions by Sieving and Thermogravimetric Measurements Gennaro Bufalo† and Luigi Ambrosone*,‡ †

INAIL-Sector Research, Certification and Verification, Department of Naples, I-80121 Naples, Italy (NanoBeM)-Nanomedicine Center, Department of Bioscience and Territory (DIBT), University of Molise, Pesche, 86090 Isernia, Italy

‡

ABSTRACT: A method for studying the kinetics of thermal degradation of complex compounds is suggested. Although the method is applicable to any matrix whose grain size can be measured, herein we focus our investigation on thermogravimetric analysis, under a nitrogen atmosphere, of ground soft wheat and ground maize. The thermogravimetric curves reveal that there are two well-distinct jumps of mass loss. They correspond to volatilization, which is in the temperature range 298−433 K, and decomposition regions go from 450 to 1073 K. Thermal degradation is schematized as a reaction in the solid state whose kinetics is analyzed separately in each of the two regions. By means of a sieving analysis different size fractions of the material are separated and studied. A quasi-Newton fitting algorithm is used to obtain the grain size distribution as best fit to experimental data. The individual fractions are thermogravimetrically analyzed for deriving the functional relationship between activation energy of the degradation reactions and the particle size. Such functional relationship turns out to be crucial to evaluate the moments of the activation energy distribution, which is unknown in terms of the distribution calculated by sieve analysis. From the knowledge of moments one can reconstruct the reaction conversion. The method is applied first to the volatilization region, then to the decomposition region. The comparison with the experimental data reveals that the method reproduces the experimental conversion with an accuracy of 5−10% in the volatilization region and of 3−5% in the decomposition region.

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droplet-size polydispersity of a system have been used.7−10 Each of them takes advantage of a particular physical property to follow and monitor. The kinetics of thermal degradation, in both the absence and the presence of air, are so complex that they cannot be monitored directly, and sophisticated mathematical models have to be used.11 Herein a semiempirical approach is employed to evaluate a continuous distribution function of activation energy. The correlation between particle size and activation energy is found experimentally via thermal analysis on samples of different sizes. On the basis of the thermogravimetric measurements, thermal degradation is divided into two independent parts: the first corresponds to the volatilization region, and the second one corresponds to the pyrolytic decomposition of the material. Both regions are studied with calculating the moments of the distribution function and the conversion of thermal degradation. The model is validated, in both regions, by comparing the predictive results with the experimental data.

INTRODUCTION Cereals are the most cultivated plants in the world, and the products of their processing are a food base for the human body.1 Therefore, large quantities of cereals are stored in silos. An inadequate storage and handling of the stored material may result in fires and explosions, possibly causing injuries to employees, sometimes loss of lives, and also considerable economic loss together with environmental loss.2 Recently we investigated calorimetrically different cereals and discovered that the products containing bran exhibit exothermic reactions of decomposition in the absence of air. This means that the simultaneous presence of seeds (or granules) and grain dusts, under conditions where only dust would be inert, can become an explosive mixture.3 Particle size reduction increases the total surface area of the material and the number of contact points for interparticle in the compaction process.4 In addition, the size polydispersity has a considerable effect on endothermic or exothermic reactions, and the aim of this paper is to provide more understanding to correlation between particle size and activation energy. In general, to describe complex reactions, such as pyrolysis of biomass, the distributed activation energy model (DAEM) is used.5 According to this model an infinite number of first-order parallel reactions take place concurrently6 and a continuous distribution function is used to represent the activation energies from various reactions. Several techniques for determining the © XXXX American Chemical Society

Received: October 25, 2015 Revised: December 15, 2015

A

DOI: 10.1021/acs.jpcb.5b10448 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

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EXPERIMENTAL SECTION Materials. Ground whole soft wheat (SWG) and ground maize (GM) have been kindly offered by the mill Filangieri of Campobasso, Italy. Methods. The thermal behavior of the treated cereal samples was examined using a TA Instruments model Q-600. A 90 μL empty platinum crucible was used as a reference. A heating rate of 10 °C·min−1 was used during all experiments. The samples were kept at 30 °C for 6 h, then analyzed in dry nitrogen at 100 mL·min−1 in the temperature range 20−800 °C. The sample mass was around 15−25 mg in all experiments. Particle Size Analysis. A sample grind of 100 g was placed in a stack of sieves arranged from the largest to smallest opening. The sieve selected was based on the range of particles in the samples. For the grinds, sieve sizes 90, 125, 212, 250, 300, 425, 500, 600, 710, 850, and 1000 μm were used. After sieving, the mass retained on each sieve was weighed. Sieve analysis was repeated three times for each ground sample.

