Mass conservation transport

The humidity and contaminant transport calculation is based on the previously calculated airflows, applying again the principle of mass conservation for the species under consideration. For each time step, the concentrations are calculated on the basis of the airflows, the source and sink strengths in the zones, and the concentration values at the previous time step. In contrast to the airflow calculation, which is a steady-state calculation at each time step, the contaminant transport calculation is dynamic. Therefore, the accuracy of the concentration results depends on the selected time-step interval. 

In PBPK models tissue blood perfusion and tissue composition can be characterized independently of the drug thus such a model can be created once and reused for many different drugs. Furthermore, because physical laws (mass conservation, diffusion, or facilitated transport mechanisms) are incor-... 

Via Eq. (136) the kinematic condition Eq. (131) is fulfilled automatically. Furthermore, a conservative discretization of the transport equation such as achieved with the FVM method guarantees local mass conservation for the two phases separately. With a description based on the volume fraction fimction, the two fluids can be regarded as a single fluid with spatially varying density and viscosity, according to... 

Biofilms adhere to surfaces, hence in nearly all systems of interest, whether a medical device or geological media, transport of mass from bulk fluid to the biofilm-fluid interface is impacted by the velocity field [24, 25]. Coupling of the velocity field to mass transport is a fundamental aspect of mass conservation [2]. The concentration of a species c(r,t) satisfies the advection diffusion equation... 

Transport of component i in a binary system is described by the equation of continuity [2], which is an expression for mass conservation of the subject component in the system, i.e.,... 

Compartmental soil modeling is a new concept and can apply to both modules. For the solute fate module, for example, it consists of the application of the law of pollutant mass conservation to a representative user specified soil element. The mass conservation principle is applied over a specific time step, either to the entire soil matrix or to the subelements of the matrix such as the soil-solids, the soil-moisture and the soil-air. These phases can be assumed in equilibrium at all times thus once the concentration in one phase is known, the concentration in the other phases can be calculated. Single or multiple soil compartments can be considered whereas phases and subcompartments can be interrelated (Figure 2) with transport, transformation and interactive equations. 

Equation (9.41) constitutes a fundamental solution for purely convective mass burning flux in a stagnant layer. Sorting through the S-Z transformation will allow us to obtain specific stagnant layer solutions for T and Yr However, the introduction of a new variable - the mixture fraction - will allow us to express these profiles in mixture fraction space where they are universal. They only require a spatial and temporal determination of the mixture fraction/. The mixture fraction is defined as the mass fraction of original fuel atoms. It is as if the fuel atoms are all painted red in their evolved state, and as they are transported and chemically recombined, we track their mass relative to the gas phase mixture mass. Since these fuel atoms cannot be destroyed, the governing equation for their mass conservation must be... 

As in all transported PDF codes, the numerical algorithm for mixing and chemical reactions is straightforward. If interactions between particles are restricted to the same cell (i.e., local estimation), local mass conservation is guaranteed. 

This is the fundamental transport equation for the species i, which does not depend on any assumption other than mass conservation. 

The gas channels contain various gas species including reactants (i.e., oxygen and hydrogen), products (i.e., water), and possibly inerts (e.g., nitrogen and carbon dioxide). Almost every model assumes that, if liquid water exists in the gas channels, then it is either as droplets suspended in the gas flow or as a water film. In either case, the liquid water has no affect on the transport of the gases. The only way it may affect the gas species is through evaporation or condensation. The mass balance of each species is obtained from a mass conservation equation, eq 23, where evaporation/condensation are the only reactions considered. 

Clearly, the actual pressure head in each phase depends on the fluid configuration within the pores. Hux equations for each of the three phases can be combined with mass conservation equations to derive governing transport equations. 

The substance being transported can be either dissolved (part of the same phase as the water) or particulate substances. We will develop the diffusion equation by considering mass conservation in a fixed control volume. The mass conservation equation can be written as... 

We will begin with domain discretization into control volumes. Consider our box used in Chapter 2 to derive the mass transport equation. Now, assume that this box does not become infinitely small and is a control volume of dimensions Ax, Ay, and Az. A similar operation on the entire domain, shown in Figure 7.1, will discretize the domain into control volumes of boxes. Each box is identified by an integer i, j, k), corresponding to the box number in the x-, y-, and z-coordinate system. Our differential domain has become a discrete domain, with each box acting as a complete mixed tank. Then, we will apply our general mass conservation equation from Chapter 2 ... 

