Flux terms

Square brackets around a molecular species indicate atmospheric concentration. The rate constants k times the reactant concentration product refers to the rates of the chemical reactions of the indicated number. The photolytic flux term /l4 refers to the photodissociation rate of N02 in Reaction R14, its value is proportional to solar intensity.]. RO2 stands for an organic peroxyl radical (R is an organic group) that is capable of oxidizing NO to NO2. Hydrocarbons oxidize to form a very large number of different RO2 species the simplest of the family is methylperoxyl radical involved in R5, R6 and R8. 

Since in most situations the perturbation quantities (V and c() are not explicitly resolved, it is not possible to evaluate the turbulent flux term directly. Instead, it must be related to the distribution of averaged quantities - a process referred to as parameterization. A common assumption is to relate the turbulent flux vector to the gradient of the averaged tracer distribution, which is analogous with the molecular diffusion expression. Equation (35). 

The transfer of heat by radiation in general can be said to occur simultaneously with heat transfer by convection and conduction. Transfer by radiation tends to become more important than that by the other two mechanisms as the temperature increases. It is useful to gain an appreciation of the basic definitions of the energy flux terms, the surface property terms and their relationships while discussing radiative heat transfer. With this objective, reference may be made to Table 3.4 in which these are presented. 

Compartment 1 represents the vascular space of the tissue and can be identified as either a blood, plasma, or serum space. Compartment 2 is the interstitial space, and compartment 3 the intracellular tissue volume. Assuming first-order or linear membrane transport for the drug, the flux terms can be expanded as... 

For concentric jets and gas-solid two-phase jets, the jet momentum flux term pf(73 can be evaluated as follows, as suggested by Yang et al. (1984b) ... 

Equation (9.106) can be modified by eliminating all of the mass terms in favor of the burning rate (flux) term. This gives... 

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 joint-dissipation-flux term ujcap and the molecular-transport term T f, defined by... 

Models are required to close the flux terms on the left-hand side, and the final two terms on the right-hand side of (4.52). 

The flux terms are usually modeled by invoking a gradient-diffusion hypothesis. For example, one of the simplest models is... 

It is now obvious that the addition of other random variables simply leads to additional conditional flux terms on the right-hand side of the PDF transport equation. The major challenge is thus to find appropriate closures for the conditional fluxes. 

The second term on the right-hand side of (A.41) represents the flux of scalar dissipation into the scalar dissipation range, and can be rewritten in terms of known quantities. From (A.39), it can be seen that Taa(icD, t) = TD(t). Likewise, using the definition of kd, it follows that 2raK = CD(e/v)1/2. The scalar-dissipation flux term can thus be expressed as... 

Let us split the flux term pv C into the purely advective term pvC and a diffusive term, hereafter also referred to as T , describing the movement of the species i relative to the center of mass p(v —v)C ... 

The diffusion model and the hydraulic permeation model differ decisively in their predictions of water content profiles and critical current densities. The origin of this discrepancy is the difference in the functions D (T) and /Cp (T). This point was illustrated in Eikerling et al., where both flux terms occurring in Equation (6.46) were converted into flux terms with gradients in water content (i.e., VA) as the driving force and effective transport coefficients for diffusion, A), and hydraulic permeation,... 

The diffusion equation will be developed by considering each term in equation (2.1) separately. In addition, the flux terms wiU be divided into diffusive and convective flux rates. 

We will begin by assigning lengths to the sides of our box of dx, dy, and dz, as shown in Figure 2.4. Then, for simplicity in this mass balance, we will arbitrarily designate the flux as positive in the + x-direction, + y-direction, and + z-direction. The x-direction flux, so designated, is illustrated in Figure 2.5. Then, the two flux terms in equation (2.1) become... 

Control Volume Mass Balance. We can now combine equations (2.1), (2.5), (2.6), (2.11), and (2.13) into a mass balance on our box for Cartesian coordinates. After dividing hyV = dx dy dz and moving the diffusive flux terms to the right-hand side, this mass balance is... 

The flux terms are discretized at the cell interface. Because diffusion occurs in both directions, we will use central differences. In addition, we will use an explicit solution technique by discretizing our flux terms at only the n time step ... 

The diffusive fluxes are still appropriately deflned with central differences. However, in two- and three-dimensional computational domains, where there are two and three independent variables, it is difficult to know in which direction the flow is moving at all locations and times. For example, in our one-dimensional case, if m is a negative value, then our flux terms at the interfaces would become... 

Viability of Starch Derivatives as Flavoring Encapsulants. The capillary GC vapor phase flux term (defined by a percent external standard or ZEStD flux) previously described (34) was used to screen starch derivatives (oxidized, dextrinized and/or covalent amino acid linkage) as to their flavor encapsulation potential. The samples were prepared as previously described (34) with the exception of an added reduced pressure deaeration step, thus allowing the use of the headspace diffusivity versus retention standard curves to predict volatile lemon oil retention following spray drying. 

The model is linear, if all terms are linear functions of the concentrations Cn and Ci2. For the transfer mass fluxes, Tm and T,21, this implies that the fluxes are proportional to the concentration in the box from which the flux originates. These flux terms couple the two differential equations. 

Let us first develop the dynamic equation of the total concentration in the SMWL, Ctmjx (t). Not much additional effort is needed. We can just copy the structure of Eqs. 1 to 3 ofBox 23.1, although the boundary flux terms have a slightly different form. The numbers shown below the different terms in Eq. 23-42 correspond to the notation introduced in Table 23.9 and also used in Fig. 23.6. 

For the purpose of illustration, take a simple level of theory representing diffusion coefficients as mixture-averaged values (Eq. 3.89). Accordingly the flux term in Eq. 3.124 depends on the mole-fraction gradient... 

Since the dependent variable in this mass-conservation equation is logically the mass fraction Yk, it would be more convenient if the flux term were represented in terms of mass fraction, rather than mole fraction. This can be done, but at the expense of introducing... 

On expanding the enthalpy-flux term on the right-hand side as... 

The summation on the right-hand side runs over all surface and bulk species. It is interesting to note that by substituting Eq. 11.123 into the flux term on the left-hand side, the energy balance can be written in a more compact form as... 

Note that the radial-flux term now involves a factor V = v/r, which in the similarity formulation of the stagnation-flow problem is a function only of z. Thus, per unit-area, this mass balance is independent of r, so long as the similarity assumptions apply. 

Substituting for the flux terms using Equation 2.7 gives us... 

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