Diffusivity numerical

BOUNDARY VALUE PROBLEMS OF HEAT CONDUCTION, M. Necati Ozisik. Systematic, comprehensive treatment of modern mathematical methfxls of solving problems in heat conduction and diffusion. Numerous examples and problems. Selected references. Appendices. 505pp. 5X x 8K. 65990-9 Pa. 11.95... 

Once the possibility of non-physical values of volume fraction is eliminated, solving phase continuity equations does not exhibit any other peculiarities, and the methods discussed in the previous chapter can be applied. One more point that must be mentioned while discussing the solution of phase continuity equations is of numerical or false diffusion. Numerical diffusion or false diffusion is not specific to multiphase flows and is related to any fixed-grid numerical solution procedure. However, it becomes very important in simulating multiphase flows. For example, suppose that in a field of uniform velocity, a front exists across which phase volume fraction exhibits a discontinuity. In the absence of diffusion, such a front will move within the... 

Dispersion theory is concerned with the dispersal of a solute in a flowing liquid owing to the combined action of a nonuniform velocity profile and molecular diffusion. Numerous authors have discussed flow... 

Ironic charge injection is facilitated (thus the efficiency increases), while in the bulk of the sample the electronic charge motion is diffusive. Numerical simulations for devices with equal electron and hole mobilities reveal a recombination profile with a maximum at the middle of the sample (contrary to the case of ordinary OLEDs see Fig. 13-12), and a width that depends on the sample thickness. In Figure 13-14, recombination is shown to take place in a thin zone in the middle of the 100 pm thick sample. The electrodynamic model is very successful in the description of the steady-state and transient electrical characteristics of the cells, however, the prediction of a uniformly low electric field in the bulk of the sample is at odds with a previous experimental observation [158]. 

Various drugs have been immobilized in radiation polymerized HEMA in order to produce drug delivery devices, such as ergota-mine,[ll3] salicylic acid,[114,115] and hormones.[116] The ability of diffusing numerous drugs into polymers may be used in different types of biotechnologies such as membrane separation and drug... 

S-3.3.5 Numerical Diffusion. Numerical diffusion is a source of error that is always present in finite volume CFD, owing to the fact that approximations are made during the process of discretization of the equations. It is so named because it presents itself as equivalent to an increase in the diffusion coefficient. Thus, in the solution of the momentum equation, the fluid will appear more viscous in the solution of the energy equation, the solution will appear to have a higher conductivity in the solution of the species equation, it will appear that the species diffusion coefficient is larger than in actual fact. These errors are most noticeable when diffusion is small in the actual problem definition. 

Ambrosini, W., Ferreri, J.C., 2003. Prediction of stabiUty of one-dimensional natural circulation with a low diffusion numerical scheme. Annals of Nuclear Energy 30, 1505—1537. 

The cleaning process proceeds by one of three primary mechanisms solubilization, emulsification, and roll-up [229]. In solubilization the oily phase partitions into surfactant micelles that desorb from the solid surface and diffuse into the bulk. As mentioned above, there is a body of theoretical work on solubilization [146, 147] and numerous experimental studies by a variety of spectroscopic techniques [143-145,230]. Emulsification involves the formation and removal of an emulsion at the oil-water interface the removal step may involve hydrodynamic as well as surface chemical forces. Emulsion formation is covered in Chapter XIV. In roll-up the surfactant reduces the contact angle of the liquid soil or the surface free energy of a solid particle aiding its detachment and subsequent removal by hydrodynamic forces. Adam and Stevenson s beautiful photographs illustrate roll-up of lanoline on wood fibers [231]. In order to achieve roll-up, one requires the surface free energies for soil detachment illustrated in Fig. XIII-14 to obey... 

Straub J E and Berne B J 1986 Energy diffusion in many dimensional Markovian systems the consequences of the competition between inter- and intra-molecular vibrational energy transfer J. Chem. Phys. 85 2999 Straub J E, Borkovec M and Berne B J 1987 Numerical simulation of rate constants for a two degree of freedom system in the weak collision limit J. Chem. Phys. 86 4296... 

Frori tier Orbital theory supplies an additional asstim piion to ih is calculation. It considers on ly the interactions between the h ighest occupied molecular orbital (HOMO) and the lowest unoccupied rn olecular orbital (I.UMO). These orbitals h ave th e sin a 1 lest energy separation, lead in g to a sin all den oin in a tor in th e Klopinan -.Salem ct uation, fhe Hronticr orbitals are generally diffuse, so the numerator in the equation has large terms. 

The finite element results obtained for various values of (3 are compared with the analytical solution in Figure 2.27. As can be seen using a value of /3 = 0.5 a stable numerical solution is obtained. However, this solution is over-damped and inaccurate. Therefore the main problem is to find a value of upwinding parameter that eliminates oscillations without generating over-damped results. To illustrate this concept let us consider the following convection-diffusion equation... 

Petera, J., Nassehi, V. and Pittman, J.F.T., 1989. Petrov-Galerkiii methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem. Ini.. J. Numer. Meth. Heat Fluid Flow 3, 205-222,... 

Morton, K. W., 1996. Numerical Solution of Convection Diffusion Problems, Chapman Hall, London. 

Hannart, B. and Hoplinger, E.J., 1998. Laminar flow in a rectangular diffuser near Hele-Sliaw conditions - a two dinien.sioiial numerical simulation. In Bush, A. W., Lewis, B. A. and Warren, M.D. (eds), Flow Modelling in Industrial Processes, cli. 9, Ellis Horwood, Chichester, pp. 110-118. 

The diffusion and Greens function Monte Carlo methods use numerical wave functions. In this case, care must be taken to ensure that the wave function has the nodal properties of an antisymmetric function. Often, nodal sur-... 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

Cyclohexane, produced from the partial hydrogenation of benzene [71-43-2] also can be used as the feedstock for A manufacture. Such a process involves selective hydrogenation of benzene to cyclohexene, separation of the cyclohexene from unreacted benzene and cyclohexane (produced from over-hydrogenation of the benzene), and hydration of the cyclohexane to A. Asahi has obtained numerous patents on such a process and is in the process of commercialization (85,86). Indicated reaction conditions for the partial hydrogenation are 100—200°C and 1—10 kPa (0.1—1.5 psi) with a Ru or zinc-promoted Ru catalyst (87—90). The hydration reaction uses zeotites as catalyst in a two-phase system. Cyclohexene diffuses into an aqueous phase containing the zeotites and there is hydrated to A. The A then is extracted back into the organic phase. Reaction temperature is 90—150°C and reactor residence time is 30 min (91—94). 

Validation and Application. VaUdated CFD examples are emerging (30) as are examples of limitations and misappHcations (31). ReaUsm depends on the adequacy of the physical and chemical representations, the scale of resolution for the appHcation, numerical accuracy of the solution algorithms, and skills appHed in execution. Data are available on performance characteristics of industrial furnaces and gas turbines systems operating with turbulent diffusion flames have been studied for simple two-dimensional geometries and selected conditions (32). Turbulent diffusion flames are produced when fuel and air are injected separately into the reactor. Second-order and infinitely fast reactions coupled with mixing have been analyzed with the k—Z model to describe the macromixing process. 

Reviews of concentration polarization have been reported (14,38,39). Because solute wall concentration may not be experimentally measurable, models relating solute and solvent fluxes to hydrodynamic parameters are needed for system design. The Navier-Stokes diffusion—convection equation has been numerically solved to calculate wall concentration, and thus the water flux and permeate quaUty (40). 

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