The char reaction is determined by both the rate of diffusion to the surface and the rate of chemical reaction at the surface. The surface reaction is assumed to be first-order in the oxygen mole fraction.

The rate of diffusion of oxygen per unit area of particle surface is given by k_d(X_g-X_S), where X_g is the mole fraction of oxygen in the furnace gases far from the particle boundary layer and X_S is the mole fraction of oxygen at the particle surface. The value of k_d is given by:

(6–67)

where:

The char oxidation rate per unit area of particle surface is given by k_cX_S. The chemical rate coefficient k_c is given by:

(6–68)

where:

and is controlled by the smaller of the rates k_d and k_c.

The oxidation mechanism of carbon can be characterized by the parameter so that oxides are produced according to the equation:

(6–70)

The value of is assumed to depend on the particle temperature T_P:

(6–71)

where the constants are given by Gibb as A_S = 2500 and T_S = 6240 K.

By solving the oxygen diffusion equation analytically, the following equation is obtained for the rate of decrease in the char mass m_c:

(6–72)

The far field oxygen concentration ρ _∞ is taken to be the time-averaged value obtained from the gas phase calculation, and ρ _c is the density of the char. Physically, k₁ is the rate of external diffusion, k₂ is the surface reaction rate, and k₃ represents the rate of internal diffusion and surface reaction. These are defined as follows:

(6–73)

where D is the external diffusion coefficient of oxygen in the surrounding gas. The coefficient is calculated in the same way as for the Field model, except in this model, kinematic diffusivity is used instead of dynamic diffusivity:

(6–74)

where:

where k_c is the carbon oxidation rate, defined by the modified Arrhenius equation

(6–76)

The default values of the model constants are A_c = 14 [m s^-1 K^-1] and T_c = 21580 K. Further:

(6–77)

where:

(6–78)

The pore diffusivity, D_p, is computed from external diffusivity, D, according to:

(6–79)

Note that the units of k₁, k₂ and k₃ for this model are s^-1, and the units for k_c in Equation 6–76 differ from those in Equation 6–68 in the Field model. "effic" is the so-called "internal diffusion efficiency" and describes the effective internal diffusion coefficient of oxygen within the particle pores. It is a user-accessible CCL parameter.

Coal combustion is calculated by combining a particle transport calculation of the coal particles with an eddy dissipation calculation for the combustion of the volatile gases in the gas phase. Two separate gases are given off by the particles, the volatiles and the char products that come from the burning of carbon within the particle.

Gas phase combustion is modeled by means of regular single phase reactions. A transport equation is solved in the fluid for each material given off by the particles. The ‘volatiles’ may be either a pure substance, a fixed composition mixture, or several independent materials.

Pulverized coal particles are treated in CFX as non-interacting spheres with internal reactions, heat transfer and full coupling of mass, momentum and energy with the gaseous phase. The combustion of a coal particle is a two stage process: the devolatilization of the raw coal particle followed by the oxidation of the residual char to leave incombustible ash. The devolatilization is usually modeled as a single step or a two step process. The char oxidation is modeled either as a global reaction, or using an analytical solution for the diffusion and reaction of oxygen within the pores of the char particle.

The devolatilization and char oxidation processes can occur on time scales on the order of milliseconds, which are several orders of magnitude less than the typical residence time of the particle in the furnace. Large variations in time scales can result in numerically stiff equations, which can cause accuracy problems with explicit integration algorithms. The CFX implementation of the particle transport model bases its timestep on the reaction rate to ensure that the solution has the required accuracy.

Devolatilization can be modeled by one or more reaction steps using the generic Arrhenius multiphase reactions capability, although normally the process is represented by one or two reaction steps. The simpler model is the single reaction model of Badzioch and Hawksley [78]. If, at time t, the coal particle consists of mass fractions C_O of raw (that is, unreacted) coal, C_ch of residual char, and C_A of ash, then the rate of conversion of the raw coal is:

(6–81)

and the rate of production of volatiles in the gas phase is given by:

(6–82)

where Y is the actual yield of volatiles (that is, the proximate yield multiplied by a factor to correct for the enhancement of yield due to rapid heating), so that the rate of char formation is:

(6–83)

The rate constant k_V is expressed in Arrhenius form as:

(6–84)

where T_P is the temperature of coal particles (assumed uniform), R is the gas constant, and A_V and E_V are constants determined experimentally for the particular coal.

Often the volatiles yield of a particular type of coal is known only from laboratory proximate analysis, where the heating rate is low and the volatiles escaping from the coal may undergo secondary reactions including cracking and carbon deposition on solid surfaces. It has been found experimentally that volatiles released from pulverized coal particles widely dispersed in a gas and heated quickly to typical furnace temperatures can produce a yield greater by as much as a factor of two than the proximate value. If the single reaction model is used, it is difficult to know which data to use because the coal particles experience a wide range of temperatures as they disperse in the furnace.

Bituminous coals generally have a volatiles yield that depends strongly on temperature and heating rates. In such cases, it is important to take account of this dependence, for example by using a multiple reaction model of devolatilization. As an alternative to the single reaction model, it is better to use the model of Ubhayakar et al. [79] in which two reactions with different rate parameters and volatiles yields compete to pyrolyse the raw coal. The first reaction dominates at lower particle temperatures and has a lower yield Y₁ than the yield Y₂ of the second reaction, which dominates at higher temperatures. As a result, the final yield of volatiles will depend on the temperature history of the particle, will increase with temperature, and will lie somewhere between Y₁ and Y₂. In this model, the mass fraction of combustible material (the raw coal) is specified as the mass fraction of volatiles because all this material could be converted to volatiles.

Again it is assumed that a coal particle consists of mass fraction C_O of raw coal, C_ch of residual char after devolatilization has occurred, and C_A of ash. The rate constants k₁ and k₂ of the two reactions determine the rate of conversion of the raw coal:

(6–85)

the rate of volatiles production is given by:

(6–86)

and so the rate of char formation is:

(6–87)

The initial value of C_O is equal to (1-C_A). The value of Y₁, and parameters A₁ and E₁ which define k₁ in the Arrhenius equation, analogous to Equation 6–84, are obtained from proximate analysis of the coal. Y₂, A₂ and E₂ are obtained from high temperature pyrolysis. Note that the yields are defined on a dry ash-free (DAF) basis.

Typically, the coal particle will swell due to the gas release during the devolatilization phase. The model assumes that the particle diameter changes in proportion to the volatiles released and the fractional increase in the mean diameter of the coal particle after complete devolatilization must be specified as input to the model. The particle diameter change due to swelling is calculated using the following relation:

(6–88)

where:

When the swelling coefficient is equal to 0.0, the particle diameter stays constant during devolatilization.