73 Simulation Of Reactive Flows

Various approaches to modeling reactive flows are discussed in Chapter 5. For most approaches, based on RANS equations, the usual finite volume method can be applied to solve the required transport equations of mean or variance of relevant quantities. However, obtaining a converged solution in a reacting flow can be difficult for a number of reasons. The chemical reactions and corresponding energy changes may have a strong impact on the basic flow patterns. This is especially true for gaseous combustion reactions, in which reactions release a large amount of energy causing significant changes in density, and large accelerations. Strong coupling between the mass/momentum transport equations and the species transport equations exists for such systems, which may lead to difficulties in convergence. Even when there is no significant coupling between momentum and species transport equations, non-linear coupling between different species equations due to chemical kinetics may make the solution task more complicated. In many cases, the reaction source term of any single species depends on concentrations of other species. Unless the solver is solving all the species equations simultaneously (which is not usually the case), this interdependence can lead to convergence difficulties. Another convergence issue in reacting flows is related to the magnitude of reaction source terms. For fast chemical reactions, the reaction source term may dominate the discretized species transport equations and it may no longer remain diagonally dominant. Such source-dominated equations may lead to rapid and unstable changes in species concentrations.

Some of the modeling approaches used for dealing with very rapid chemistry are discussed in Chapter 5. Apart from these modeling approaches, several techniques have been evolved to tackle convergence difficulties. In many cases, it is advantageous to start simulations from a simplified case. For example, it is often beneficial to first carry out cold flow simulations without considering chemical reactions. Using these results as an initial guess, complete model including chemical reactions and energy equations can be solved. For reactive flows, it is essential to carry out the time scale analysis of the processes under consideration (as discussed in Chapter 5). The relative magnitudes of characteristic time scales of convection, mixing and chemical reactions often provide guidelines for selecting suitable parameters of the numerical method. If such a time scale analysis reveals very different time scales for different processes, it may be necessary to use a fractional time step method, which uses different time steps for different processes. For example, when implementing the multi-environment model, Ranade and Bourne (1991) used two different time steps, one for simulating large-scale processes such as convection and turbulent diffusion and the other for simulating micromixing and reactions occurring within each computational cell.

When using a conserved scalar approach with the equilibrium assumption, it is necessary to carry out computations of: (1) equilibrium composition of gas from

FIGURE 7.19 Generic coupling between a PDF solver and a CFD solver.

the atomic composition and the total enthalpy; and (2) mean scalar values from the instantaneous mixture fraction. Often these calculations are carried out using a separate program by executing it outside the main flow solver. The results of these computations can be stored in look-up tables. These look-up tables along with an efficient interpolation algorithm, are used by the flow solvers to estimate the required quantities. The generic coupling of such external codes with CFD solvers is shown in Fig. 7.19. Details of numerical methods required for interpolation, integration etc. are not discussed here and can be found in Press et al. (1992). For full PDF methods, Monte Carlo methods may be used (Pope, 1981; 1985). Roekaerts (1989) extended these methods using a Monte Carlo ensemble associated with a fixed grid, instead of using stochastic differential equations. This approach can conveniently be implemented in conventional turbulent flow solvers. The above cited papers and a recent review by Fox (1996) can be referred to for more details of full PDF methods and their relationship with other approaches to modeling reactive flows.

To simulate multiphase reactive flows, it is necessary to devise suitable partial or full elimination algorithms to treat coupling between multiple phases due to interphase mass transfer and chemical reactions. If the reactions are slow, the standard partial elimination algorithm, discussed earlier, can be applied since in such a case, the interphase mass transfer terms are linear. When chemical reactions interact with mass transfer, interphase transfer terms become non-linear and special algorithms need to be developed on a case-by-case basis. If enhancement in mass transfer rate (multiplier to the standard linear interphase mass transfer term, see Eq. (5.32)) is accounted for by developing a look-up table (following the strategy used for PDF models), the standard elimination algorithm discussed earlier can be used. Computationally, however, it is more efficient to develop a case-specific linearization to account for the functional dependence of mass transfer enhancement on species concentrations. For very complex multiphase reactive flows, it may be useful to use a multiscale modeling approach rather than developing an all-encompassing comprehensive CFD model. In the multiscale approach, complex chemistry and its interaction with mass transfer and mixing are modeled using a simpler (than CFD) modeling framework comprising fewer computational cells or sub-regions. Such a model, therefore, requires several empirical inputs regarding the underlying flow patterns, degree of mixing, phase volume fractions and so on. Detailed flow simulations using a suitably simplified (with respect to chemistry) CFD model is used to generate the necessary information.

Inlet

Outlet

Molecular weights of A and B = 20

Liquid density = 1000 kg m-3

Reactor volume = 0.01 m3

Conversion based on ideal mixing: Xmix = 1/1 + kt

Initial conditions: mass fractions, mA = 1.0, mB = 0.0

Sample of simulated results:

Sr. No.

Rate constant, l/s

Impeller speed, m s 1

(1- — XA)mix

1 - Xa

0 0

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