Reactive transport model governing equation

For reactive flows the governing equations used by Lindborg et al [92] resemble those in sect 3.4.3, but the solid phase momentum equation contains several additional terms derived from kinetic theory and a frictional stress closure for slow quasi-static flow conditions based on concepts developed in soil mechanics. Moreover, to close the kinetic theory model the granular temperature is calculated from a separate transport equation. To avoid misconception the model equations are given below (in which the averaging symbols are disregarded for convenience) ... 

Figure2 Ohnishi et al. (1985) and Chijimatsu et al. (2000)) and reactive-mass transport model (inside the box named Chemical in Figure 2). This is a system of governing equations composed of Equations (l)-(9), which couple heat flow, fluid flow, deformation, mass transport and geochemical reaction in terms of following primary variables temperature T, pressure head y/, displacement u total dissolved concentration of the n master species C< > and total dissolved and precipitated concentration of the n" master species T,. Here we set master species as the linear independent basis for geochemical reactions, and speciation in solution and dissolution/precipitation of minerals are calculated by a series of governing equations for geochemical reaction. Now we adopt equilibrium model for geochemical reaction (Parkhurst et al. (1980)), mainly because of reliability and abundance of thermodynamic data for geochemical reaction.

Below, a reactive fiow system will be discussed which can be described by the one-dimensional governing equations using the geometry of the problem. So, the resulting independent variables are the time and the distance normal to the catalytic surface. Detailed models for the chemical reactions as well as for the molecular transport are used. In order to include surface chemistry, the gas-phase problem is closely coupled with the transport to the gas-surface interface and the reaction thereon. The elementary-reactions concept is extended to heterogeneous reactions. Therefore, the boundary conditions for the governing equations at the catalyst become more complex compared to the pure gas-phase problem. 

The solution found when the rate equations are pul equal to zero corresponds to equilibrium in the case of a uniform reaction environment, but also characterizes the steady state if it is assumed that the linear lattice separates two two-dimensional spaces such that on the one side the reaction is all 0 —> 1 according to ku k2, and k3 and on the other all 1 —> 0 according to k2 k2 and k3. As the k s can include functions of the environment within them such as the concentrations of a transported substance with which the lattice reacts, this model can be used to discuss transport through membranes with reactions governed by near neighbor effects. It will be clear that the reactivity of the linear lattice must be defined in an asymmetric fashion in order to obtain transport. 

Of course, for reactive flow calculations a new model would have to be constructed based on these techniques which used instead the equations governing compressible fluids and which contained the added chemical reactions and diffusive transport effects. 

From the large number of mathematical models for the transport of transformation products with kinetic reactions that can be considered in the Rockflow system we have chosen a first-order chemical nonequilibrium model to simulate the sorption reaction. It can be described by the governing solute transport equation with rate-limited sorption and first-order decay in aqueous and sorbed phases. This model includes the processes of advection, dispersion, sorption, biological degradation or radioactive decay of the contaminant in the aqueous and/or sorbed phases. Figure 6.1 illustrates the conceptual model for sequential decay of a reactive species. 

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