In nonstoichiometric modeling, no knowledge of a particular reaction mecha­nism is required to solve the problem. In a reacting system, a stable equilibrium condition is reached when the Gibbs free energy of the system is at the minimum. So, this method is based on minimizing the total Gibbs free energy. The only input needed is the elemental composition of the feed, which is known from its ultimate analysis. This method is particularly suitable for fuels like biomass, the exact chemical formula of which is not clearly known.

The Gibbs free energy, Gtotai for the gasification product comprising N species (i = 1.. .N) is given by

where AG0,; is the Gibbs free energy of formation of species i at standard pres­sure of 1 bar.

Equation (5.73) is to be solved for unknown values of ni to minimize Gtotal, bearing in mind that it is subject to the overall mass balance of individual

elements. For example, irrespective of the reaction path, type, or chemical formula of the fuel, the amount of carbon determined by ultimate analysis must be equal to the sum total of all carbon in the gas mixture produced. Thus, for each jth element we can write

where ai, j is the number of atoms of the jth element in the ith species, and Aj is the total number of atoms of element j entering the reactor. The value of n should be found such that Gtotai will be minimum. We can use the Lagrange multiplier methods to solve these equations.

The Lagrange function (L) is defined as

Xai>ni — Aj kJ/mol

where Л is the Lagrangian multiplier for the jth element.

To find the extreme point, we divide Eq. (5.75) by RT and take the derivative,

£ > ° (576)

Substituting the value of Gtotal from Eq. (5.73) in Eq. (5.75), and then taking its partial derivative, the final equation is of the form given by

Kinetic Models

Gas composition measurements for gasifiers often vary significantly from those predicted by equilibrium models (Peterson and Werther, 2005; Li et al., 2001; Kersten, 2002). This shows the inadequacy of equilibrium models and under­scores the need of kinetic models to simulate gasifier behavior.

A kinetic model gives the gas yield and product composition a gasifier achieves after a finite time (or in a finite volume in a flowing medium). Thus, it involves parameters such as reaction rate, residence time of particles, and reactor hydrodynamics. For a given operating condition and gasifier configura­tion, the kinetic model can predict the profiles of gas composition and tempera­ture inside the gasifier and overall gasifier performance.

The model couples the hydrodynamics of the gasifier reactor with the kinet­ics of gasification reactions inside the gasifier. At low reaction temperatures, the reaction rate is very slow, so the residence time required for complete conversion is long. Therefore, kinetic modeling is more suitable and accurate

at relatively low operating temperatures (<800 °C) (Altafini et al., 2003). For higher temperatures, where the reaction rate is faster, the equilibrium model may be of greater use.

Kinetic modeling has two components: (1) reaction kinetics and (2) reactor hydrodynamics.