The biomass consists of several components including hemicellulose, cellulose and lignin. Since the activation energy is related to the bond energy and bond energy varies widely within biomass fuels having multiple components, it can be assumed that the pyrolysis process consists of an infinite number of reactions proceeding in parallel with E ranging from 0 to infinity (Anthony et al., 1973). The parallel reaction model (PRM), can also be called the distributed activation energy model with a large number of reactions in order to avoid confusion with the two or three single reactions proceeding in parallel. If Smv E is the mass change within a short period of time dt having activation energy between E and E + dE, the rate of liberation of volatiles for a first order pyrolysis can be written as:

where the specific reaction constant kE [1/s] is given by the Arrhenius expression:

where B and E are the pre-exponential factor and activation energy, respectively.

Assuming a Gaussian distribution, the fraction of initial total volatiles mass having activation energy betweenE and E + dE can be expressed as:

and:

J f (E)dE = 1

0

where Em is the mean activation energy, and a is the standard deviation of activation energy. The Gaussian distribution indicates that 1% mass has activation energy within E < Em — 2.3a; these E values refer to low activation energy components of the volatiles. Similarly, 1% of mass corresponds to high activation energy components with E > Em + 2.3a. Thus 98% of mass is covered for Em — 2.3a < E < Em + 2.3a while about 99.9% of the mass is located for E such that Em — 3a < E < Em + 3a. Assuming pre-exponential factor B is the same for all volatiles having activation energy 0 < E < ж and equal to B and integrating over all possible positive values of E will provide the volatile fraction:

(3.30)

With further rearranging, equation (3.30) becomes:

Note: the limits of integration have been changed from 0, <x to Em ± 3a that covers 99.9% of total mass. Defining:

G can be represented as a 2D matrix for values of E between Em — 3a and Em + 3a and values of T between T q and T n, where T 0 corresponds to temperature at the beginning of pyrolysis (99% VM remaining) and Tn corresponds to temperature at the end of pyrolysis (1% VM remaining), respectively (Martin, 2006; Chen, 2012b). With equation (3.32) in equation (3.31):

(3.33)

The equation (3.33) can be broken down into:

G(Em — 3a, Tq) G(Em — 3a, Tq + AT)

G(Em — 3a + AE, Tq)

G(E, T)

G(Em + 3a, Tq + nAT)

(3.34)

where T = Tq, Tq + AT, Tq + 2AT, T0 + 3AT,…, Tn = T0 + nAT, Em — 3a < E < Em + 3a.

Note that total number of terms in the G matrix will increase as the temperature T is increased or as AT is reduced. The value forB was set at 1.67 x 1013 [1/s] from transition state theory (Anthony et al., 1973). Assuming Em and a, volatile mass fraction {mv(T)/mv0} can now be calculated at a selected T by using G (E, T) in equation (3.33). Let the error between the calculated and measured mv(T)/mv0 fromTGAbe Sj at selected T = T j. The values for Em and a were calculated by minimizing the summed squared error Ejej at all selected data points within the domain of pyrolysis.

A spread sheet program was developed to determine the values of Em and a for the minimum most Ejs2. In the spread sheet, first the value of Em is fixed and a is varied. For a fixed value of Em, there is a value of sigma that will produce the minimum amount of error Ejs2. Then Em is varied and the combination of Em and a that produces the minimum most error can be determined.