assumes the thermal degradation to be the summation of an unlimited number of parallel single-step reactions.12 Consequently, eq 4 assumes the form −β

μn =

−

∫0

∞

E n Q (E ) d E

(6)

k d ln(1 − α) = 0 β dT

∞

∑ n=0

( −1)n μn n! (RT )n

(7)

(8)

where Ψ(T ) = (T − T0) −

μ1 R

ln

T 1 + T0 R

∞

∑ n=2

⎡ 1 1 ⎤ ⎢ ⎥ n − (RT0)n ⎦ ⎣ (RT )

( −1)n μn (1 − n)n!

(9)

Equation 7 is a differential equation of the first order; therefore, the only knowledge of the conversion at T0 is necessary and sufficient to determine the conversion to any other temperature;15 however, experimentally speaking, the constant k0 is unknown and then it returns to be useful to treat the term k0/β as a constant to be determined by an auxiliary condition on the conversion, that is

(2)

(3)

α(T1) = α1

where k is the rate constant. When k follows Arrhenius law, the activation energy can be easily calculated by defining a function Yβ(E,T) given by3 d ln(1 − α) = Yβ(E , T ) = k 0e−E / RT dT

(5)

α(T ) = 1 − e−k 0/ β Ψ(T )

If a first-order kinetics accurately describes the experimental results, the differential equation for the thermal degradation takes the following form

−β

Q (E)Yβ(E , T ) dE

We integrate the right-hand side from T = T0 to T and the lefthand side from the initial condition α(T0) = 0 to α. Accordingly

where νG and νS are the number of grams of volatiles and solid residues formed in the process. It is possible to determine the mass of various components at any time during the reaction in terms of the dimensionless progress variable α. Indeed, if mw0 is the mass of compound initially when α = 0 and m∞ S at the end when the degradation is completed, then

dα = k(1 − α) dt

∞

It is also normalized so that μ0 = 1. Although the distribution Q(E) can be calculated by the method described elsewhere,13,14 here we prefer to expose a simpler and practical procedure. To this end we expand Yβ(E,T) in Taylor series and rewrite eq 4 terms of μn

THEORETICAL SECTION Thermal degradation of a compound, W, can be schematized with a chemical reaction in the solid state, where one of the products is volatile (G) and all other substances are in the condensed state (S) W → νGG + νSS (1)

m w (t ) − mS∞ m w0 − mS∞

∫0

where Q(E) dE can be interpreted as the probability that the activation energy is between E and E + dE. The distribution function Q(E) is assumed to be unimodal with moments given by

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α=

d ln(1 − α) = dT

(10)

Combination of eqs 10 and eq 8 results in k0 −ln(1 − α1) = β Ψ(T1)

(4)

where E and k0 are the activation energy and Arrhenius preexponential factor, respectively, and β symbolizes the heating rate that is kept constant for each experiment. Decomposition reactions are intimately related to the complex molecular structure of cereals; they lead to the disintegration in numerous products with smaller molecules, which, in turn, reduce further through numerous reaction pathways. Describing all of these discrete reactions individually is next to impossible; therefore, a continuous distribution of species with respect to a set of continuous variables can be assumed. In a discrete reaction, the rate is expressed in terms of a constant activation energy. In a continuum of reactions the rate will be expressed in terms of an activation energy function which will, in general, depend on all species involved in the reaction. DAEM is widely used to investigate the kinetic mechanism of biomass pyrolysis. It

(11)

Thermogravimetric experiments provide a direct measure of the function α(T) so that α0 and α1 are extractable directly from experimental results. The temperatures T0 and T1 must lie within the thermal regime considered. The temperature T0 is easily determined inasmuch it coincides with the beginning of the ramp of the curve α(T) in the temperature range considered. The choice of T1 is somewhat more critical because for a too high value the assumption of parallel first-order kinetics may cease to be valid. We propose to choose T1 as the temperature at the inflection point of the experimental curve α(T). This issue deserves further note. The experiments are carried out at constant heating rate (β); then, the inflection point of the experimental curve α(T) is the maximum of the reaction rate. This choice therefore ensures us to use the largest rate range to determine the activation energy. The coordinates B


