The existence of slag film on the wall due to droplets deposition from the flow and heterogeneous vapour condensation on the film surface is taken into account. The wall of the plane channel is considered to be metal and water cooled, which leads to formation of frozen slag layer and slowly moving liquid slag film on it.

The solution is obtained by means of splitting of the whole problem as follows: at first we predict the mean flow parameters, Stanton number and friction in the channel with chosen wall temperature; then the characteristics of homogeneous condensation are predicted and then we employ the obtained data for prediction of the deposition film thickness and radiative-convective heat transfer in it.

The first problem of radiative-convective heat transfer of multi-phase flow was solved as two-dimensional in the narrow channel approach. The turbulence characteristics of the flow were predicted by means of semi-empirical dependence. The characteristics of homogeneous condensation were predicted on the basis of kinetics equation solution with known flow velocity and bulk temperature variation along the channel. Then the first problem could be resolved with known sub micron droplets concentration and size distribution, and the following influence of flow parameters on droplets characteristics is small due to the predicted fact, that distribution function rapidly establishes after abrupt homogeneous condensation process and slowly changes along the channel (without taking the coagulation process into consideration).

In radiative fluxes predictions the particles optical properties were defined by means of Mie theory for spectral complex refraction indexes (for large droplets - on the basis of data for slag, for small droplets - on the basis of data for optial silicon glass). In view of the predicted factor of primary influence of large droplets deposition on film formation its absorption coefficient was defined on the basis of data for slag. The input of both droplets fractions was taken into account.

Mass transfer rate of the flow with film surface was predicted on the basis of semi-empirical approach. The main mechanism of large droplets deposition was considered to be the turbulent migration, of small droplets- the thermophoresis.

The problem of combined heat transfer in two-layers liquid/frozen film was solved by means of iterative algorithm for definition of film thickness, corresponding to heat- and mass transfer rates in the channel. The discrete form of energy equation and system of two-point algebraic equations for one-side radiative fluxes were solved with the aid of vector sweep method.

The results of predictions of heat transfer and film characteristics are presented for the case when radiative heat transfer rate is substantially higher, then the convective one. In the considered case of low inlet vapour and slag droplets mass concentrations (C_{v 0}=0.01, C_{d 0}=0.02) the contribution of small silicon condensed droplets in heat - and mass transfer is small, but micron size slag droplets make a substantial input in radiative fluxes, as well as in mass transfer rate of the flow, and hence strongly influence the film characteristics and heat transfer in the channel.

The legitimacy of the problem solution splitting method is discussed.

In addition, it was assumed that the forces of friction between the gas and the particles are great. Only in this case, the velocity of the gas is equal to the velocity of the particles. The system of differential equations was derived in Lagrangian form. The solutions of differential equations are characterized by two different characteristic time and spatial dimensions. This results in the formation of the time and spatial boundary layers of a complicated structure that changes with variation parameters. Numerical solutions involved an adaptive implicit, finite difference scheme.

Depending on the characteristics of the condensed matter (sizes and ignition temperature of the particles, a kinetic law of heterogeneous reaction), dynamic regimes of unsteady process of combustion for the suspension were investigated. Our theoretical studies are focused on the dynamic regime of combustion. The unsteady combustion regime is determined by combination of the following factors: radiative, convective and conductive heat transfer. If the chemical reaction is accompanied by the gasification of condensed phase and the radius of particles exceed the critical value, then the maximum temperature of particles, front velocity and velocity of convective flow increase. At the stage of formation of combustion processes, the convective heat heat transfer results in a decrease of combustion wave. Transition from slow conductive combustion to the fast radiative one has an explosive character.

If the particles absorption the gaseous oxidizer, then the rate of heterogeneous reaction, the velocity of combustion wave and maximum temperature of particles decrease. If the particle size is lower than the critical one, the convective flux leads to an increase in duration of time initiation and in limiting case this time tends to infinity.

For the first time a Cartesian based method is proposed for combustion problems involving one ordinate symmetry and absorbing - emitting media with isotropic scattering. This method collapses the 3-D radiative information of the problem into its 2-D solution plane and is applicable for all ranges of optical thicknesses while maintaining almost analytic accuracy. To accomplish this task, this method makes use of effective intensity rays (EIR) each of which contains the information of a plane of real intensity rays perpendicular to the solution plane. As demonstrated in the reference A, this novel procedure thus eliminates the use of solid angles from the formulation and reduces both the complexity and computational expense involved in the solution of such problems.