Diffusive/advective transport /reaction

This chapter has developed the basic concepts of modeling diffusive transport and coupled diffusion, advection, and reaction in physiological systems. The emphasis... 

In water and sediments, the time to chemical steady-states is controlled by the magnitude of transport mechanisms (diffusion, advection), transport distances, and reaction rates of chemical species. When advection (water flow, rate of sedimentation) is weak, diffusion controls the solute dispersal and, hence, the time to steady-state. Models of transient and stationary states include transport of conservative chemical species in two- and three-layer lakes, transport of salt between brine layers in the Dead Sea, oxygen and radium-226 in the oceanic water column, and reacting and conservative species in sediment. 

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

Other models directly couple chemical reaction with mass transport and fluid flow. The UNSATCHEM model (Suarez and Simunek, 1996) describes the chemical evolution of solutes in soils and includes kinetic expressions for a limited number of silicate phases. The model mathematically combines one- and two-dimensional chemical transport with saturated and unsaturated pore-water flow based on optimization of water retention, pressure head, and saturated conductivity. Heat transport is also considered in the model. The IDREAT and GIMRT codes (Steefel and Lasaga, 1994) and Geochemist s Workbench (Bethke, 2001) also contain coupled chemical reaction and fluid transport with input parameters including diffusion, advection, and dispersivity. These models also consider the coupled effects of chemical reaction and changes in porosity and permeability due to mass transport. 

Figure 13.5. Transport vs surface controlled dissolution. Schematic representation of concentration in solution, C, as a function of distance from the surface of the dissolving mineral. In the lower part of the figure, the change in concentration (e.g., in a batch dissolution experiment) is given as a function of time, (a) Transport controlled dissolution. The concentration immediately adjacent to the mineral reflects the solubility equilibrium. Dissolution is then limited by the rate at which dissolved dissolution products are transported (diffusion, advection) to the bulk of the solution. Faster dissolution results from increased flow velocities or increased stirring. The supply of a reactant to the surface may also control the dissolution rate, (b) Pure surface controlled dissolution results when detachment from the mineral surface via surface reactions is so slow that concentrations adjacent to the surface build up to values essentially the same as in the surrounding bulk solution. Dissolution is not affected by increased flow velocities or stirring. A situation, intermediate between (a) and (b)—a mixed transport-surface reaction controlled kinetics—may develop.

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... 

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... 

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. 

Both models simulate the complex situation of combining diffusive, advective and biotur-bational transports in pore water and in the solid phase. They contain the various reactions of early diagenesis and require a large number of reliable measurements. Otherwise, too many degrees of freedom would remain. 

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. 

Reaction rates of nonconservative chemicals in marine sediments can be estimated from porewater concentration profiles using a mathematical model similar to the onedimensional advection-diffusion model for the water column presented in Section 4.3.4. As with the water column, horizontal concentration gradients are assumed to be negligible as compared to the vertical gradients. In contrast to the water column, solute transport in the pore waters is controlled by molecular diffusion and advection, with the effects of turbulent mixing being negligible. 

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. 

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