Article

The Journal of Physical Chemistry B of the inflection point (T1, α1) identify the condition to determine k0/β through eq 11. Of course, the procedure can be applied successfully only if the moments μn are known. Unfortunately the distribution function Q(E) is unknown, and its moments are inaccessible; however, if it is possible to establish experimentally that a functional relation of the type

E = E(+)

F(+) =

dF =

(15)

Table 1. Parameters of Grain Size Distribution for SWG and GM Samples

(12)

d E = Q (E ) d E (13)

sample

ζ

σ

χ2

R

SWG GM

5.79 ± 0.04 6.61 ± 0.03

0.65 ± 0.04 0.40 ± 0.09

0.003 0.01

0.999 0.994

An important characteristic of these materials is the endosperm texture that is related to the ability of the starch and protein components of the endosperm to bind together into a strong cohesive mass.19 The intrinsic hardness of the cereal particles is the same for both starch granules and protein fragments derived from endosperm, but the energy input required to mill the cereal particles to a size similar to that of the starch granules is much greater for the materials with hard endosperm textures. The samples were ground with the same mill, then results of Table 1 indicate that the endosperm texture of GM is much harder than SWG. Thermal Degradation. Volatilization Region. Thermogravimetric curves obtained on ground cereals exhibit two distinct jumps corresponding to two large mass losses of material.3 The shape of the curves is maintained even when the material is sieved, and the aliquots collected on each sieve tray were subjected to thermogravimetric analysis. In Figure 2, the

As one sees, the eq 13 connects the function F(+), calculated experimentally, to the unknown distribution Q(E). Then, the moments μn can be calculated if the functional relationship between E and + is known. In general, a power law E = h+s

ζ⎞ ⎟ ⎠

where + the particle diameter, ζ and σ are two adjustable parameters, and erf is the error function. The evaluated parameters are collected in Table 1.

exists, it can be exploited to evaluate the moments μn. Indeed, by virtue of eq 12 we can write dF d+ dE d+

1 1 ⎛ ln + − + erf⎜ 2 2 ⎝ 2σ

(14)

is a very versatile function that can be adapted in many circumstances, and therefore it was used therein.

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RESULTS AND DISCUSSION Sieve Analysis. Results of the sieve analysis were summed to give a cumulant passing each sieve tray. The grain size distributions from this analysis are presented as a histogram in Figure 1. The advantage of this type to display is that it allows

Figure 1. Cumulant passing each sieve tray. The dashed curves are the result of a quasi-Newton fitting algorithm to adjust the parameter to fit a log-normal distribution function. Figure 2. Thermogravimetric curve obtained on sample SWG from sieve tray with nominal maximum size 90 μm. Volatilization and decomposition regions are well-distinguished.

easy fitting of discrete data and interpolation to read off particular value of the distribution.16 For instance, +50 is 750 μm for GM and 250 μm for SWG. In other words, 50% of GM is made from particles much larger than the sample SWG. It is well known that maize does not contain structuring proteins such as gluten, which, on the contrary, are very abundant in soft wheat.17 Because of this different chemical composition, materials react differently to the grinding and exhibit different values of +50 . A quasi-Newton fitting algorithm was used to obtain the best fit to experimental data.18 Of course, a large number of functions can be used for fitting to the same experimental data set so that a criterion for selection must be adopted. We chose as the best distribution the curves that had the lowest values of the correlation coefficient, R, and χ2. The procedure, applied to both samples, provides the following distribution

thermogravimetric curve for SWG from sieve tray with nominal maximum size 90 μm is shown as an example. Two mass loss jumps are well-distinguished, and this allows us to analyze the two regions separately. The first region starts immediately when the temperature rises and stops at 433 K, representing the dehydration−volatilization of the sample. The percentage change, Δm/m0, is a good indicator for evaluating the correlation level between the volatilization process and particle size. Plots of Δm/m0 versus + displayed in Figure 3 reveal a strong correlation between size particle and mass loss in the volatilization region. Indeed, the oscillation of 1 mm in diameter causes a mass variation of ∼4% in SWG and 6% in C


Article


Figure 3. Percentage change of mass in the volatilization region. The linear trend reveals a strong correlation between size particle and mass loss.

