Dimensionless interface

Molecular dynamics and density functional theory studies (see Section IX-2) of the Lennard-Jones 6-12 system determine the interfacial tension for the solid-liquid and solid-vapor interfaces [47-49]. The dimensionless interfacial tension ya /kT, where a is the Lennard-Jones molecular size, increases from about 0.83 for the solid-liquid interface to 2.38 for the solid-vapor at the triple point [49], reflecting the large energy associated with a solid-vapor interface. 

Here r is the distance between the centers of two atoms in dimensionless units r = R/a, where R is the actual distance and a defines the effective range of the potential. Uq sets the energy scale of the pair-interaction. A number of crystal growth processes have been investigated by this type of potential, for example [28-31]. An alternative way of calculating solid-liquid interface structures on an atomic level is via classical density-functional methods [32,33]. 

Chapter 9 is devoted to regimes of capillary flow with a distinct interface. The effect of certain dimensionless parameters on the velocity, temperature and pressure within the liquid and vapor domains are considered. The parameters corresponding to the steady flow regimes, as well as the domains of flow instability are defined. 

Two-phase flows in micro-channels with an evaporating meniscus, which separates the liquid and vapor regions, have been considered by Khrustalev and Faghri (1996) and Peles et al. (1998, 2000). In the latter a quasi-one-dimensional model was used to analyze the thermohydrodynamic characteristics of the flow in a heated capillary, with a distinct interface. This model takes into account the multi-stage character of the process, as well as the effect of capillary, friction and gravity forces on the flow development. The theoretical and experimental studies of the steady forced flow in a micro-channel with evaporating meniscus were carried out by Peles et al. (2001). These studies revealed the effect of a number of dimensionless parameters such as the Peclet and Jacob numbers, dimensionless heat transfer flux, etc., on the velocity, temperature and pressure distributions in the liquid and vapor regions. The structure of flow in heated micro-channels is determined by a number of factors the physical properties of fluid, its velocity, heat flux on... 

The measured growth rates are illustrated by the circles in Fig. 7. The interface velocity is plotted versus the interface temperature T. The value of T is always greater than Tq because of the release of the latent heat at the interface. Dimensionless units for T and the velocity are used here. The maximum velocity corresponds to 80m /s for argon. The most surprising aspect is the rapid crystallization at low temperatures. Most materials exhibit sharply reduced rates at low temperatures, as expected for an activated growth process. That is, the kinetics can be represented as the product of an Arrhenius factor F(T) and a term that accounts for the net production of crystalline material as a result of the atoms ordering and disordering at the interface,... 

Figure 7. Families of cellular interfaces computed for System I with k = 0.865 as a function of increasing P in a A /2 sample size. The cells are represented by the dimensionless arc length. The letters refer to sample interface shapes shown in Figure 8.

The reaction (Eqn. 5.4-65) takes place in the liquid phase. The molecules are transferred away from the interface to the bulk of the liquid, while reaction takes place simultaneously. Two limiting cases can be envisaged (1) reaction is very fast compared to mass transfer, which means that reaction only takes place in the film, and (2) reaction is very slow compared to mass transfer, and reaction only takes place in the liquid bulk. A convenient dimensionless group, the Hatta number, has been defined, which characterizes the situation compared to the limiting cases. For a reaction that is first order in the gaseous reactant and zero order in the liquid reactant (cm = 1, as = 0), Hatta is ... 

Recently [7] we constructed an example showing that interfacial flexibility can cause instability of the uniform state. Two elastic capacitors, C and C2, were connected in parallel. The total charge was fixed, but it was allowed to redistribute between C and C2. It was shown that if the interface was absolutely soft , i.e., contraction of the two gaps was not coupled, the uniform distribution became unstable at precisely the point where the dimensionless charge density s reached the critical value, = (2/3). In other words, the uniform distribution became unstable at the point where, under a control,... 

Routh and Russel [10] proposed a dimensionless Peclet number to gauge the balance between the two dominant processes controlling the uniformity of drying of a colloidal dispersion layer evaporation of solvent from the air interface, which serves to concentrate particles at the surface, and particle diffusion which serves to equilibrate the concentration across the depth of the layer. The Peclet number, Pe is defined for a film of initial thickness H with an evaporation rate E (units of velocity) as HE/D0, where D0 = kBT/6jT ir- the Stokes-Einstein diffusion coefficient for the particles in the colloid. Here, r is the particle radius, p is the viscosity of the continuous phase, T is the absolute temperature and kB is the Boltzmann constant. When Pe 1, evaporation dominates and particles concentrate near the surface and a skin forms, Figure 2.3.5, lower left. Conversely, when Pe l, diffusion dominates and a more uniform distribution of particles is expected, Figure 2.3.5, upper left. 

