Coupled Mass Transport Models

Coupled mass transport models can also include heat transport (e.g., Raffensperger and Garven, 1995) or fluid flow. Coupled reactive transport models represent the desired tools for evaluating fate and transport of contaminants. 

Mass transport models for multicomponent systems have been developed where the equilibrium interaction chemistry is solved independently of the mass transport equations which leads to a set of algebraic equations for the chemistry coupled to a set of differential equations for the mass transport. (Cederberg et al., 1985). 

In general, geochemical models can be divided according to their levels of complexity (Figure 2.3). Speciation-solubility models contain no spatial or temporal information and are sometimes called zero-dimension models. Reaction path models simulate the successive reaction steps of a system in response to the mass or energy flux. Some temporal information is included in terms of reaction progress, f, but no spatial information is contained. Coupled reactive mass transport models contain both temporal and spatial information about chemical reactions, a complexity that is desired for environmental applications, but these models are complex and expensive to use. 

In this book, we use the term coupled model to describe models in which two sets of equations that describe two types of processes are solved together. For example, multi-component, multi-species coupled reactive mass transport models (this is a mouthful,... 

As discussed in Chapter 2, we reserve the term coupled transport model to multiple component-multiple species reactive mass transport models such as phreeqc described above. In coupled models, two set of equations are solved together through some coupling schemes. In the case of coupled reactive transport models, two sets of mathe-... 

As discussed in previous sections, coupled reactive transport models generate numerous data and the predicted mass transport is complex. Here, we only give some basic information from this modeling exercise. Interested readers are referred to Zhu and Burden (2001) and Zhu et al. (2001a, 2002) for details. 

While the mechanistic treatment of chemical reactions in the coupled multi-component, multi-species mass transport has obvious advantages over the empirical isotherm-based transport models, we can also easily compile a long list of shortcomings for coupled reactive transport models. We choose a few and list them here. 

Giles, M. R., Indrelid, S. L., Beynon, G. V. Amthor, J. 2000. The origin of large-scale quartz cementation evidence from large data sets and coupled heat-fluid mass transport modelling. In Worden, R. H. Morad. S. (eds) Quartz Cementation in Sandstones. International Association of Sedimcntologists, Special Publication, 29, 21-38. 

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.

Electroactive polymers have a number of attractive features that account for this continuing interest. First they present a distributed array of catalytic sites. Thus in contrast to monolayer chemically modified electrodes, there are potentially a much greater number of reactive sites that can contribute to the catalytic current. Since these sites are distributed throughout the film, it is essential to consider the mass transport of reagents into the film and the mass transport of products out of these films when studying the overall kinetics of these processes. The coupled mass transport and kinetics in redox polymer films have been investigated in some detail, and good models exist for these processes. ... 

Prediction models Prediction models provide data on flow, subsidence, and mass transport of pollutants. They utilize data on aquifer parameters such as direction of flow, rate of flow, changes in water level, stream-aquifer interactions, and effects of wells. The mass transport models may be conservative in which reactions are considered or nonconservative Most effort to date has been in this area Predictions are deterministic rather than probabilistic Spatially these models may be two-dimensional or multilayered Chemical reactions can be considered Most flow models do not consider interactions at boundaries with steam or seawater Few models can handle both ground and surface water Difficult to model highly variable flow conditions Computer support required Few models couple flow and quality... 

Steefel et al. ([23] and references therein) noted that the approach does not account for pH, competitive ion effects or oxidation-reduction reactions. As a consequence, values may vary by orders of magnitude from one set of conditions to another. Chen [25] also highlighted these limitations by comparing numerical modeling results of contaminant transport using a multi-component coupled reactive mass transport model and a based transport model. The conclusion from this work was that values vary with location and time and this variation could not be accounted for in the model. 