Regardless of what other conservation equations may be appropriate, a bulk-fluid mass-conservation equation is invariably required in any fluid-flow situation. When N is the mass m, the associated intensive variable (extensive variable per unit mass) is r) = 1. That is, r) is the mass per unit mass is unity. For the circumstances considered here, there is no mass created or destroyed within a control volume. Chemical reaction, for example, may produce or consume individual species, but overall no mass is created or destroyed. Furthermore the only way that net mass can be transported across the control surfaces is by convection. While individual species may diffuse across the control surfaces by molecular actions, there can be no net transport by such processes. This fact will be developed in much depth in subsequent sections where mass transport is discussed. 

In addition to overall mass conservation, we are concerned with the conservation laws for individual chemical species. Beginning in a way analogous to the approach for the overall mass-conservation equation, we seek an equation for the rate of change of the mass of species k, mk. Here the extensive variable is N = mu and the intensive variable is the mass fraction, T = mk/m. Homogeneous chemical reaction can produce species within the system, and species can be transported into the system by molecular diffusion. There is convective transport as well, but it represented on the left-hand side through the substantial derivative. Thus, in the Eulerian framework, using the relationship between the system and the control volume yields... 

Beginning with a mass-conservation law, the Reynolds transport theorem, and a differential control volume (Fig. 4.30), derive a steady-state mass-continuity equation for the mean circumferential velocity W in the annular shroud. Remember that the pressure p 6) (and hence the density p(6) and velocity V(6)) are functions of 6 in the annulus. 

Deriving the mass-continuity equation begins with a mass-conservation principle and the Reynolds transport theorem. Unlike the channel with chemically inert walls, when surface chemistry is included the mass-conservation law for the system may have a source term,... 

To complete the set of kinetic equations we observe that ub = (A/ /Ac)b where Acb can be expressed in terms of <5 ,b. Finally, the requirement of mass conservation yields a further equation. Considering the inherent nonlinearities, this problem contains the possibility of oscillatory solutions as has been observed experimentally. Let us repeat the general conclusion. Reactions at moving boundaries are relaxation processes between regular and irregular SE s. Coupled with the transport in the untransformed and the transformed phases, the nonlinear problem may, in principle, lead to pulsating motions of the driven interfaces. 

Model Description. The transport and fate of PCBs in Twelve Mile Creek and the upper portion of Lake Hartwell are described by a series of mass conservation equations for solids and PCBs in the water column and in the active sediment layer (given as the top 10 cm of sediment) following the approach described by O Connor (20-22). For solids, the equations for the water column and active sediment layer are given as ... 

The mass conservation equation for a gas channel can be obtained by substituting 4> = 1 in the general scalar transport equation, Equation (5.25), resulting in ... 

The transport Equations (1) are supplemented by mass conservation equations. Mass conservation for each conservative ion species is expressed by,... 

As contaminant transport occurs over times much greater than the times over which groundwater flow fluctuates, steady flow is frequently assumed. For steady groundwater flow in three dimensions, the following vector equation, developed based on mass conservation principles, is typically used to model advective/ dispersive transport of a dissolved reactive contaminant (after [53]) ... 

In general, the equations governing various transport processes, like equations for chemical kinetics in well mixed systems, are built upon the foundation of mass conservation [23]. 

In these equations fi is the coluirm mass of dry air, V is the velocity (u, v, w), and (jf) is a scalar mixing ratio. These equations are discretized in a finite volume formulation, and as a result the model exactly (to machine roundoff) conserves mass and scalar mass. The discrete model transport is also consistent (the discrete scalar conservation equation collapses to the mass conservation equation when = 1) and preserves tracer correlations (c.f. Lin and Rood (1996)). The ARW model uses a spatially 5th order evaluation of the horizontal flux divergence (advection) in the scalar conservation equation and a 3rd order evaluation of the vertical flux divergence coupled with the 3rd order Runge-Kutta time integration scheme. The time integration scheme and the advection scheme is described in Wicker and Skamarock (2002). Skamarock et al. (2005) also modified the advection to allow for positive definite transport. 

It is rather straight forward to check the ability of mass conservation of the transport scheme, by initializing the model domain with a constant mixing ratio internally and on the boundaries, and during model run the mixing ratio should stay constant. 

We have illustrated the need for initialization of non-hydrostatic as well as hydrostatic driving meteorological data for off-line atmospheric chemistry and transport models. The impact on the wind field from initialization is of the same magnitudes for both non-hydrostatic and hydrostatic data (i.e. less than a few dm/s), that indicates that no specific problem concerning initialization for mass conservation of non-hydrostatic data. 

Kaas E (2008) A simple and efficient locally mass conserving semi-Lagrangian transport scheme. Tellus 60A 305-320... 

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