Diffusive/advective transport /reaction equation

So far, the concept of mass conservation has been applied to large, easily measurable control volumes such as lakes. Mass conservation also can be usefully expressed in an infinitesimal control volume, mathematically considered to be a point. Conservation of mass is expressed in such a volume with the advection—dispersion-reaction equation. This equation states that the rate of change of chemical storage at any point in space, dC/dt, equals the sum of both the rates of chemical input and output by physical means and the rate of net internal production (sources minus sinks). The inputs and outputs that occur by physical means (advection and Fickian transport) are expressed in terms of the fluid velocity (V), the diffusion/dispersion coefficient (D), and the chemical concentration gradient in the fluid (dC/dx). The input or output associated with internal sources or sinks of the chemical is represented by r. In one dimension, the equation for a fixed point is... 

If there are no or very few irrigated burrows present in the sediment, lateral diffusion is not significant and the r dependence of Eq. (6.12) can be ignored. In that case, the equation becomes the more traditional onedimensional transport-reaction equation used to model pore-water solute profiles where advection is relatively unimportant (Berner, 1971 1980 Lerman, 1979). Both the cylindrical microenvironment model and the onedimensional Cartesian coordinate model will be used here to quantify the Mn distributions at NWC and DEEP. 

Transport Processes and Gauss Theorem One-Dimensional Diffusion/Advection/Reaction Equation Box 22.1 One-Dimensional Diffusion/Advection/Reaction Equation at Steady-State... 

Imboden and Schwarzenbach (1985) have illustrated how the mass-balance equation is a means of accounting for chemical and biological reactions that produce or consume a chemical within a test volume, and for transport processes dial import or export the chemical across the boundaries. Each process acting on a chemical can be characterized by an environmental first-order rate constant, expressed in units of time-1. Transport mechanisms include water renewal by nvers, horizontal and vertical turbulent diffusion, advection by lake particles, and settling of particles (Imboden and Schwarzenbach, 1985). Chemical reaction i ales and reaction half-lives for a wide variety of reactions have been summarized by I loffmann (1981), Pankow and Morgan(1981), Morgan and Stone(1985),and Santsehi (1988). 

In the case of the fast binary reaction we could eliminate the reaction term from the reaction-diffusion-advection equation. But in general this is not possible. In this chapter we consider another class of chemical and biological activity for which some explicit analysis is still feasible. We consider the case in which the local-reaction dynamics has a unique stable steady state at every point in space. If this steady state concentration was the same everywhere, then it would be a trivial spatially uniform solution of the full reaction-diffusion-advection problem. However, when the local chemical equilibrium is not uniform in space, due to an imposed external inhomogeneity, the competition between the chemical and transport dynamics may lead to a complex spatial structure of the concentration field. As we will see in this chapter, for this class of chemical or biological systems the dominant processes that determine the main characteristics of the solutions are the advection and the reaction dynamics, while diffusion does not play a major role in the large Peclet number limit considered here. Thus diffusion can be neglected in a first approximation. 

The condensed phase density p, specific heat C, thermal conductivity A c, and radiation absorption coefficient Ka are assumed to be constant. The species-A equation includes only advective transport and depletion of species-A (generation of species-B) by chemical reaction. The species-B balance equation is redundant in this binary system since the total mass equation, m = constant, has been included the mass fraction of B is 1-T. The energy equation includes advective transport, thermal diffusion, chemical reaction, and in-depth absorption of radiation. Species diffusion d Y/cbfl term) and mass/energy transport by turbulence or multi-phase advection (bubbling) which might potentially be important in a sufficiently thick liquid layer are neglected. The radiant flux term qr... 

The species-B balance equation includes advective transport, Fickian diffusion, and depletion by chemical reaction. The binary diffusion coefficient D represents downstream diffusion of reactant species-B relative to upstream diffusion of product species-C. The expression for Ys, the surface mass fraction of B (gas side), is obtained from a species balance at the surface on B which includes advective transport of pure B to the interface on the condensed phase side and both advective and diffusive transport of B away from the surface on the gas side. The downstream condition K(oo)=0 represents the assumption of complete conversion... 

