One dimensional transport

Parkhurst DL, Appelo CAJ (1999) User s guide to PHREEQC (version 2) - a computer program for speciation, batchreaction, one-dimensional transport, and inverse geochemical calculations U.S. Geological Survey Water-Resources Investigations Report 99-4259, 312 pp... 

Let us consider a case of steady evaporation. We will assume a one-dimensional transport of heat in the liquid whose bulk temperature is maintained at the atmospheric temperature, 7 X. This would apply to a deep pool of liquid with no edge or container effects. The process is shown in Figure 6.9. We select a differential control volume between x and x + dx, moving with a surface velocity (—(dxo/df) i). Our coordinate system is selected with respect to the moving, regressing, evaporating liquid surface. Although the control volume moves, the liquid velocity is zero, with respect to a stationary observer, since no circulation is considered in the contained liquid. 

Parkhurst, D.L. Appelo, C.A.J. 1999. User s Guide to PHREEQC (Version 2.15.0)-A Computer Program for Speciation, Batch-Reaction, One-Dimensional Transport and Inverse Geochemical Calculations. U.S. Geological Survey, Water Resources Investigation Report 99-4259. 

LEACH - One-Dimensional Transport of Solute Through Soil System... 

Figure 5.18. The diffusion flamelet can be approximated by a one-dimensional transport equation that describes the change in the direction normal to the stoichiometric surface. The rate of change in the tangent direction is assumed to be negligible since the flamelet thickness is small compared with the Kolmogorov length scale. The flamelet approximation is valid when the reaction separates regions of unmixed fluid. Thus, the boundary conditions on each side are known, and can be uniquely expressed in terms of .

PAH chemistry is of practical as well as theoretical interest. PAHs can be regarded as well defined subunits of graphite, an important industrial material, which is so far not totally understood at the macroscopic level. In this context, it is our aim to delineate the molecular size at which the electronic properties of PAHs converge to those of graphite. Furthermore, alkyl substituted derivatives of hexabenzocoronene (HBC) form discotic mesophases and, therefore, provide opportunities for materials which allow one-dimensional transport processes along their columnar axis [83,84]. Their application for photovoltaics and Xerox processes is also of current interest. 

EXAMPLE 7.2 Effect of fall turnover on oxygen concentration in lake sediments (unsteady, one-dimensional transport with step boundary conditions and a first-order sink, solved using explicit, central differences)... 

In the case of one-dimensional transport there follows from the conservation principle (K7) ... 

In order to solve the transport problem we have to complete the set of necessary equations and, therefore, boundary conditions must be formulated. Depending on the boundary conditions we impose, quite different transport situations will arise. Let us analyze the one-dimensional transport in a binary electrolyte as an illustration. Two different boundary conditions will be introduced. 1) AX is brought between different chemical potentials relative to one of its component (open electrical circuit). 2) AX is brought between two inert electrodes to which a voltage A U is applied. Figures 4-3a and 4-3b show the experimental schemes. Let us examine them separately. 

This equation can be used to describe one-dimensional transport of radionuclides through porous media (e.g. radionuclide elution curves from laboratory columns packed with interbed solids) assuming instantaneous sorption and desorption. Van Genuchten and coworkers have demonstrated the importance of using both sorption and desorption isotherms in this equation when hysteresis is significant. Isotherm data for sorption and desorption reactions of radionuclides with interbed materials are presented in this paper which can be used to predict radionuclide transport. 

Defining the cross-section average of u as u = Jf u(t, x, z)dz and letting e 0, the triple u,v,w converges to the unique weak solution triple U, V, W of the upscaled one-dimensional transport-reaction model introduced in [3],... 

Figure 15 shows the calculated DOS for electrons in a 40-nm bismuth nanowire compared to that of bulk bismuth. The DOS in nanowires is a superposition of one-dimensional transport channels, each located at a quantized subband energy snm. We note that the DOS in nanowires has sharp peaks at the subband edges, whereas that in a bulk material is a smooth monotonic function of energy. The enhanced DOS at the subband edges of nanowires has important implications for many applications, such as in optics (Black et al, 2000) and thermoelectrics (Hicks and Dresselhaus, 1993). 

By using Fick s and Fourier s laws in one-dimensional transport in a slab catalyst pellet (Figure 9.1) with, equimolar counter-diffusion under mechanical equilibrium, Eqs. (9.14) and (9.15) become... 

If we consider Oj" and h as carriers, we refer, e.g. to chemical diffusion of oxygen in La2CuC>4. Writing down the one-dimensional transport equations (Eq. 103) for ionic and electronic carriers and considering flux coupling and electroneutrality, we immediately obtain for this case (see, e.g., Ref.173)... 

Percolation theory is helpful for analyzing disorder-induced M-NM transitions (recall the classical percolation model that was used to describe grain-boundary transport phenomena in Chapter 2). In this model, the M-NM transition corresponds to the percolation threshold. Perhaps the most important result comes from the very influential work by Abrahams (Abrahams et al., 1979), based on scaling arguments from quantum percolation theory. This is the prediction that no percolation occurs in a one-dimensional or two-dimensional system with nonzero disorder concentration at 0 K in the absence of a magnetic field. It has been confirmed in a mathematically rigorous way that all states will be localized in the case of disordered one-dimensional transport systems (i.e. chain structures). 

Again assuming one-dimensional transport, continuity for the combined reactive and inert species can be expressed as... 

One-dimensional transport through soils of calcium affected by equilibrium-controlled self-exchange is described by ... 

Geochemical modeling of reactants in flowing mountainous stream systems can be done with the USGS codes OTIS (Runkel, 1998) and OTEQ (Runkel et al., 1996, 1999) that model solute transport and reactive transport, respectively. OTIS, or one-dimensional transport with inflow and storage, is based on the earlier work of Bencala (1983) and Bencala and Walters (1983). The OTEQ code combines the OTIS code with MEMTEQA2 for chemical reaction at each... 

Parkhurst D. L. and Appelo C. A. J. (1999) User s Guide to PHREEQC (version 2 —A Computer Program for Specia-tion. Batch-reaction, One-dimensional Transport, and Inverse Geochemical Calculations. US Geol. Surv. Water-Resour. Invest. Report 99-4259, 312pp. 

Generally, the concentration profile of analytes in FFF can be obtained from the solution of the general transport equation. For the sake of simplicity, the concentration profile of the steady-state zone of the analyte along the axis of the applied field is calculated from the one-dimensional transport equation ... 

Complexation of a polymer main chain by CDs differs significantly from complex-ations of polymer side chains. Complexations of side chains occur in parallel, while complexation of a main chain is a serial process in which consecutive steps are dependent on each other. Since complexation of a main chain polymer, so-called threading, requires a one-dimensional transport of CD rings along the chain, it requires much more time than complexation of a side chain polymer. While the first segments of a polymer chain are rapidly complexed, migration along the polymer is slow and a molecular version of a traffic jam can occur. 

A complete derivation of the mass transport equation is presented in detail elsewhere, and an abbreviated derivation is outlined here for one-dimensional transport of a chemical species in porous media saturated with water (e.g.. Ref. P °l). As with the flow equation, the transport equation begins with the mass conservation principle the change in mass within a control... 

---