The diffusion-limited surface reaction rate model, which is the default model in Ansys Fluent, assumes that the surface reaction proceeds at a rate determined by the diffusion of the gaseous oxidant to the surface of the particle:

(12–146)

Equation 12–146 is derived from the model of Baum and Street [50] with the kinetic contribution to the surface reaction rate ignored. The diffusion-limited rate model assumes that the diameter of the particles does not change. Since the mass of the particles is decreasing, the effective density decreases, and the char particles become more porous.

The kinetic/diffusion-limited rate model assumes that the surface reaction rate is determined either by kinetics or by a diffusion rate. Ansys Fluent uses the model of Baum and Street [50] and Field [179], in which a diffusion rate coefficient

(12–147)

and a kinetic rate

(12–148)

are weighted to yield a char combustion rate of

(12–149)

where is the surface area of the droplet (), is the partial pressure of oxidant species in the gas surrounding the combusting particle, and the kinetic rate, , incorporates the effects of chemical reaction on the internal surface of the char particle (intrinsic reaction) and pore diffusion. In Ansys Fluent, Equation 12–149 is recast in terms of the oxidant mass fraction, , as

(12–150)

The particle size is assumed to remain constant in this model while the density is allowed to decrease.

When this model is enabled, the rate constants used in Equation 12–147 and Equation 12–148 are entered in the Create/Edit Materials Dialog Box, as described in Setting Material Properties for the Discrete Phase in the User's Guide.

The intrinsic model in Ansys Fluent is based on Smith’s model [609], assuming the order of reaction is equal to unity. Like the kinetic/diffusion model, the intrinsic model assumes that the surface reaction rate includes the effects of both bulk diffusion and chemical reaction (see Equation 12–150). The intrinsic model uses Equation 12–147 to compute the diffusion rate coefficient, , but the chemical rate, , is explicitly expressed in terms of the intrinsic chemical and pore diffusion rates:

(12–151)

is the effectiveness factor, or the ratio of the actual combustion rate to the rate attainable if no pore diffusion resistance existed [344]:

(12–152)

where is the Thiele modulus:

(12–153)

is the density of oxidant in the bulk gas (kg/) and is the effective diffusion coefficient in the particle pores. Assuming that the pore size distribution is unimodal and the bulk and Knudsen diffusion proceed in parallel, is given by

(12–154)

where is the bulk molecular diffusion coefficient and is the porosity of the char particle:

(12–155)

and are, respectively, the apparent and true densities of the pyrolysis char.

(in Equation 12–154) is the tortuosity of the pores. The default value for in Ansys Fluent is , which corresponds to an average intersecting angle between the pores and the external surface of 45^° [344].

is the Knudsen diffusion coefficient:

(12–156)

where is the particle temperature and is the mean pore radius of the char particle, which can be measured by mercury porosimetry. Note that macropores ( Å) dominate in low-rank chars while micropores ( Å) dominate in high-rank chars [344].

(in Equation 12–151 and Equation 12–153) is the specific internal surface area of the char particle, which is assumed in this model to remain constant during char combustion. Internal surface area data for various pyrolysis chars can be found in [608]. The mean value of the internal surface area during char combustion is higher than that of the pyrolysis char [344]. For example, an estimated mean value for bituminous chars is 300 /g [101].

(in Equation 12–151 and Equation 12–153) is the intrinsic reactivity, which is of Arrhenius form:

(12–157)

where the pre-exponential factor and the activation energy can be measured for each char. In the absence of such measurements, the default values provided by Ansys Fluent (which are taken from a least squares fit of data of a wide range of porous carbons, including chars [608]) can be used.

To allow a more adequate description of the char particle size (and hence density) variation during combustion, you can specify the burning mode , relating the char particle diameter to the fractional degree of burnout (where ) by [607]

(12–158)

where is the char particle mass and the subscript zero refers to initial conditions (that is, at the start of char combustion). Note that where the limiting values 0 and correspond, respectively, to a constant size with decreasing density (zone 1) and a decreasing size with constant density (zone 3) during burnout. In zone 2, an intermediate value of , corresponding to a decrease of both size and density, has been found to work well for a variety of chars [607].

When this model is enabled, the rate constants used in Equation 12–147, Equation 12–151, Equation 12–153, Equation 12–154, Equation 12–156, Equation 12–157, and Equation 12–158 are entered in the Create/Edit Materials Dialog Box, as described in Setting Material Properties for the Discrete Phase in the User’s Guide.

Modeling multiple particle surface reactions follows a pattern similar to the wall surface reaction models, where the surface species is now a "particle surface species". For the mixture material defined in the Species Model dialog box, the particle surface species can be depleted or produced by the stoichiometry of the particle surface reaction (defined in the Reactions dialog box). The particle surface species constitutes the reactive char mass of the particle, hence, if a particle surface species is depleted, the reactive "char" content of the particle is consumed, and in turn, when a surface species is produced, it is added to the particle "char" mass. Any number of particle surface species and any number of particle surface reactions can be defined for any given combusting particle.

Multiple injections can be accommodated, and combusting particles reacting according to the multiple surface reactions model can coexist in the calculation, with combusting particles following other char combustion laws. The model is based on oxidation studies of char particles, but it is also applicable to gas-solid reactions in general, not only to char oxidation reactions.

In cases where the multicomponent particle type is used to define multiple reactions, and none of its components are specified as an evaporating species, you can use all correlations for heat exchange available in Ansys Fluent. Refer to Inert Heating or Cooling (Law 1/Law 6) for more information on different heat exchange correlations.

See Combusting Particle Surface Reactions for information about particle surface reactions.

The surface reaction consumes the oxidant species in the gas phase; that is, it supplies a (negative) source term during the computation of the transport equation for this species. Similarly, the surface reaction is a source of species in the gas phase: the product of the heterogeneous surface reaction appears in the gas phase as a user-selected chemical species. The surface reaction also consumes or produces energy, in an amount determined by the heat of reaction defined by you.

The particle heat balance during surface reaction is

(12–159)

where is the heat released by the surface reaction. Note that only a portion () of the energy produced by the surface reaction appears as a heat source in the gas-phase energy equation: the particle absorbs a fraction of this heat directly. For coal combustion, it is recommended that be set to 1.0 if the char burnout product is and 0.3 if the char burnout product is [73].

For heat exchange coefficient correlations available in Ansys Fluent, refer to Inert Heating or Cooling (Law 1/Law 6).

Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation heat transfer to particles using the Particle Radiation Interaction option in the Discrete Phase Model Dialog Box.

By default, Equation 12–159 is solved analytically, by assuming that the temperature and mass of the particle do not change significantly between time steps. Ansys Fluent can also solve Equation 12–159 in conjunction with the equivalent mass transfer equation using a stiff coupled solver. See Including Coupled Heat-Mass Solution Effects on the Particles in the User's Guide for details.