Indicator function volume

Note Rp-values (p-carotene = 100) are used for systems 1 to 3 and Rp values for system 4. Solvent compositions by volume, p.e. = petroleum ether (40 to 60°C) DB indicates number of in chain conjugated double bonds FG indicates functional groups E = epoxy, H = hydroxyl, K = ketone. 

Note that packing contributions are most obvious in the outer shell contribution to Eq. (33) because the indicator function 0 (1 — bj) constrains that term to the case where no occupancy of the defined inner shell is permitted. These excluded volume interactions are the essense of packing contributions. To study those contributions, we consider a fictitious solute that does not interact with the solvent at all e AU/kT = 1. In that case, of course, p7x is zero and we write... 

Euler-Euler models assume interpenetrating continua to derive averaged continuum equations for both phases. The probability that a phase exists at a certain position at a certain time is given by a phase indicator function, which, for steady-state processes, is equivalent to the volume of fraction of the correspondent phase (volume-of-fluid technique). The phase-averaging process introduces further unknowns into the basic conservation equations their description requires empirical and problem-dependent input (94). In principal, Euler-Euler models are applicable to all multiphase flows. Advantages and disadvantages of both methods are compared, e.g., in Refs. 95 and 96. 

The new introduced parameter CLir = VA/5 is called clearance, and it has dimensions of flow, volume Xtime-1. The clearance has a bidirectional use and indicates the volume of the solution that is cleared from drug per unit of time because of the drug movement across the plane. For an isotropic membrane, structural and functional characteristics are identical at both sides of the membrane, CLir = CLri. In practice, the term clearance is rarely used except for the irreversible removal of a material from a compartment by unidirectional pathways of metabolism, storage, or excretion. The other new parameter P = T>/8 characterizes the diffusing ability of a given solute for a given membrane, and it is called permeability. Permeability has dimensions of length xtime-1. 

Fig. 9.4. Specific volume of Snase as a function of pressure at 40°C [15], The protein is folded up to 50 MPa, and the slope up to that pressure is indicative of the isothermal compressibility of the folded state. The arrow at 100 MPa indicates the volume change of unfolding assuming constant compressibility of the folded state and nearly complete unfolding by 100 MPa. Unfortunately, the high-pressure densitometer was limited to 100 MPa, so the compressibility of the unfolded state could not be determined...

Figure 4.5 Measurements of effective Flory-Huggins x parameter for deuterated polystyrene-poly(vinylmethylether) blends (Han et al, 1988) as a function of l/T. The temperature range is typically 100-150 °C, and the numbers near each set of data indicate the volume fraction of polystyrene. The observed linear dependence on 1/r provides the physical conclusion that first-order perturbation theory is a satisfactory treatment of attractive interactions here.

Fig. 3. Viscosity of water-ethylene glycol-methanol mixtures as a function of temperature. Numbers indicate the volume ratio for each mixture. From Travers et al. (1975). Reprinted with permission of Biochimie. Copyright by die Societe de Chimie Biologique.

Hirt and Nichols [12] demonstrated the volume of fluid (VOF) method and started a new trend in multiphase flow simulation. It relies on the definition of an indicator function y. This function allows us to know whether one fluid or another occupies the cell, or a mix of both. In the conventional volume of fluid method [12], the transport equation for an indicator function y, representing the volume fraction of one phase, is solved simultaneously with the continuity and momentum equations as follows ... 

Fig. 2.1 Coefficients of mutual diffusion of tracers in various carrier gases at 1 bar and 500 K as a function of the molar volume of the condensed tracer see text for explanations. Arrows indicate molar volumes of the particular compounds.

In the case of two immiscible fluids, a characteristic phase indicator function, Xj, may be defined that is equal to 1 In one of the phases and 0 in the other phase. Then Xi and nj are related by nidi = VXi analogue to the relations used In standard volume averaging procedures [54] [164]. An averaged representation of this relation may be given as f mSida = —Vai... 

