Tracers transport equations

Pudykiewicz, J.A. (1998) Application of adjiont tracer transport equations for evaluating sourse parameters, Atmos.Environ. 32, 3039-3050. 

Russell GL, Lerner JA (1981) A Finite-Difference Scheme for the Tracer Transport Equation. Journal of Applied Meteorology 20 1483-1498... 

Developing concentration profiles in a soil column for cyclical boundary loading functions is important for several reasons. One reason is that the solution increases the repertoire of mathematical models that are available for which someone may find a use. A second reason is that the solution can be used by those who set up experiments to estimate parameters such as the dispersion coefficient. Another reason is that analytical solutions to tracer transport equations are desired because they can be used easily lot some simple flow cases to quickly estimate what... 

T), may depend on pol3nner or ion concentration, temperature etc is the dispersion of component i in the aqueous phase and q are the source/sink terms for component i through chemical reaction and injection/ production respectively. Polymer adsorption, as described by the term in equation (2), may feed back onto the mobility term in equation (1) through permeability reduction. In addition to the polymer/tracer transport equation above, a pressure equation must be solved (5-8), in order to find the velocity fields for each of the phases present ie aqueous, oleic and micellar (if there is a surfactant present). If thermal effects are also to be included, then a heat balance equation is also required. The SCORPIO simulator (26, 27), which is used in our studies allows for all of these effects. 

In gridpoint models, transport processes such as speed and direction of wind and ocean currents, and turbulent diffusivities (see Section 4.8.1) normally have to be prescribed. Information on these physical quantities may come from observations or from other (dynamic) models, which calculate the flow patterns from basic hydrodynamic equations. Tracer transport models, in which the transport processes are prescribed in this way, are often referred to as off-line models. An on-line model, on the other hand, is one where the tracers have been incorporated directly into a d3mamic model such that the tracer concentrations and the motions are calculated simultaneously. A major advantage of an on-line model is that feedbacks of the tracer on the energy balance can be described... 

Via a passive scalar method [6] where or, denotes the volume fraction of the i-th phase, while T, represents the diffusivity coefiBcient of the tracer in the i-th phase. The transient form of the scalar transport equation was utilized to track the pulse of tracer through the computational domain. The exit age distribution was evaluated from the normalized concentration curve obtained via measurements at the reactor outlet at 1 second intervals. This was subsequently used to determine the mean residence time, tm and Peclet number, Pe [7]. 

In whichever approach, the common denominator of most operations in stirred vessels is the common notion that the rate e of dissipation of turbulent kinetic energy is a reliable measure for the effect of the turbulent-flow characteristics on the operations of interest such as carrying out chemical reactions, suspending solids, or dispersing bubbles. As this e may be conceived as a concentration of a passive tracer, i.e., in terms of W/kg rather than of m2/s3, the spatial variations in e may be calculated by means of a usual transport equation. 

In the ideal plug flow reactor, the flow traverses through the reactor hke a plug, with a uniform velocity profile and no diffusion in the longitudinal direction, as illustrated in Figure 6.2. A nonreactive tracer would travel through the reactor and leave with the same concentration versus time curve, except later. The mass transport equation is... 

Convection, diffusion, and dispersion can only describe part of the processes occurring during transport. Only the transport of species that do not react at all with the solid, liquid or gaseous phase (ideal tracers) can be described adequately by the simplified transport equation (Eq. 94). Tritium as well as chloride and bromide can be called ideal tracers in that sense. Their transport can be modeled by the general transport equation as long as no double-porosity aquifers are modeled. Almost all other species in water somehow react with other species or a solid phase. These reactions can be subdivided into the following groups, some of which have already been considered in the previous part of the book. 

Equation (12), when used in large-scale coarse resolution (non-eddy resolving) models, is incomplete in one other respect. The effect of mesoscale eddies on tracer transport includes an additional, advective-like mechanism sometimes referred to as bolus transport (see Gent and McWilliams, 1990 Gent et al., 1995). This can be included in Equation (12) as... 