Here Npe > 1 means that transport in the chemical isolation layer is dominated by advection while Wpe < 1 implies that transport is dominated by diffusion. Advection and diffusion in either the cap isolation layer or bioturbation layer are not independent because advection tends to reduce diffusion gradients and diffusion tends to reduce the advective flux. In the cap isolation layer, a reasonable approximation is to assume that the flux is well-estimated by the dominant flux (either advection or diffusion). Solutions to the steady-state transport equations considering both diffusion and advection with and without reaction are feasible, but are algebraically more complicated and deviate significantly from solutions assuming only the dominant process in the relatively narrow range of approximately 0.3 < Npe < 3. Even within this range, the dominant process correctly estimates the flux within a factor of 2. 

A number of different approaches are proposed and used in modeling flow through porous media. Some of the most popular approaches include (i) Darcy s law, (ii) Brinkman equation, and (iii) a modified Navier-Stokes equation. In the absence of the bulk fluid motion or advection transport, the reaction gas species can only transport through the GDL and CL by the diffusion mechanisms, which we will discuss in a later section. 

In this situation, transport equations similar to those discussed previously can be applied. For example, by assuming sorption to be essentially instantaneous, the advective-dispersion equation with a reaction term (Saiers and Hornberger 1996) can be considered. Alternatively, CTRW transport equations with a single ti/Ci, t) can be applied or two different time spectra (for the dispersive transport and for the distribution of transfer times between mobile and immobile—diffusion, sorption— states can be treated Berkowitz et al. 2008). 

In Section 3.2 we introduced the basic processes of advection, diffusion, and drift, by which material is transported in biophysical systems. In this chapter we focus on a specialized class of transport transport across biological membranes. Transport of a substance across a membrane may be driven by passive permeation, as described by Equation (3.60), or it may be facilitated by a carrier protein or transporter that is embedded in the membrane. Thus transport of substances across membranes mediated by transporters is termed carrier-mediated transport. The most basic way to think about carrier proteins or transporters is as enzymes that catalyze reactions that involve transport. 

The problem of reacting transport is different due to the presence of the production term/ (0, ..., 0N) [see Eq. (1)] that makes it to be nonlinear. Here we consider the simplest nontrivial case of Eq. (1) a unique scalar field 0(x, t) evolving according to the advection-reaction-diffusion equation... 

If transport occurs much faster than sorption, sorption processes may not reach equilibrium conditions. Nonequilibrium sorption may result from physical causes such as intraparticle rate-limited diffusion, chemical causes such as rate-limiting reaction kinetics, or a combination of the two. One approach used to model rate-limited sorption is bi-continuum models consisting of one region where transport is described by the advection-dispersion equation with equilibrium sorption, and another region where transport is diffusion limited with equilibrium sorption, or another region where sorption is chemically rate limited. 

A natural response to the limitations of both geochemical equilibrium models and the solute transport models (see 10.3 for a discussion) is to couple the two. Over the last two decades, a number of models that couple advective-dispersive-diffusive transport with fully speciated chemical reactions have been developed (see reviews by Engesgaard and Christensen, 1988 Grove and Stollenwerk, 1987 Mangold and Tsang, 1991). In the coupled models, the solute transport and chemical equilibrium equations are simultaneously evaluated. 

The mixing of a product into a fluid flow results from two mechanisms stirring, which regards to the advection of fluid particles, and molecular diffusion, which is characterized by a molecular diffusion coefficient Dp,. In the absence of a chemical reaction, the evolution in time of the concentration field Ca x,1) of a product A inside a fluid domain is governed by the transport equation ... 

The species continuity equation (CE) is an expression of the Lavoisier general law of conservation of mass. Equation 2.1 presents the CE in vector form and provides the proper context for the various types of chemical mass transport processes needed for chemical modeling and fate analysis. In Section 2.2.2, the mass accumulation portion of the CE is highlighted as the principal term for assessing chemical fate in the media compartments. This term includes reaction, advection, diffusion, and turbulent transport and dispersion processes. Because the magnitude and direction of this term reflect the sum total of all processes, this term uniquely defines chemical fate. In Equation 2.2, the steady-state CE minus the reaction term is commonly referred to as the advective-diffusive (AD) equation. It provides the appropriate starting point for addressing the various transport processes associated with the mobile phases in near-surface soils. 

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