FIG. 12 Adsorption of a poly electrolyte (zp = — 1) on an uncharged surface as a function of time for various 1-1 electrolyte concentrations (indicated as volume fractions Balt). In (a) the adsorption is given for short times, in (b) for long times (on a logarithmic scale). Endpoints in (b) are for equilibrium adsorption and a polymer concentration of 300 mg L 4 (poiymer = 1CT4). (Calculated data from Ref. 14.)... 

In the general model the state variables are chosen to be the molar volume fractions intrinsic densities a. and strains 6k f-j of the components, and the common temperature T [Kj. The indicator function taking care of the restriction for the molar volume fractions is... 

Pioneering work of Tool [1946, 1948] on inorganic glasses using dilatometry indicated that volume relaxation after a temperature jump from an initial equilibrium state could not be described simply by a kinetic model in which the relaxation time T was solely dependent on the temperature. Tool therefore proposed that r was also a function of the structure of the glass, and this led to the definition of the Active temperature Tf. 

In the volume-of-fluid (VOF) method [6, 9, 10], a discretized analog of the phase-indicator function X is used to track or capture the location of the interface. This analog is the volume fraction field, defined as the fraction of each computational cell that is occupied by fluid 1, say, at any given instant. For instance, in a three-dimensional grid with cell having volume Vijk, the volume fraction Cijk is defined by... 

Numerical interface capturing methods consist of various techniques for integrating the above system of conservation of mass and momentum equations, together with advection of an appropriate level set or phase-indicator function, to enable an approximate localization of the interface and proper assignation of fluid properties. We now describe two widely used methods to accomplish this volume-of-fluid and level set methods. 

Figure 8. Left The position of the first (FDP, squares) and second diffraction peak (SDP, circles) in 5 (g) for a—Si as a function of pressure during compression. Experimental and simulation results are shown as filled versus open symbols respectively. Right V(P) relations for a—Si polyamorphs obtained from MD simulations compared to DFT calculations. The curve marked LDA is that obtained using the modified SW potential this shows a clear pressure-driven transition at 11 GPa. The curve marked SW was obtained using the unmodified SW potential, that preferentially returns a hlgh-density form for a—Si at ambient conditions. The vertical lines indicated by A and o represent the volume changes observed by Durandurdu et al. [119] and Morishita [121], respectively. The solid line labeled Si-I shows the V(P) behavior of the ideal simulated diamond crystal. The lines marked Si-II and Si -V indicate the volumes of these two polymorphs at their experimental transition pressures.

The evolution of the interface and advection of the phase-indicator function is accomplished by reconstructing the interface within each computational cell and computing the volume flux that occurs from each cell to its immediate neighbors under the prevaihng flow. The surface reconstruction problem [11] is one of finding an interface with the correct unit normal vector which divides the computational cell into two regions, each occupied by the respective fluid phase. One popular way to accomplish this is using the Piecewise Linear Interface Calculation or Con-... 

If the relation between the density of the reaction mixture and the reactant concentration is known, the latter can be calculated as a function of time. However, if we describe the progress of the reaction in terms of the degree of conversion, see eq. (3.8), we d the same solution as eq. (3.15), which indicates that volume changes during reaction do not influence the rate of conversion of first order reactions. 

In the VOF method, the volume fraction of fluid A, Ca, is defined as the integral of indicator function of fluid A in each mesh grid (control volume). Obviously,... 

In this section the basis elements of the volume of fluid (VOF) method are described. In general the VOF model is composed of a set of continuity and momentum equations, as well as a transport equation for the evolution of a phase indicator function which is used to determine the location and orientation of the interface. We distinguish between the jump condition—and the whole field formulations of the method, in which both forms are based on a macroscopic view defining the interface as a 2D surface. The jump condition form is especially convenient for free-surface flow simulations, whereas the whole field formulation is commonly used for interfacial flow calculations in which the internal flow of all the phases are of interest. 

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