To determine surface species site fractions, we begin with a conservation equation that balances the time rate of change of each surface species concentration with the net production rate of that surface species through chemical reactions at the surface. For each surface material there may be more than one surface phase; for example, sp ² and sp ³ structures may be treated as two different phases in the surface mechanism. In addition, there may be several species defined for each phase representing, for example, open surface sites, physisorbed species, or chemisorbed species. Each surface phase contains an independent set of surface species; in other words, a given surface species exists in only one phase on only one material. The surface species conservation equation is applied to every species in each surface phase contained on each surface material , as:

(8–4)

Here is the surface area of the m th material in the reactor and is the molar concentration of the k th surface species (mole/cm ² ).

Within the CHEMKIN reactor models, we use a strict ordering of species, species types, and surface materials that allows very general descriptions of the chemistry between species of all types in the reactor. The subscript is then used for all species, whether gas-phase, surface-phase, or bulk-phase. For this reason, in Equation 8–4 and subsequent equations, we introduce somewhat complicated indices for the range of surface species. Specifically, refers to the first surface-phase species in the first surface phase of the material , while refers to the last surface-phase species of the last surface phase of the m th material. The total number of surface materials is , with the total number of surface phases on each material given by and the total number of bulk phases by . The surface phase index range for a material is given by . The first phase defined in the system is always the gas-phase, with , such that and in all cases where surface species are defined. In each phase, , there are species, whose indices are in the range, . When the species index range includes all surface species, as in Equation 8–4 , we introduce a shorthand notation, where and . Although this nomenclature is somewhat confusing on first glance, it need not unduly concern the user; all of this formalism is strictly maintained internally in the Ansys Chemkin software structure. For further insight into this nomenclature, see the Introduction explanation .

The molar concentration of a surface species is related to the total site density of a surface phase, , to the site fraction of the k th surface species, and to the species coverage factor, , as follows:

(8–5)

Assuming the surface areas of each material are constant, substituting Equation 8–5 into Equation 8–4 , and expanding the derivatives, we obtain:

(8–6)

The net change in surface-phase site densities is defined as:

(8–7)

Equation 8–7 is the surface site conservation equation that must hold true for each surface phase in each material. Here we note that, for the steady-state case, applying Equation 8–6 to all surface species will lead to a singular Jacobian matrix. To obtain a well posed system of equations, one must introduce the additional requirement that all surface site fractions, , sum to one:

(8–8)

Equation 8–8 may then be solved directly in place of one of the surface species balances in Equation 8–6 . However, this approach results in the numerical round-off error in the calculation of the all of the ‘s to be assigned to one equation (Equation 8–8 ). This may cause problems in the case when that surface-site fraction is small. Instead, we chose to apportion the additional constraint represented by Equation 8–8 over all of the equations represented by Equation 8–6 for the surface phase n according to the size of the surface site fraction, as follows:

(8–9)

Note that the time constant for the last term on the right-hand-side of Equation 8–9 is arbitrary, and we have chosen because it is dimensionally correct and physically significant for the well mixed reactor model. For transient cases, the last term on the right-hand side is set to zero and a true transient for all species is solved. If Equation 8–9 is summed over all surface species in phase and combined with Equation 8–7 , then the following relation results:

(8–10)

which is equivalent to the sum of the ’s being equal to one, with a false transient.