Reaction source term, time

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. 

In many applications, due to the large value of k, the first reaction is essentially instantaneous compared to the characteristic flow time scales. Thus, if the transport equation is used to solve for Y, the chemical-source term iS) will make the CFD code converge slowly. To avoid this problem, Y can be written in terms of by setting the corresponding reaction-rate expression (S ) equal to zero as follows ... 

In this section, we first introduce the standard form of the chemical source term for both elementary and non-elementary reactions. We then show how to transform the composition vector into reacting and conserved vectors based on the form of the reaction coefficient matrix. We conclude by looking at how the chemical source term is affected by Reynolds averaging, and define the chemical time scales based on the Jacobian of the chemical source term. 

Fractional time stepping is widely used in reacting-flow simulations (Boris and Oran 2000) in order to isolate terms in the transport equations so that they can be treated with the most efficient numerical methods. For non-premixed reactions, the fractional-time-stepping approach will yield acceptable accuracy if A t r . Note that since the exact solution to the mixing step is known (see (6.248)), the stiff ODE solver is only needed for (6.249), which, because it can be solved independently for each notional particle, is uncoupled. This fact can be exploited to treat the chemical source term efficiently using chemical lookup tables. 

In a transported PDF simulation, the chemical source term, (6.249), is integrated over and over again with each new set of initial conditions. For fixed inlet flow conditions, it is often the case that, for most of the time, the initial conditions that occur in a particular simulation occupy only a small sub-volume of composition space. This is especially true with fast chemical kinetics, where many of the reactions attain a quasi-steady state within the small time step At. Since solving the stiff ODE system is computationally expensive, this observation suggests that it would be more efficient first to solve the chemical source term for a set of representative initial conditions in composition space,156 and then to store the results in a pre-computed chemical lookup table. This operation can be described mathematically by a non-linear reaction map ... 

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