MARCO CONCEPTUAL

With its high precision and well-controlled experimental conditions, TGA is a useful tool for studying the devolatilization, gasification and combustion of biomass under a kinetic

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regime86-90. However, TGA can be employed only at relatively low heating rates because the temperature of small samples is unknown at high heating rates. Accordingly, the results of TGA studies cannot be used alone in the modeling of industrial reactors; they serve as basic research to direct further development in the field. Heat and mass transfer limitations must be included in an overall model of industrial reactors. It is assumed that the samples are under kinetic control, meaning that heat and mass transport processes do not generate significant macroscopic heterogeneity in samples. Hence, product formation is governed by chemical reactions88.

Rate Equations

It is well known that thermogravimetric curves can be analyzed mathematically using the following type of kinetic equation91:

( ) ( ) α = α d k T f dt (2-1)

where α is the reacted fraction, f(α) is a continuous function representing the reaction model and k(T) is the temperature-dependent rate constant defined by the Arrhenius equation92:

/

( ) eE RT

k T =A − (2-2)

where A is the pre-exponential factor, E is the activation energy and R is the universal gas constant. Various forms of f(α) and g(α), the integral of 1/f(α), are listed in Table 2-292.

Isothermal and non-isothermal kinetics

General expectations for a good kinetic model include: description of the behavior of samples under a wide range of experimental conditions; prediction of behavior outside the domain of the observations; characteristics that can reveal similarities and differences between the samples; and finally a deeper insight into the processes taking place93.

TGA experiments can be conducted under either isothermal conditions at a particular temperature, or with dynamic/non-isothermal heating programs that involve different heating rates. Non-isothermal heating resolves a major defect of isothermal experiments, which is that a sample requires some time to reach the experimental temperature. During the non- isothermal period of an isothermal experiment, the sample undergoes transformations that are likely to affect the results of the following kinetic analysis. This problem restricts the use of high temperatures in isothermal experiments92, 94.

It is generally believed that kinetic analysis yields an adequate kinetic description of thermal decomposition in terms of the reaction model and Arrhenius parameters. These three

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components (f(α), E and ln A) are also called the ‘kinetic triplet’. To determine Arrhenius parameters using equation 2-1, one has to separate the temperature k(T) and conversion dependence f(α) of the reaction rate. The most popular way to do this is by fitting experimental data to different reaction models. This is also referred to as model fitting. Using this method, k(T) is determined by the form of f(α) chosen from Table 2-292.

Table 2-2: Alternate reaction models92

In isothermal kinetics, k(T) and f(α) are separated by the conditions of the experiment (k(T) is constant at constant T). The f(α) term is determined by fitting reaction models from Table 2-2 to the experimental data. After the f(α) term has been established for a series of temperatures, k(T) can be evaluated. Note that this procedure involves two sequential constrained fits: the first finds f(α) from data obtained at constant temperature, and the second finds E and A based on a fixed form of f(α)92.

A single non-isothermal experiment also provides information on both k(T) and f(α), but not separately. The model fitting approach attempts to determine all three members of the kinetic triplet simultaneously. Therefore, almost any f(α) can fit data satisfactorily, at the cost of dramatic variations in the Arrhenius parameters, which compensate for the difference between the assumed form of f(α) and the true but unknown kinetic model. Isothermal experiments include temperature as an experimental variable, whereas non-isothermal experiments allow fits that vary temperature sensitivity (E, ln A) and reaction model f(α) simultaneously. This allows errors in the reaction model to be concealed by other compensating errors89, 91-93. To overcome this, one should collect experimental data under a broad range of experimental conditions and evaluate them21, 90, 93, 95.

For both isothermal and non-isothermal studies, statistical methods are used in most cases to choose a unique kinetic triplet. The method of least squares is the most commonly used to

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characterize the goodness of fit; the minimum value of the residual sum of squares is used to choose the unique kinetic triplet91, 92. For non-isothermal data, many studies have used Coats- Redfern linearization as follows92, 93:

2 ) ( lng lnAR E T

β

where g(α) is the integral of 1/f(α) and β is the heating rate. This method is one of the most frequently used to process non-isothermal data. Inserting various g(α) into the equation results in a set of Arrhenius parameters. The left-hand side can be regarded as an experimental quantity, and this equation can then be used in least squares methods. However, this method is restricted to models containing only one reaction step, and the reaction should obey a variant of equation 2-193. Additionally, the evaluation is restricted to experiments with linear heating programs; f(α) should not contain unknown kinetic parameters, and α should be directly calculated from the experimental data. These criteria are rarely applicable to kinetic models involving wood. Therefore, these linearization techniques or ‘‘model-free’’ approaches may help to find initial values for iterations in the method of least squares, but these goals are better fulfilled by a method aiming directly at the description of the experimental data under a wide range of experimental conditions93. The least squares method works well for both isothermal and non-isothermal kinetics; there is no need to replace it with methods requiring simpler programming or less computation time. The method of least squares can be applied successfully to models of more than one partial process: competitive, consecutive and parallel reactions. Additionally, simultaneous evaluation of a series of experiments can be achieved91, 93.

Heterogeneous samples

Biomass is too complex a material to be described by only one chemical reaction. However, as an approximation, we can regard it as composed of pseudocomponents, where a pseudocomponent is a fraction of a reactive species exhibiting similar reactivities. A kinetic equation in the form of equation 2-1is assumed for each pseudocomponent21, 90, 93, 95. Each reaction includes values for E, A and the weighting factors for the pseudocomponents. The resulting mass loss rate curve is the weighted sum of the individual reaction rates93:

1 comp N j j j dm dt d c dt α = = −

∑

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where m is the normalized sample mass, Ncomp is the number of pseudocomponents and cj is

the normalized mass of volatiles formed from pseudocomponent j. Obviously one experiment cannot provide enough information for so many parameters. Therefore, one should evaluate a series of experiments. Evaluation can be done in two ways: either the experiments are evaluated simultaneously, looking for a set of kinetic parameters describing all of the experiments, or the thermogravimetric curves are evaluated one by one, independently from each other, and the results are compared after evaluation. The latter approach works well if the information content of an experiment is sufficient for the determination of all unknown parameters, and if it is followed up to verify that the kinetic parameters vary with the experimental conditions93, 96.

Recently, distributed activation energy models (DAEM) have been applied to the decomposition kinetics of biomass21, 90, 93, 97, 98. Several variants of DAEMs are known; usually a Gaussian distribution of activation energy is employed. Due to the complexity of the investigated materials, the model was expanded to simultaneous parallel reactions (pseudocomponents) that were described by separate DAEMs99-102. According to this model, the sample is regarded as a sum of M pseudocomponents, where M is usually between 2 and 4. The reactivity differences between species in a pseudocomponent are described by different activation energy values. On a molecular level, each species in pseudocomponent j is assumed to undergo first-order decay. The distribution of the species differing by E within a given pseudocomponent is approximated by a Gaussian function, with a mean activation energy value and a width parameter (variation)97.