Figure 5. Log−Log plot of activation energy versus mean diameter of particles in the volatilization region. The straight lines indicate that a power law is a good functional relationship that links the activation energy to particle size.

GM; that is, volatilization of GM is 1.5 times faster. Doubtless this effect is due to the different chemical structure of the materials; however, a mechanism of the process of volatilization is hardly obtainable by simple methods because moisture, aldehydes, ketones, alcohols, and many other volatile substances are released.20 Furthermore, the situation is chemically made more complex by the presence of starch. This latter is a polymer composed of amylose and amylopectin such that if it is heating in the presence of water undergoes to ordered to disordered state phase transition, known as gelatinizaztion.21,22 Many complex reactions can frequently be approximated by first or pseudo- first-order reactions. In the kinetic analysis of such reactions it is convenient and practical to consider a continuous distribution of components. On the contrary, eq 4 states that for first-order kinetics a semilog plot of Yβ versus T−1 can be used to extract from the experimental data the activation energy of the process.3 Figure 4 highlights

range. Parameters h and s of eq 14 were calculated by nonlinear regression, and the results are collected in Table 2. Substitution Table 2. Correlation between Activation Energy and Particle Size for SWG and GM Samples s

s

sample

h

volatilization region

h

decomposition region

SWG GM

84 ± 2 55 ± 2

−0.195 ± 0.005 −0.106 ± 0.008

212 ± 23 314 ± 12

−0.10 ± 0.02 −0.12 ± 0.02

of eqs 13, 14, and 15 into eq 6 and integration of the resulting expression yield ⎛ ⎞ 1 μn = h2 exp⎜nsζ + n2s 2σ 2⎟ ⎝ ⎠ 2

(16)

As one can see, we have obtained an expression for μn that is a function of ζ, σ, h, and s, that is, quantities experimentally accessible. The first five moments were used to evaluate the conversion of both materials, and results are displayed in Figure 6. The accuracy of the method to reproduce the experimental data is in the range 5−10%. Basically, the result indicates that the volatilization process can be kinetically described as a continuous process. Decomposition Region. The region of decomposition is very wide and extends to 1073−1173 K. Herein the material undergoes a strong mass loss due to the formation of heavy volatile, pyrolysis, and carbonation. Figure 7 shows that the percentage loss of mass is a linear function of particle size even in this region. Therefore, a strong correlation between material decomposition and particle size exists, and this makes us confident of being able to apply the method described above also in this region. The thermogravimetric curves were scaled to the interval [0, 1] and the functions Yβ(E, T) were computed for all dimensions taken from the sieves. In Figure 8 values of both samples, collected on sieve tray with nominal maximum size 90 μm, are shown as an example. As one sees, unlike the region of volatilization, here the straight lines are almost coincident, indicating that the Arrhenius equation can be used to describe the process and the pyrolysis is less sensitive to the material local structure than volatilization process. By repeating this method for all of the collected fractions we obtain results plotted in Figure 9. As it can be seen, the power law fits the experimental data well, and applying a procedure of leastsquares, we evaluated parameters h and s collected in Table 2.

Figure 4. Semilog plot of Yβ(E,T) vs T−1 for SWG and GM samples collected on sieve tray with nominal maximum size 90 μm, in the volatilization region.

results obtained for material collected on a sieve tray with nominal maximum size of 90 μm. The procedure was repeated for all each sieve trays, and we checked that the linear regression correlation coefficient was within the range [0.98, 1]. This assured us that the volatilization for each particle size of material was well-described by a first-order reaction. Then, values of E obtained were plotted as a function of the mean diameter to estimate the functional dependence between E and + in the volatilization region. The Log−Log plot shown in Figure 5 indicates that a power law accurately describes the activation energy of thermal processes in this temperature D


Article


Figure 9. Log−Log plot of activation energy versus mean diameter of particles in the decomposition region. The straight lines indicate that a power law is a good functional relationship that links the activation energy to particle size.

Figure 10 for both cereal samples. The experimental conversion is reproduced with an accuracy of 3−5%; however, at high temperatures the computed value tends to deviate from the experimental one. This is because the system is no longer analyzable by first-order kinetics.

Figure 6. Calculated (red curve) and experimental (black curve) conversion as a function of temperature in the volatilization region. Experimental results are based on measurements carried out on a ground and not sieved product. The largest recorded deviations are within 5−10%.

