This model is based on the premise that at equilibrium stage, the total Gibbs free energy has to be minimized. The procedure mentioned below is developed for spouted bed gasification by Jarungthammachote and Dutta (2008) and for steam gasification with in-process carbon dioxide capture by Acharya and Dutta (2008). The total Gibbs free energy is given by:

N /=1

where ni = number of moles of species i,

Pi = chemical potential of species i given by,

( f ‘

Pi=G°i + RTIn -5-

I/ i)

f = fugacity of species i and Go; andfo i = standard Gibbs free energy and standard fugacity of species i.

Equation [16.20] can be written in terms of pressure as

[16.21]

where ф = fugacity coefficient.

For the ideal gas case at atmospheric conditions

P, = AGf + RT ln(y), [16.22]

where yi = mole fraction of gas species i

n

У =———————— 1——————- ‘

‘ Total moles in the mixture,/? .

7 total

AGof, i is the standard Gibbs free energy of formation of species i and is set equal to zero for all chemical elements.

Now, substituting equation [16.22] in equation [16.19], we get

[16.23]

The value of n; should be found such that the G* will be minimum. Lagrange multiplier methods can be used for this purpose. To use this method, the constraints need to be defined. Thus, the constraints can be defined in terms of the elemental balance on both the reactant and product side as:

N

Yjaljni=Aj, J = ,2,X—;K [16.24]

/ = 1

where aij = number of atoms of jth element in a mole of ith species. A} = total number of atoms of jth element in the reaction mixtures.

Thus, the Lagrange function (L) is defined as:

К f N

і V i=l

where Я = lagrangian multiplier. So, to find the extreme point,

Substituting the value of Gt from equation [16.23] to equation [16.25] and then taking its partial derivative as defined by equation [16.26]; the final equation will be of the form given by equation [16.27]: