Transport term

Simulation of aerosol processes within an air quaUty model begins with the fundamental equation of aerosol dynamics which describes aerosol transport (term 2), growth (term 3), coagulation (terms 4 and 5), and sedimentation (term 6) ... 

Transport term , i.e. the rate at which molecules arrive at the surface Apportioning factor proportion of the bulk free energy released during stem deposition Lateral growth rate of a sector Strain surface free energy... 

After phase separation, two sets of equations such as those in Table A-1 describe the polymerization but now the interphase transport terms I, must be included which couples the two sets of equations. We assume that an equilibrium partitioning of the monomers is always maintained. Under these conditions, it is possible, following some work of Kilkson (17) on a simpler interfacial nylon polymerization, to express the transfer rates I in terms of the monomer partition coefficients, and the iJolume fraction X. We assume that no interphase transport of any polymer occurs. Thus, from this coupled set of eighteen equations, we can compute the overall conversions in each phase vs. time. We can then go back to the statistical derived equations in Table 1 and predict the average values of the distribution. The overall average values are the sums of those in each phase. 

In most cases, the term expressing the divergence of the molecular flux in Equation (40) (DV c,) can be neglected compared to the other two transport terms. Important excep-... 

This solution is composed of simple radiative and conductive heat transport terms. 

As indicated earlier, the axial conduction term is almost always negligible compared to the convective enthalpy transport term. Therefore, equation 12.7.47 is usually simplified to give... 

Note that the Eqs. (1), (2), and (8) are really and essentially different due to the absence or presence of different turbulent transport terms. Only by incorporating dedicated formulations for the SGS eddy viscosity can one attain that LES yield the same flow field as DNS. RANS-based simulations with their turbulent viscosity coefficient, however, essentially deliver steady flow fields and as such are never capable of delivering the same velocity fields as the inherently transient LES or DNS, irrespectively of the refinement of the computational grid ... 

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... 

At each time step, instead of evaluating the entire governing equation at once, as we did for the transport equation, we treat first the transport terms, then the reaction terms. The process is shown in Figure 21.1. We have split each step in the... 

The two terms on the right-hand side of this expression appear in closed form. However, the molecular transport term vV2 (Ut) is of order Re 1, and thus will be negligible at high Reynolds numbers. 

The molecular transport term vV2(m m ) is closed, but negligible (order ReL 1) in high-Reynolds-number turbulent flows. The production term... 

The first term on the right-hand side of this expression is the molecular transport term that scales as Sc Re 1. Thus, at high Reynolds numbers,26 it can be neglected. The two new unclosed terms in (3.88) are the scalar flux (u.ja), and the mean chemical source term (Sa(chemical reacting flows, the modeling of (Sa(0)) is of greatest concern, and we discuss this aspect in detail in Chapter 5. 

Thus, the closure problem reduces to finding an appropriate expression for the scalar flux (Ujtp). In high-Reynolds-number turbulent flows, the molecular transport term is again negligible. Thus, the scalar-flux term is responsible for the rapid mixing observed in turbulent flows. 

The triple-correlation term (u xix/)") and the molecular-transport term T -, defined by... 

The molecular-transport term rV2( 2) will be negligible at high Reynolds number. The scalar-variance-production term V4, is defined by... 

This type of model is usually referred to as an algebraic scalar-flux model. Similarmodels for the Reynolds-stress tensor are referred to as algebraic second-moment (ASM) closures. They can be derived from the scalar-flux transport equation by ignoring time-dependent and spatial-transport terms. 

If one then neglects the accumulation and transport terms in (4.83),22 an algebraic second-moment (ASM) model for the scalar flux results ... 

In order to simplify the discussion further, we will only consider the case where the molecular diffusivities of all chemical species are identical. We can then write the linear accumulation and transport terms as a linear operator ... 

In other words, if Y is really independent of , then the turbulent transport terms in its transport equation should not depend on . 

In Section 3.3, the general transport equations for the means, (3.88), and covariances, (3.136), of 0 are derived. These equations contain a number of unclosed terms that must be modeled. For high-Reynolds-number flows, we have seen that simple models are available for the turbulent transport terms (e.g., the gradient-diffusion model for the scalar fluxes). Invoking these models,134 the transport equations become... 

Alternatively, one can attempt to formulate an algebraic model by assuming that the spatial/temporal transport terms are null for the conditional reaction-progress variables. However, care must be taken to ensure that the correct filtered reaction-progress variables are predicted by the resulting model. 

In an effort to improve the description of the Reynolds stresses in the rapid distortion turbulence (RDT) limit, the velocity PDF description has been extended to include directional information in the form of a random wave vector by Van Slooten and Pope (1997). The added directional information results in a transported PDF model that corresponds to the directional spectrum of the velocity field in wavenumber space. The model thus represents a bridge between Reynolds-stress models and more detailed spectral turbulence models. Due to the exact representation of spatial transport terms in the PDF formulation, the extension to inhomogeneous flows is straightforward (Van Slooten et al. 1998), and maintains the exact solution in the RDT limit. The model has yet to be extensively tested in complex flows (see Van Slooten and Pope 1999) however, it has the potential to improve greatly the turbulence description for high-shear flows. More details on this modeling approach can be found in Pope (2000). 

The estimate of (U X > will also contain statistical error. However, at high Reynolds numbers, the molecular transport term in (6.178) will be small, and thus noise in this term is less problematic. 

Keeping only the accumulation and spatial-transport terms, the FV code solves a discretized form of... 

The spectral transport term for the first wavenumber band is defined by... 

In the SR model, forward and backscatter rate constants are employed to model the spectral transport terms ... 

Comparing this result with (B.15), we can observe that the terms in (B.15) involving a , bn, and cn represent the transport terms for the moments, (B.17), predicted by the... 

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