Chemical source term

Despite the progress in CFD for inert-scalar transport, it was recognized early on that the treatment of turbulent reacting flows offers unique challenges (Corrsin 1958 Danckwerts 1958). Indeed, while turbulent transport of an inert scalar can often be successfully described by a small set of statistical moments (e.g., (U), k, e, p, and (0 2)), the same is not true for scalar fields, which are strongly coupled through the chemical source term in (1.28). Nevertheless, it has also been recognized that because the chemical source term depends only on the local molar concentrations c and temperature T  

As discussed in Chapter 2, a fully developed turbulent flow field contains flow structures with length scales much smaller than the grid cells used in most CFD codes (Daly and Harlow 1970).29 Thus, CFD models based on moment methods do not contain the information needed to predict / (0 x, /). Indeed, only the direct numerical simulation (DNS) of (1.27)-(1.29) uses a fine enough grid to resolve completely all flow structures, and thereby avoids the need to predict / (0 x, / ). In the CFD literature, the small-scale structures that control the chemical source term are called sub-grid-scale (SGS) fields, as illustrated in Fig. 1.7. 

Heuristically, the SGS distribution of a scalar field 0(x, t) can be used to estimate the composition PDF by constructing a histogram from all SGS points within a particular CFD grid cell.30 Moreover, because the important statistics needed to describe a scalar field (e.g., its expected value ()) or its variance ( //2)) are nearly constant on sub-grid 

29 Only direct numerical simulation (DNS) resolves all scales (Moin and Mahesh 1998). However, DNS is computationally intractable for chemical reactor modeling. 

30 The reader familiar with the various forms of averaging (Pope 2000) will recognize this as a spatial average over a locally statistically homogeneous field. 

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Output mass rate and vapor temperature of release, mass rate of air entrained, density of mixtu >1 mass fraction in cloud. Limitations single chemical source terms, limited chemical... 

ALOHA has a comprehensive chemical source term library (>700 pure chemicals). The code can address many types of pipe and tank releases, including two-phased flows from pressurized/ cryogenic chemicals. The user may enter a constant or variable vapor source rate and duration of... 

FIRAC is a computer code designed to estimate radioactive and chemical source-terms as.sociaied with a fire and predict fire-induced flows and thermal and material transport within facilities, especially transport through a ventilation system. It includes a fire compartment module based on the FIRIN computer code, which calculates fuel mass loss rates and energy generation rates within the fire compartment. A second fire module, FIRAC2, based on the CFAST computer code, is in the code to model fire growth and smoke transport in multicompartment stmetures. 

In chemical reacting systems, the Reynolds number of the flow is not the only source of computational challenges. Indeed, even for laminar reacting flows the chemical source term can be extremely stiff and tightly coupled to the diffusive transport terms. Averaging, as done above to treat turbulent flows, does not... 

The CFD model developed above is an example of a moment closure. Unfortunately, when applied to reacting scalars such as those considered in Section III, moment closures for the chemical source term are not usually accurate (Fox, 2003). An alternative approach that yields the same moments can be formulated in terms of a presumed PDF method (Fox, 1998). Here we will consider only the simplest version of a multi-environment micromixing model. Readers interested in further details on other versions of the model can consult Wang and Fox (2004). 

Acid-base reactions are the archetypical instantaneous reactions. If we let A denote the acid concentration and B the base concentration, the chemical source term for both the acid and base can be expressed as... 

To overcome this difficulty, we can introduce a new variable defined in terms of a linear combination of A and B such that the chemical source term for ( is null. Consider an acid-base reaction of the form... 

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 ... 

Fio. 5. Region in ,-Y2 phase space with non-zero chemical source term and the mixing line. 

The region in - Y2 composition space where this chemical source term is nonzero is shown in Fig. 5. Note that the maximum conversion of C occurs when , = < i and corresponds to T2max = d s2 or (using Eq. 63) to c( — 0 (i.e., complete conversion). 

Except for the chemical source term, these equations have the same form as those used for the mixture fraction. Note that the chemical source term (S oo) is evaluated using the mixture fraction and reaction-progress variable in the particular environment. The average chemical source term (S2oo(Y2) will thus not be equal to S2cc, (( ), (Y2)) unless micromixing occurs much faster than the second reaction. 

By definition of the reaction-progress variables, Y2 and T22 are zero for the inlet streams, and nonnegative inside the reactor due to the chemical source term. Once the CFD model has been solved, the reactant concentrations in each environment n are found from... 

In practice, it may be difficult to determine in advance which method is best to use for a particular application. For example, the CFD results may be more sensitive to large-scale inhomogeneities in the flow field than to the chemical source term closure. A rational approach to determine whether a more detailed SGS model is needed might be to start with N — 1 (laminar-chemistry approximation) and compare the predicted mean chemical species fields to the two-environment model (N — 2). If the differences are small, then the simpler model is adequate. However, if the differences are large, then the CFD simulation can be repeated with N — 3 and the results compared to N — 2. Naturally, once this procedure has converged, it will still be necessary to validate the CFD results with experimental data whenever possible. Indeed, it may be necessary to... 

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

The CSTR model can be derived from the fundamental scalar transport equation (1.28) by integrating the spatial variable over the entire reactor volume. This process results in an integral for the volume-average chemical source term of the form ... 

Because die outlet concentrations will not depend on it, micromixing between duid particles can be neglected. The reader can verify this statement by showing that die micromixing term in the poorly micromixed CSTR and the poorly micromixed PFR falls out when die mean outlet concentration is computed for a first-order chemical reaction. More generally, one can show that die chemical source term appears in closed form in die transport equation for die scalar means. 

Similar remarks apply for CFD models that ignore sub-grid-scale mixing. The problem of closing the chemical source term is discussed in detail in Chapter 5. 

Note that in order to close (1.16), the micromixing time must be related to the underlying flow field. Nevertheless, because the IEM model is formulated in aLagrangian framework, the chemical source term in (1.16) appears in closed form. This is not the case for the chemical source term in (1.17). 

The last term on the right-hand side of (1.28) is the chemical source term. As will be seen in Chapter 5, the chemical source term is often a complex, non-linear function of the scalar fields , and thus solutions to (1.28) are very different than those for the z nm-scalar transport equation wherein S is null. 

Similarly, turbulent scalar transport models based on (1.28) for the case where the chemical source term is null have been widely studied. Because (1.28) in the absence... 

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