Polycyclic Aromatic Hydrocarbons (PAHs) are a group of chemicals that occur naturally in coal, crude oil and gasoline. The nearly endless growth process of PAHs, where the six carbon atoms form an aromatic ring, necessitates a detailed understanding of chemical reaction pathways responsible for their formation and sequential growth, followed by the nucleation or inception of small soot particles, soot growth by coagulation and mass addition from gas phase species etc. The transition process from planar PAHs to spherical nascent soot particles is accompanied with stocking of PAHs on top of each other. Because the growth of PAHs involves an unlimited number of different species and reactions, a detailed description of the entire process is yet not possible. But to still make a proposition about the amount of PAHs formed and their size distribution, the statistical method of Linear Lumping  that describes the chemical source terms of PAHs was proposed by Frenklach. Therefore the growth of PAHs can be approximately described as a rapid polymerisation  process. In this study, it is assumed that the rates of polymerisation of PAHs and that of particle formation from both PAHs and condensation of PAHs at the soot surface are faster than the rates of convection and diffusion. As a result, the Steady State Assumption can be introduced for every single PAH that leads to a systematical reduction and consequently a clear decrease of numerical effort. In general, this assumption is valid, if there are chemical reactions which consume a chemical specie faster than it is formed by other reactions. In multidimensional steady systems it means that the convective and diffusive terms in the conservation equations are essential smaller than the chemical terms.
According to the steady state condition, we investigate in this work the Particle Size Distribution Functions (PSDF) of the PAHs and incipient soot within a range of temperatures and compositions. Instead of endless conservation equations for every single PAH a limited number of the moments of the size distribution functions of the PAHs can be solved. In the HACA mechanism described by Frenklach  the accumulation of acetylenes and the aromatic ring closure is described in one reaction only, whereas in this study we divide it into two reactions in a modified HACARC mechanism . This division is introduced, because the connection between the attached acetylenes and the soot surface element can be broken at high flame temperature before a new ring closure occurs. Furthermore in contrast to Mauss et al. , in this work, the ring closure is assumed to be irreversible.
In general, the well-known bimodality of the incipient soot PSDF can be described by using a complex log function . In this work an algebraic description of the target particle diameters region based on one Difference Equation System is established and then solved with the help of Z-transform. The primary results have been evaluated against published experimental study from Wang et al.  and the satisfactory agreement in target range has been achieved.