### 6.3.5. Reactions

The only reactions allowed are generic Arrhenius reactions, and some specific char reactions for coal combustion. The reactions determine the rate of change of species in the particle and the continuous phase, and the amount of heat released in the two phases.

#### 6.3.5.1. Arrhenius Reactions

For the Arrhenius reactions, all reactants must be in the particle, but products can be in both the particle and the continuous phase. The rate of reaction is

| (6–63)

where is the mass of reactant , V is the particle volume, is the order of reactant , and

| (6–64)

where A has units of

| (6–65)

Note that the Arrhenius expression is evaluated at the particle temperature,

The rate of change of mass of a reactant A is then , and of a product B is , where is the mass coefficient of that species. The reaction orders and the temperature exponent may be fractional numbers, and not necessarily integers.

#### 6.3.5.2. Heat Release

For any reaction, the heat release depends upon the reference specific enthalpies of the various reactants and products, if these are all provided. Alternatively, the heat release of a reaction can be specified directly and is then given by

| (6–66)

where refers to the reference material, and is the latent heat at the particle temperature.

#### 6.3.5.3. Char Oxidation

##### 6.3.5.3.1. Field

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 kd(Xg-XS), where Xg is the mole fraction of oxygen in the furnace gases far from the particle boundary layer and XS is the mole fraction of oxygen at the particle surface. The value of kd is given by:

| (6–67)

where:

- rp is the particle radius

- TP is the particle temperature

- Tg is the far-field gas temperature

- P is the local pressure

- PA is atmospheric pressure

- Dref is the dynamic diffusivity (recommended value is 1.8e-5 [kg m^-1 s^-1])

- Tref is the reference temperature (recommended value is 293 [K])

- is the exponent with value 0.75

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

| (6–68)

where:

- The parameters Ac and Tc depend on the type of coal, and are specified as input parameters.

- The default value of n is 0.0.

- For this model, kd and kc are in units of [kg m^-2 s^-1],

- Recommended values for Ac and Tc are 497 [kg m^-2 s^-1] and 8540 K [80].

  By equating the diffusion rate and the chemical reaction rate to eliminate XS, the overall char reaction rate of a particle is given by:

  | (6–69)

and is controlled by the smaller of the rates kd and kc.

##### 6.3.5.3.2. Gibb

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 TP:

| (6–71)

where the constants are given by Gibb as AS = 2500 and TS = 6240 K.

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

| (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:

- Dref is the dynamic diffusivity (recommended value is 1.8e-5 [kg m^-1 s^-1])

- Tref is the reference temperature (recommended value is 293 [K])

- is the exponent with value 0.75.

  | (6–75)

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

| (6–76)

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

| (6–77)

where:

| (6–78)

The pore diffusivity, Dp, 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 kc 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.

#### 6.3.5.4. Radiative Preheating

The stability of a pulverized coal flame depends on the feedback of heat from the flame zone to the raw coal as it emerges from the burner. In general, this preheating is supplied by a combination of convective heating by recirculating product gases, as well as by the absorption of radiation. CFX calculates the radiative heating using Equation 6–40.

The value of the particle emissivity is expected to change as pyrolysis proceeds (that is, it varies depending upon the mass fractions of coal and char). The present model assumes a linear variation in from the raw coal value (coal) to the value for char (char). That is:

| (6–80)

where fv is the fractional yield of volatiles. Typical values for are 1 for coal and 0.6 for char.

#### 6.3.5.5. Coal Combustion

##### 6.3.5.5.1. Coal Combustion - Gas Phase

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.

##### 6.3.5.5.2. Coal Decomposition

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.

##### 6.3.5.5.3. Devolatilization

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 CO of raw (that is, unreacted) coal, Cch of residual char, and CA 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 kV is expressed in Arrhenius form as:

| (6–84)

where TP is the temperature of coal particles (assumed uniform), R is the gas constant, and AV and EV 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 CO of raw coal, Cch of residual char after devolatilization has occurred, and CA 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 CO is equal to (1-CA). 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:

- is the current particle diameter

- is the swelling coefficient

- is the particle diameter at the start of devolatilization

- is the rate of change of mass of the reference material

- is the mass of the reference material at the start of the devolatilization

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

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