The success of SECM methodologies in providing quantitative information on the kinetics of interfacial processes relies on the availability of accurate theoretical models for mass transport and coupled kinetics, to allow the analysis of experimental data. The geometry of SECM is not conducive to exact analytical solution and hence a number of semiana-lytical [40,41], and numerical [8,10,42 46], methods have been introduced for a variety of problems. 

The previous models were developed for Brownian particles, i.e. particles that are smaller than about 1 pm. Since most times particles that are industrially codeposited are larger than this, Fransaer developed a model for the codeposition of non-Brownian particles [38, 50], This model is based on a trajectory analysis of particles, including convective mass transport, geometrical interception, and migration under specific forces, coupled to a surface immobilization reaction. The codeposition process was separated in two sub-processes the reduction of metal ions and the concurrent deposition of particles. The rate of metal deposition was obtained from the diffusion... 

Since publication of the first edition, the held of reaction modeling has continued to grow and hnd increasingly broad application. In particular, the description of microbial activity, surface chemistry, and redox chemistry within reaction models has become broader and more rigorous. Reaction models are commonly coupled to numerical models of mass and heat transport, producing a classification now known as reactive transport modeling. These areas are covered in detail in this new edihon. 

At the same time, reaction modeling is now commonly coupled to the problem of mass transport in groundwater flows, producing a subfield known as reactive transport modeling. Whereas a decade ago such modeling was the domain of specialists, improvements in mathematical formulations and the development of more accessible software codes have thrust it squarely into the mainstream. 

Step-mobility-limited models can be further separated into two limits conserved and non-conserved [20]. This terminology refers to the local conservation of mass transport is said to be conserved if a surface defect generated at a step edge eventually annihilates at the same step or at one of the two adjacent steps. Thus, the motion of adjacent steps is coupled. The 1-D conserved model of Nozieres [21] predicts T a L, independent of Zo. On the other hand, in a non-conserved model the motion of adjacent steps is uncorrelated surface defects generated at a step edge can annihilate at any step edge on the surface. Uwaha [22] has considered this case and found x a L L/zay. In the discussion below, we will use these two limiting cases of step-mobility-limited models [21, 221 to extract the step-mobilities on Si(OOl) and Ge(OOl) surfaces from experiments on relaxation kinetics. 

The simplest approach used for autocatalyst modelling is the so-called look up table (Laing et al., 1999). Essentially, the model is populated with a database of conversions for various species as a function of temperature and space velocity, from which conversions can be predicted by interpolation. This, coupled with a simple thermal model for catalyst temperature and some way of allowing for mass transport control, constitutes the simplest type of model. Once this sort of model has been written, adapting to another formulation is a relatively quick process of measuring new conversion curves and adding these to the model. 

EHD impedances have been measured on the diffusion plateau at 0.7 V/SCE. The mass transport time constant of the redox couple in solution, which is one of the terms implied in the impedance expression is independent of the interface nature. The Schmidt number Sc was first determined on a bare electrode, and this value of 1540 is further used as a fixed parameter in the analysis of the diagrams obtained on pECBZ films at different Q (Fig. 6-14). The different diagrams are analyzed in the light of the theoretical model predicted by expression (6-34). 

S //Asa mediator between CFD calculations and macro-scale process simulations, the reactor geometry is represented by a relatively small number of cells which are assumed to be ideally mixed. The basic equations for mass, impulse and energy balance are calculated for these cells. Mass transport between the cells is considered in a network-of-cells model by coupling equations which account for convection and dispersion. The software is capable of optimizing a process in iterative simulation cycles in a short time on a standard PC, but it also requires experimentally-based data to calibrate the software modules to a specific micro reactor. 

Figure 12(b) shows the local current distribution of first and second order reactions and applied over potentials ° for the coupled anode model without the mass transfer parameter y. The figure also shows the effect of a change in the electrode kinetics, in terms of an increase in the reaction order (with respect to reactant concentration) to 2.0, on the current distribution. Essentially a similar variation in current density distribution is produced, to that of a first order reaction, although the influence of mass transport limitations is more severe in terms of reducing the local current densities. 

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