The spectral absorption coefficient of a particle cloud can be modeled by the Mie solution reformulated in terms of moments of particle size distribution [101]. The absorption coefficient of an ensemble of spherical particles is defined as

(13–15)

Using the Penndorf expansion of the Mie solution for scattering and extinction efficiencies[101] , the absorption coefficient takes the form

(13–16)

where

and the K' s are the optical coefficients given in Frenklach and Wang[101] .

For the absorption coefficients of agglomerates, the equivalent-sphere model [102] is employed. For this model, we let be the average class of the primary particle in the , which is assumed to be constant throughout the particle population. The equivalent effective diameter for a class i agglomerate is given as

(13–17)

Substituting the effective diameter into the expression for K _abs (λ) (Equation 13–15 ) yields the same formulation in terms of moments (Equation 13–16 ).

The total emissivity of the particle ensemble can be obtained by integrating the spectral radiation as

(13–18)

where the Planck function is given as

(13–19)

Since computing the overall absorption (or emission) coefficient of a particle cloud is computationally expensive, a semi-empirical model can be derived by dropping terms of high-order d ₀ in Equation 13–16 :

(13–20)

where f _v is the volume fraction of the particle ensemble.

Accordingly, the overall emissivity of a particle cloud can be expressed as a function of temperature T and particle volume fraction f _v :

(13–21)

The model parameter c _part is a user input and has dimensions of [length * temperature]^-1. (See the empar keyword in the Chemkin Input Manual Input Manual.)