The final dimensionless group to be evaluated is the interfacial heat-transfer number, and therefore the interfacial heat-transfer coefficient and the interfacial area must be determined. The interface is easily described for this regime, and, with a knowledge of the holdup and the tube geometry, the interfacial area can be calculated. The interfacial heat trasfer coefficient is not readily evaluated, since experimental values for U are not available. A conservative estimate for U is found by treating the interface as a stationary wall and calculating U from the relationship... 

On the submicron scale, the current distribution is determined by the diffusive transport of metal ion and additives under the influence of local conditions at the interface. Transport of additives in solution may be non-locally controlled if they are consumed at a mass-transfer limited rate at the deposit surface. The diffusion of additives in solution must then be solved simultaneously with the flux of reactive ion. Diffusive transport of inhibitors forms the basis for leveling [144-147] where a diffusion-limited inhibitor reduces the current density on protrusions. West has treated the theory of filling based on leveling alone [148], In his model, the controlling dimensionless groups are equivalent to and D divided by the trench aspect ratio. They determine the ranges of concentration within which filling can be achieved. 

External humidification, 12 213 External interface management, in technology transfer, 24 366 External loop airlift bioreactors, 1 741, 742 Externally manifolded fuel cells, 12 200 External magnetic field, 23 835 External mass transfer, 15 728-729 External mass transfer resistance dimensionless parameter and,... 

This number is conceptually an energy ratio, but independent of the interface heat extraction rate and thus the contact area. Since the interface heat transfer is assumed to control the solidification process of an impacting droplet, the choice of a dimensionless number should involve an evaluation of the influence exerted by this key factor. Therefore, the use of this newly defined dimensionless number is limited to an initial decision on which of the Impact number and the Freezing number is most appropriate for the application to a given material system at a know impact velocity. 

The concentration profiles for >,-=0.1 and Dt=5 have been depicted in Figure 9.16 as a function of the dimensionless distance vx/. Accumulation of incompatible elements and depletion of compatible elements in the vicinity of the interface are the remarkable features of this model. Concentrations at the interface are given by... 

In the gas/vapour phase the dimensionless distance tj ranges from 0 to 1, where tj — 1 corresponds to the position of the interface. In the liquid phase this parameter ranges from 0 to 1 for the mass transfer film and from 0 to Le for the heat transfer film. Hence, rj = 0 corresponds to the position of the interface and rj = I and t] = Le correspond, respectively, to the boundaries of the mass and heat transfer film. The mass and energy fluxes can now be calculated by solving the differential equations (4) and (8)-(12) subject to the boundary conditions (15). Due to the non-linearities a numerical solution procedure has been used which will be discussed subsequently. 

Figure 11. Tafel plot of flooded porous-electrode simulation results for the cathode at three different values of xp = 2.2nFIfQ 2 02, z=dbK. The z coordinate ranges from 0 (catalyst layer/membrane interface) to L (catalyst layer/diffusion medium interface), the dimensionless overpotential is defined as // = —o FIRT r]oRR, - ), and the ORR rate constant is defined as A = hFFq 2 (Reproduced with permission from ref 36. Copyright 1998 The Electrochemical Society, Inc.)...

Mass spectrometry is a sensitive analytical technique which is able to quantify known analytes and to identify unknown molecules at the picomoles or femto-moles level. A fundamental requirement is that atoms or molecules are ionized and analyzed as gas phase ions which are characterized by their mass (m) and charge (z). A mass spectrometer is an instrument which measures precisely the abundance of molecules which have been converted to ions. In a mass spectrum m/z is used as the dimensionless quantity that is an independent variable. There is still some ambiguity how the x-axis of the mass spectrum should be defined. Mass to charge ratio should not lo longer be used because the quantity measured is not the quotient of the ion s mass to its electric charge. Also, the use of the Thomson unit (Th) is considered obsolete [15, 16]. Typically, a mass spectrometer is formed by the following components (i) a sample introduction device (direct probe inlet, liquid interface), (ii) a source to produce ions, (iii) one or several mass analyzers, (iv) a detector to measure the abundance of ions, (v) a computerized system for data treatment (Fig. 1.1). 

For most of the results, the values of yt+ (dimensionless distance parameter), at the gas/liquid interface correspond to the buffer region... 

For clarity, use w to denote mass fraction (dimensionless, i.e., the concentration unit is not kg/m or mol/m ) in the melt Wqtz denotes mass fraction in quartz. The mass flux toward the interface (in the interface-fixed reference frame) is... 

These equations have been solved for rigid (Nl) and circulating spheres (Jl, K6, W3, W4) in creeping flow. Since the dimensionless velocities within the particle are proportional to (1 + k) (see Eq. (3-8)), F is a function only of Tp and PCp/(l + k). In presenting the results, it is instructive to consider the instantaneous overall Sherwood number, Shp, as well as F. The driving force is taken as the difference between the concentration inside the interface, and the mixed mean particle concentration, Cp, giving... 

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