Figure 7. Percentage change of mass in the decomposition region. The linear trend reveals a strong correlation between size particle and mass loss.

Figure 10. Calculated (red curve) and experimental (black curve) conversion as a function of temperature in the decomposition region. Experimental results are based on measurements carried out on a ground and not sieved product. The largest recorded deviations are within 3−5%.

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CONCLUSIONS It is difficult to investigate the kinetics of complex compounds because of the lack of a sufficient number of experimental data. Then, we have made us of kinetic models with distributed activation energy. In general, the distribution function is assumed to be Gaussian. This function is symmetric and valid only for a narrow range of activation energy, ΔE. For complex matrices, where a sieving analysis can be performed, we propose a method based on experimental results. Although this method can be applied to different matrices, our investigation has focused on two samples of cereals, SWG and GM, which have a

Figure 8. Semilog plot of Yβ(E,T) versus T−1 for SWG and GM samples collected on sieve tray with nominal maximum size 90 μm in the decomposition region.

These values were then used to determine the first five moments of the distribution function Q(E); then, the conversion α(T) was calculated. Results are summarized in E


Article


Wood by Distribution Activation Energy Model DAEM. Chem. Biochem Eng. Q 2012, 26, 45−53. (13) Ambrosone, L.; Ceglie, A.; Colafemmina, G.; Palazzo, G. A novel Approach for Determining the Droplet Size Distribution in Emulsion Systems by Generating Function. J. Chem. Phys. 1997, 107, 10756−10763. (14) Ambrosone, L.; Ceglie, A.; Colafemmina, G.; Palazzo, G. General Methods for Determining the Droplet Size Distribution in Emulsion Systems. J. Chem. Phys. 1999, 110, 797−804. (15) Bert, C. W.; Malik, M. Differential Quadrature Method in Computational Mechanics: A Review. Appl. Mech. Rev. 1996, 49, 1− 28. (16) Seward Thompson, B.; Hails, J. An Appraisal of the Computation of Statistical Parameters in Grain Size Analysis. Sedimentology 1973, 20, 161−169. (17) Brites, C.; Trigo, M. J.; Santos, C.; Collar, C.; Rosell, C. M. Maize-Based Gluten-Free Bread: Influence of Processing Parameters on Sensory and Instrumental Quality. Food Bioprocess Technol. 2010, 3, 707−715. (18) Dennis, J. E., Jr.; Moré, J. J. Quasi-Newton Methods, Motivation and Theory. SIAM Rev. 1977, 19, 46−89. (19) Turnbull, K. M.; Rahman, S. Endosperm Texture in Wheat. J. Cereal Sci. 2002, 36, 327−337. (20) Maeda, T.; Kim, J.; Ubukata, Y.; Morita, N. Analysis of Volatile Compounds in Polished-Graded Wheat Flour Bread Using Headspace Sorptive Extraction. Eur. Food Res. Technol. 2009, 228, 457−465. (21) Xingxun, L.; Yu, L.; Liu, H.; Chen, L.; Li, L. Thermal Decomposition of Corn Starch with Differential Amylose/Amylopectin Ratios in Open and Sealed Systems. Cereal Chem. 2009, 86, 383− 385. (22) Hatakeyama, T.; Inui, Y.; Iijima, M.; Hatakeyama, H. Bound Water Restrained by Nanocellulose Fibres. J. Therm. Anal. Calorim. 2013, 113, 1019−1025.

different chemical composition and respond differently to the grinding. This property is exploited to determine the activation energy of the thermal degradation process. The thermogravimetric curves indicate that there are two well-separated and independent regions. The activation energy was calculated by means of the function Yβ(E, T) for which the Arrhenius equation is valid. The procedure was repeated for materials collected on all sieves so that the activation energy was evaluated by varying the particle size. Then, a nonlinear regression was used to estimate the functional relationship E(+) to be a power law. Such a procedure has the advantage of estimating the moments of the distribution function Q(E) through the distribution function F(+), which is experimentally obtained from sieving measurements. The method is equally used to calculate the conversion in volatilization and decomposition regions. The comparison with experimental results confirms that the method reproduces the data with an accuracy of 5−10% in the volatilization region and of 3−5% in the decomposition region.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Method for Determining the Activation Energy Distribution Function of

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