Advection-diffusion models, chemical

The ratio vJD can then be used to calculate a chemical reaction rate for a nonconservative solute, S. To do this, the one-dimensional advection-diffusion model is modified to include a chemical reaction term, J. This new equation is called the one-dimensional advection-diffusion-reaction model and has the following form ... 

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. 

The standard method of estimating chemical migration in a cap is via a transient advection-diffusion model as described by Palermo et al. [1]. This model is applied to the chemical isolation layer of a cap, which is the cap thickness after removing components for porewater expression via consolidation of underlying sediment, consolidation of the cap, and bioturbation of the upper cap layers. Normally, an analytical solution to the mass conservation equation, assuming that the cap is semi-infinite, is employed in such an... 

Contaminants in the soil compartment are associated with the soil, water, air, and biota phases present. Transport of the contaminant, therefore, can occur within the water and air phases by advection, diffusion, or dispersion, as previously described. In addition to these processes, chemicals dissolved in soil water are transported by wicking and percolation in the unsaturated zone.26 Chemicals can be transported in soil air by a process known as barometric pumping that is caused by sporadic changes in atmospheric pressure and soil-water displacement. Relevant physical properties of the soil matrix that are useful in modeling transport of a chemical include its hydraulic conductivity and tortuosity. The dif-fusivities of the chemicals in air and water are also used for this purpose. 

Acar and coworkers (46] and Shapiro et al. [52] have presented general models based on the first of these two approaches. These models predict that the contaminant and the electrolysis products at inert electrodes will be transported and dispersed by advection, migration, and diffusion. Modelling in this manner provides only a first-order, mathematical framework to examine the flow patterns and chemistry generated in the process adsorption/desorption kinetics, acld/base chemical reactions, complex equilibria, and precipitatlon/solubility factors may heavily influence the model accuracy and outcome of any site remediation. Two approaches for mathematic modelling are the use of analytical solutions or numerical, finite element methods (FEM). Both models require adequate definitions for the boundary conditions (nature of electrolyses, flow behaviour). 

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. 

In addition to dissipation of the substance from the model system through degradation, other dissipative mechanisms can be considered. Neely and Mackay(26) and Mackay(3) have also introduced advection (loss of the chemical from the troposphere via diffusion) and sedimentation (loss of the chemical from dynamic regions of the system by movement deep into sedimentation layers). Both of these mechanisms are then assumed to act in the unit world. This approach makes it possible to investigate the behavior of atmosphere emissions where advection can be a significant process. Therefore, from a regulatory standpoint if the emission rate exceeds the advection rate and degradation processes in a system, accumulation of material could be expected. Based on such an analysis reduction of emissions would be called for. 

Chemical mass is redistributed within a groundwater flow regime as a result of three principal transport processes advection, hydrodynamic dispersion, and molecular diffusion (e.g., Bear, 1972 Freeze and Cherry, 1979). Collectively, they are referred to as mass transport. The nature of these processes and how each can be accommodated within a transport model for a multicomponent chemical system are described in the following sections. 

The construction of a mass balance model follows the general outline of this chapter. First, one defines the spatial and temporal scales to be considered and establishes the environmental compartments or control volumes. Second, the source emissions are identified and quantified. Third, the mathematical expressions for advective and diffusive transport processes are written. And last, chemical transformation processes are quantified. This model-building process is illustrated in Figure 27.4. In this example we simply equate the change in chemical inventory (total mass in the system) with the difference between chemical inputs and outputs to the system. The inputs could include numerous point and nonpoint sources or could be a single estimate of total chemical load to the system. The outputs include all of the loss mechanisms transport... 

Since this model does not account for chemical transformation or diffusion in the axial direction, the rate of change of mass of solute at any location along the capillary is driven by advection alone. If the blood velocity were zero, then the total mass density at any location z would remain constant. 

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. 

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