Advection velocity

Now we need the corresponding expression for advective transport. Note that the advective velocity along the x-axis, vx, can be interpreted as a volume flux (of water, air, or any other fluid) per unit area and time. Thus, to calculate the flux of a dissolved chemical we must multiply the fluid volume flux with the concentration of... 

Note that the advection velocity, vx, no longer appears in this expression. Therefore, this situation is called the diffusion-reaction regime. 

Figure 22.3 One-dimensional concentration profiles at steady-state calculated from the diffusion/advec-tion/reaction equation (Eq. 22-7) for different parameter values D (diffii-sivity), x (advection velocity), and kr (first-order reaction rate constant). Boundary conditions at x = 0 and x - L are C0 and CL, respectively. Pe = 7. vx ID is the Peclet Number, Da = Dk/v] is the Damkohler Number. See text for further explanations.

Then it can be concluded that an extremely small advection velocity is sufficient to produce a significant flux enhancement and to deform the TCE profile correspondingly. Note that case (2) (Pe 1) only occurs under extremely quiet conditions. 

Explain the difference between the dispersion coefficient, dis, in a river and in the atmosphere. How is dis related to the mean advection velocity and to lateral turbulent diffusivity in each case ... 

Consider the same profile as in P 22.1. In addition to diffusion, an advective velocity v acts on the profile, (a) Calculate the corresponding additional contribution to the flux and to dC /dt. (b) Determine the relation between v and the other parameters (.D, a, C0) such that the profile, given in P22.1 between x = 0 and x = °° corresponds to a steady-state. Is such a steady-state possible if v > 0 ... 

Many of the questions about the origin of the suboxic zone and the redox reaction zones would be easier to answer if we could calculate vertical fluxes. Unfortunately, neither the mechanism nor the rate of vertical transport are well understood. Estimates of the vertical advection velocity (w) and eddy diffusion coefficients (K.) are available in the literature (e.g., 5, 32, 39, 40), but they are probably not realistic, considering the importance of horizontal ventilation discussed earlier. 

In contrast to diffusion, mechanical dispersion is attributed to variations in advective velocities over a wide range of spatial scales. On the microscale, velocity... 

The PBE is a simple continuity statement written in terms of the NDE. It can be derived as a balance for particles in some fixed subregion of phase and physical space (Ramkrishna, 2000). Let us consider a finite control volume in physical space O and in phase space with boundaries defined as dO. and dO., respectively. In the PBE, the advection velocity V is assumed to be known (e.g. equal to the local fluid velocity in the continuous phase or directly derivable from this variable). The particle-number-balance equation can be written as... 

In the case of turbulent advection velocity, the transported quantity in the PBE (i.e. the NDF) fluctuates around its mean value. These fluctuations are due to the nonlinear convection term in the momentum equation of the continuous phase. In turbulent flows usually the Reynolds average is introduced (Pope, 2000). It consists of calculating ensemble-averaged quantities of interest (usually lower-order moments). Given a fluctuating property of a turbulent flow f>(t,x), its Reynolds average at a fixed point in time and space can be written as... 

However, it would then be necessary to relate Vp and Ad to the internal coordinates used to describe the particle size (see Section 5.2.1). By dividing Eq. (5.53) by the particle mass (PpVp), the particle acceleration due to buoyancy and drag is readily calculated, and by assuming that all the particles are statistically identical the following expression for the pure advection velocity is obtained for an isolated sphere (ap = 1) ... 

We shall use the term active to refer to the velocity internal coordinate because its value directly affects the spatial transport. Other internal coordinates (e.g. size, mass, etc.) are referred to as passive if they are simply advected in space by the free-transport term. However, if the advection velocity is modeled as a function of internal coordinates such as size, then such coordinates are also referred to as active. 

Since the moment-transport equation is closed, one might be tempted to try to design a high-order scheme directly for M. However, for such a scheme it would be difficult to ensure realizability, and thus a better approach is to work directly with the NDF. Moreover, when the advection velocity depends on f the moment-transport equation is not closed, thus working with the NDF will result in more general formulas. 

Consider now the case in which the known advection velocity u(t, x, depends on For this case, the moment-transport equation is not closed, but the NDF transport equation is still given by Eq. (B.18). For the x direction, the finite-volume formula for the NDF is... 

Notice that the advection velocity depends linearly on k, so the highest velocity will be associated with the highest-order moments appearing in M. Let = max(ki) be the largest... 

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