The convective-diffusion equation for solute (e.g., tracer) transport in both the axial and radial direction is... 

In the Eulerian approach (Jankowski et al., 1996 Lou and Ridd, 1997 Holt and James, 1999 Ribbe and Holloway, 2001 Christiansen et al., 2002), a transport equation is solved for the sediment concentration. Recent studies (Kuhrts et al., 2004, 2006 Seifert et al., 2007) describe moving sedimentary material as a tracer variable in a three-dimensional circulation model. This approach combines a wave boundary layer (Grant and Madsen, 1979) and a friction layer (Smith and McLean, 1977) to compute the wave-induced and the current-induced contribution to the skin friction acting on the seabed. 

The concentration of A and B can then be related to the PDF of a passive scalar or tracer. Consider again the instantaneous transport equations for A and B, assuming equal diffusivity of both species, constant total molar concentration and that the molar average and mass average velocities are about equal we get ... 

The software is completely in ANSI-C. Dynamic memory allocation and pointer technique are particularly important for the utilization of automatic grid adaptation and the coupling of different FE kernels. The code consists of several FE kernels, which assemble the matrix equations for the specific partial differential equations of a physico-chemical problem, e.g. saturated groundwater flow, gas flow, multiphase flow, tracer transport and reactive transport. These FE kernels may be connected to each other via internal interfaces (so-called models) and with the common libraries to simu-... 

Before casting Equation 7.1 in dimensionless form, the inclusion of terms to describe adsorption and velocity enhancement of the transported species will be considered. These phenomena are, of course, more appropriate to polymer transport than tracer transport but the form of the equation is very similar. The velocity enhancement referred to concerns the effect of the excluded volume or inaccessible pore volume effect which the polymer shows (Chauveteau, 1982, Dawson and Lantz, 1972) and which is discussed in more detail below. However, the physical observation on polymer transport is that there appears to be a fraction of the pore space—either the very small pores (Dawson and Lantz, 1972) or regions close to the wall of the porous medium (Chauveteau, 1982)—which is inaccessible to polymer transport. This leads to an enhancement of the average velocity of the polymer through the porous medium. When there is both adsorption of transported polymer onto the rock matrix and a fraction of the pore volume is apparently inaccessible to the polymer, Equation 7.1 must be extended to ... 

This modified water mobility provides the link between the pressure equation, from which the flows are calculated (i.e. velocities, and UJ, and the polymer transport equation (Equation 8.13). It is this feedback that is the important feature of polymer flooding the presence of polymer changes the flows which in turn results in the polymer being transported into different regions of the reservoir compared with the transport of an inert tracer which has no feedback effect on the water mobility. 

Subtracting the latter relation from the previous one, the source term can be eliminated and we obtain a transport equation for a conserved scalar which behaves as a nonreacting tracer ... 

For instantaneous reactions the problem is thus reduced to the calculation of the presumed PDF of a passive scalar or tracer. A large number of alternative presumed PDFs have been listed and discussed by [2, 60, 67]. Each presumed PDF has its advantages and disadvantages, but none of them are generally applicable. The concept of the full PDF approaches is to formulate and solve additional transport equations for the PDFs determining the evolution of turbulent flows with chemical reactions. These models thus require modeling and solution of additional balance equations for the one-point joint velocity-composition PDF. The full PDF models are thus much more CPU intensive than the moment closures and hardly tractable for process engineering calculations. These theories are comprehensive and well covered by others (e.g., [2, 8, 26]), thus these closures are not examined further in this book. For Unite rate chemical reaction processes neither of the asymptotic simplifications explained above are applicable and appropriate elosures for 5c (w) are very difficult to achieve. 

The chemical species were treated as passive scalar tracers in the unsteady LBM equations. The reaction was simulated as being mass-transfer limited at low Re — 166, with diffusivities in the ratios DA DB Dc— 1 3 2. The concentration fields shown in Fig. 16 are different for each species due to the different diffusivities. The slow-diffusing species A is transported mainly by convection and regions of high or low concentration correspond to features of the flow field. A more uniform field is seen for the concentration of faster... 

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