Reaction source term

Finally, to conclude our discussion on coupling with chemistry, we should note that in principle fairly complex reaction schemes can be used to define the reaction source terms. However, as in single-phase flows, adding many fast chemical reactions can lead to slow convergence in CFD simulations, and the user is advised to attempt to eliminate instantaneous reaction steps whenever possible. The question of determining the rate constants (and their dependence on temperature) is also an important consideration. Ideally, this should be done under laboratory conditions for which the mass/heat-transfer rates are all faster than those likely to occur in the production-scale reactor. Note that it is not necessary to completely eliminate mass/heat-transfer limitations to determine usable rate parameters. Indeed, as long as the rate parameters found in the lab are reliable under well-mixed (vs. perfect-mixed) conditions, the actual mass/ heat-transfer rates in the reactor will be lower, leading to accurate predictions of chemical species under mass/heat-transfer-limited conditions. 

In the differential equations of the reverse combustion model, the reaction source term, dispersion coefficient, and the conductivity coefficient need to be defined. This is done based on data presented in Ref. [12]. At present, the model operates with a source term based on the combustion of coal particles. The source term is given by Eq. (5) ... 

Here, the densities of the gaseous and solid fuels are denoted by pg and ps respectively and their specific heats by cpg and cps. D and A are the dispersion coefficient and the effective heat conductivity of the bed, respectively. The gas velocity in the pores is indicated by ug. The reaction source term is indicated with R, the enthalpy of reaction with AH, and the mass based stoichiometric coefficient with u. In Ref. [12] an asymptotic solution is found for high activation energies. Since this approximation is not always valid we solved the equations numerically without further approximations. Tables 8.1 and 8.2 give details of the model. 

Results of the model for two parameters, i.e., the spatial temperature profile and the mass flux into the reaction zone as a function of gas mass flux are presented in Fig. 8.7. The temperature profile of the solid fuel flame (Fig. 8.7, left) is similar to that of a premixed laminar flame it consists of a preheat zone and a reaction zone. (The spatial profile of the reaction source term, which is not depicted here, further supports this conclusion.) The temperature in the burnt region (i.e., for large x) increases with the gas mass flux. The solid mass flux (Fig. 8.7, right) initially increases with an increase of the air flow, until a maximum is reached. For higher air flows, it decreases again until the flame is extinguished. 

The mass balance equations of the traditional multicomponent rate-based model (see, e.g., Refs. 57 and 58) are written separately for each phase. In order to give a common description to all three considered RSPs (where it is possible, of course) we will use the notion of two contacting fluid phases. The first one is always the liquid phase, whereas the second fluid phase represents the gas phase for RA, the vapor phase for RD and the liquid phase for RE. Considering homogeneous chemical reactions taking place in the fluid phases, the steady-state balance equations should include the reaction source terms ... 

Dm denotes the molecular diffusion coefficient F denotes the interphase mass exchange rate between the dense and the dilute phases and Fc = — F, which can be directly calculated with EMMS/matrix model parameters if the reaction source term, Sk, is negligible compared to the bulk gas conservation. For vaporization of A, the source term reads... 

The component mass balance equations of the traditional multicomponent rate-based model (see, e.g., Refs. [15, 16]) are written separately for each phase. As chemical reactions take place in the fluid phases, the steady-state balance equations should include the reaction source terms ... 

When chemical reactions are slow (with respect to mixing) it is not necessary to employ additional models to close the reaction source terms. For slow reactions (Da 1), turbulent mixing will be complete before the reaction can take place. The contributions of fluctuating concentrations may be neglected. Therefore, the time-averaged reaction source term can be related to the time-averaged temperature and species concentrations ... 

For fast and intermediate reactions, the time-averaged reaction source term will contain some additional terms. These additional terms need to be modeled to close the set of equations. For example, consider the case of a single second-order reaction with instantaneous rate given by... 

The non-linearity in terms of concentrations and exponential factor containing temperature, make the task of closing the reaction source term quite difficult. Even for an isothermal system, the time-averaged reaction source term will contain a new term, the time average of the product of fluctuating concentrations ( c ) of component 1 and component 2 ... 

In Eulerian-Eulerian (EE) simulations, an effective reaction source term of the form of Eq. (5.32) can be used in species conservation equations for all the participating species. The above comments related to models for local enhancement factors are applicable to the EE approach as well. It must be noted that interfacial area appearing in Eq. (5.32) will be a function of volume fraction of dispersed phase and effective particle diameter. It can be imagined that for turbulent flows, the time-averaged mass transfer source will have additional terms such as correlation of fluctuations in volume fraction of dispersed phase and fluctuations in concentration even in the absence... 

Note again that the concentrations c,- are given in moles per unit void volume, while the pseudohomogeneous rates /), defined by Eq.(2.1.26), are given in moles per unit total volume per unit time. Eq.(2.1.32) is a simplified form of the energy equation. It is a heat conduction equation with a chemical reaction source term and partly neglects the variation with temperature of the enthalpies. 

Finally the heterogeneous reaction source terms shall be analyzed that indicate the regions where the main coal-consuming reactions take place. In Figure 5.5a-d,... 

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