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Degree of macromixing

FIGURE 5.5 Extent of macromixing and micromixing and suitability of reactor models.

5.3.1. Direct Numerical Simulations (DNS)

As mentioned in Chapter 3, DNS involves full numerical simulation of the governing equations without approximations. Even in laminar reactive flow processes, space and time scales of relevant processes may range over many orders of magnitude. Such a wide range of time scales makes the governing equations quite 'stiff'. (In general, governing equations describing a flow process are termed 'stiff' if the process being modeled has a characteristic time of variation shorter than the time step one can afford.) Chemical reactions may cause very steep gradients locally. Resolving such gradients and handling the strongly coupled, non-linear governing equations may pose challenges even in laminar flow processes. Turbulence, which occurs on intermediate spatial scales, poses further problems in solving the governing equations. Despite the 'stiffness' of governing equations, DNS attempts to simulate reactive flow processes by resolving all the relevant space and time scales. Such rigorous DNS studies (see for example, Chakrabarti and Hill, 1997) have pointed out several characteristic features of reactive flow processes, such as:

(1) persistent tendency to initially segregate, with relatively thin reaction zones;

(2) scalar dissipation zone tracks well with the reaction zone, however, product concentration does not;

(3) reaction rate is highest in regions of greatest compressive strain rates;

(4) scalar variance decay is dominated by molecular dissipation and not by reaction.

Such an insight is useful for understanding the interaction of turbulence and chemical reactions and the influence of such interactions on effective rate and selectivity of chemical reactions. However, DNS may be carried out only for relatively simple geometry with moderate Reynolds number. Even then, huge computational resources are required. Some idea of the computational requirements imposed by the resolution demanded by DNS may be obtained by examining relevant dimensionless numbers. For direct simulations of reactive flow processes, in addition to resolving all the flow length scales from the integral to the Kolmogorov scale, scalar fields must be resolved up to Batchelor's scale. This means that the maximum wave number for the scalar field varies as:

To make the matters worse, chemical reactions steepen scalar gradients and often, larger values of kmax need to be used. Since (kmax x kmax x kmax) values must be stored in the computer memory for each field for each time step, application of DNS to reactive flow processes is limited to moderate Reynolds numbers and Schmidt numbers near unity. The Damkohler number (ratio of characteristic time scales of small-scale mixing and chemical reaction, see Chapter 2) is generally limited to values less than 30 to 50. Even if huge computational resources are available, the DNS approach is difficult to apply to the realistic geometry of industrial chemical reactors.

Moreover, practical reactive flow processes usually involve many interacting chemical species. These interactions are represented by many sets of coupled equations, which must be solved simultaneously. Before developing a model representing the entire reactive system, each individual process/chemical reaction must be understood and modeled separately. These sub-models can then be incorporated into the overall model, either directly or using a phenomenology. For reactive flow processes relevant to reactor engineers, it is often the case that various sub-models are not known adequately (for example, rates of chemical reactions or thermo-chemical data). It must be noted that, although DNS can provide valuable information about the interactions of flow processes and chemical reactions, it requires huge computational resources. When the accuracy of the required sub-models to carry out reactive flow simulations is inadequate, spending of huge computational resources on DNS is seldom justified. DNS, therefore, is not used to simulate complex industrial reactive flow processes. DNS, however, is an excellent tool for studying the fundamentals of turbulent reactive flow processes and for verifying other closure and phenomenological models of reactive flow processes.

5.3.2. Large Eddy Simulation (LES)

As discussed in Chapter 3, with LES, the smallest scale to be resolved is chosen to lie in the inertial sub-range of the energy spectrum, which means the so-called sub-grid scale (SGS) wave numbers are not resolved. As LES can capture transient large-scale flow structures, it has the potential to accurately predict time-dependent macromixing phenomena in the reactors. However, unlike DNS, a SGS model representing interaction of turbulence and chemical reactions will be required in order to predict the effect of operating parameters on say product yields in chemical reactor simulations. These SGS models attempt to represent an inherent loss of SGS information, such as the rate of molecular diffusion, in an LES framework. Use of such SGS models makes the LES approach much less computationally intensive than the DNS approach. DNS

studies may be used to develop a suitable SGS model, which may, in turn, be used with an LES model to simulate complex reactive flow processes. The SGS models may also be developed based on phenomenological, micromixing models. The LES approach, therefore, may serve as a link between the more simplified moment clo-sure/phenomenological models and DNS models. The overall computational demands of combining SGS models with LES simulations of complex industrial reactors may prove to be beyond the typical resources available to reactor engineers. It may be more effective to use Reynolds-averaged or phenomenological models in such cases.

5.3.3. RANS-based Models

By far, the most widely employed models for reactive flow processes are based on Reynolds-averaged Navier Stokes (RANS) equations. As discussed earlier in Chapter 3, Reynolds averaging decomposes the instantaneous value of any variable into a mean and fluctuating component. In addition to the closure equations described in Chapter 3, for reactive processes, closure of the time-averaged scalar field equations requires models for: (1) scalar flux, (2) scalar variance, (3) dissipation of scalar variance, and (4) reaction rate. Details of these equations are described in the following section. Broadly, any closure approach can be classified either as a phenomenologi-cal, non-PDF (probability density function) or as a PDF-based approach. These are also discussed in detail in the following section.

Apart from these three main approaches (DNS, LES and RANS), several attempts have been made to simulate reactive flow processes by using specialized micromix-ing models. These micromixing models are phenomenological and require empirical information to determine values of essential parameters (David and Villermaux, 1987; Ranade and Bourne, 1991). A typical model follows a lump of fluid in a Lagrangian frame that mixes with its environment following predetermined rules. Since the motion of a fluid element is tracked in a Lagrangian frame, chemical reactions occurring within the fluid element are treated without modeling. However, these micromixing models cannot be used as stand-alone models to simulate general reactive flow processes for the following reasons:

(1) these models require knowledge of mean velocity and turbulence fields;

(2) coupling between the micromixing time scale (see Section 5.1.1) and turbulence time scales is ambiguous; and

(3) extension to complex, inhomogeneous flows, where the environment contains partially reacted fluids, is difficult.

It is possible to eliminate some of these disadvantages of micromixing models by judiciously developing a composite modeling approach based on RANS and these micromixing models. Some such attempts are also discussed in the following section.

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