Local volume fraction

For example, if (p is the solids volume fraction entering the pipe at velocity V, and solid velocity is Vs, a component balance gives... 

Let us consider a dynamically symmetric binary mixture described by the scalar order parameter field < )(r) that gives the local volume fraction of component A at point r. The order parameter < )(r) should satisfy the local conservation law, which can be written as a continuity equation [143] ... 

An alternative way to find out the expression for C is to assume that the same form of the structure factor, Eq. (58), will be valid also locally for the inhomogeneous system, if the average volume fraction <[)0 in Eq. (58) is replaced by its local values < )(r). Guided by this assumption, one can allow the coefficient at the gradient term to be dependent on the local volume fractions and write down C as... 

However, in some special cases, the lost of information due to the thresholding procedure may cause a noticeable systematic error, because each lattice point such that < )(r) > < )0 contributes the same volume fraction 1/L3 regardless of the field magnitude. Consider an asymmetric binary mixture undergoing the phase separation. The local volume fraction distribution P(< >) has maxima at the equilibrium volume fractions, 1, and is asymmetric relatively to... 

Hartmann et al. (2006) reported very detailed simulation results (see also Hartmann, 2005) (Fig. 9). Their LB simulation was restricted to a lab-scale vessel 10 L in size only for which 2403 lattice cells a bit smaller than 1 mm2 were used the temporal resolution was 25 ps only. A set of 7 million mono-disperse spherical particles 0.3 mm in size was released in the upper 10% part of the vessel. At the moment of release, the local volume fraction amounted to 10%. The particle properties were those of calcium chloride. The simulation was carried out on 30 parallel processors of an SGI Altrix 3700 system and required for 6 weeks for 100 impeller revolutions. 

In the derivation of the mean-field partition function, it is necessary to know the probability for inserting a chain molecule in a given conformation into the system. The classical way to compute this quantity is by approximating it by a product of the local volume fractions of an unoccupied site (averaged over lattice layers). It was realised that, besides the density information, information on the bond distributions is also available. The bond distribution gives information on the average local order. Using this information, it becomes possible to more accurately access the vacancy probability. 

Here Ay is in units of kBT,g0 is the number density of pure components (assumed equal), (pc(z) is the local volume fraction of block copolymer at the interface and ipc is the overall volume fraction of block copolymer. For Ncy h 1 this can be approximated as (Noolandi and Hong 1984)... 

Figure 7. Local volume fractions of the monomers for two configurations of chains which end up at Z and z-i, respectively (a) simple addition of the volume fractions of the brushes (b) configuration that minimizes the Flory— Huggins mixing free energy for the conditions mentioned in the text.

Scheffold et al. estimated also the concentration variation of - df(bare) ((t>)/d(]> as function of the (local) volume fraction (]> at the surfaces of their blends. 

We can now go back to the evaluation of the adhesion energy between the grafted surface and the elastomer. In all what follows we assume that the elastomer was cross-linked in the dry state. As long as ografted chains, , is much smaller than unity. More precisely, Nc/N for o Nc/N312 (see [105]) and we expect Eq.(8) to hold. For o>Nc /Nil2, Eq. (8) should be replaced by [ 107]... 

X (z) t b c cD mean area per chain comprising brush layer 4.1 local volume fraction of blend component A at depth z 2.1 local composition with plateau in ( >(z) profile 3.2.1 critical volume fraction 2.1 critical volume fraction of very thin film between symmetric walls 3.2.2 ... 

Fluctuation in the order parameter is determined by the free energy change due to variation of the order parameter from its thermal average. For incompressible diblock block copolymer melts, the order parameter is defined as the local volume fraction of A-monomers relative to its global value, (r) = - /a- Because of the periodic nature of the ordered phases, it... 

The relationship between the maximum amount of the relative deviation dmax and the ratio x — Ha/mb of the normalities of both reaction partners before they are brought together A - additive B - receiver) can be established as follows. Upon color change both reactands are present in stoichiometric equivalence (e). For the local volume fraction the following then applies... 

It is noted that both the time fraction and the average occurrence or number fraction 7 are often treated as equivalent to the local volume fraction Ofc of phase k. However, this assumption is strictly correct for stationary and homogeneous flows only. 

Problem 2-15. Derivation of Transport Equation for a Sedimenting Suspension. There are many parallels among momentum, mass, and energy transport because all three are derived from similar conservation laws. In this problem we derive a microscopic balance describing the concentration distribution (x, t) of a very dilute suspension of small particles suspended in an incompressible fluid undergoing unsteady flow. [Note cj>(. t) is the local volume fraction of particles in the fluid (i.e. volume of particles/volume of fluid) and hence is dimensionless. ... 

A completely fresh and different approach to polymer solutions is being studied in at least two places under the auspices of the Paint Research Institute. Some years ago, Prausnitz (25) reported on polymer segment-interaction theory to predict phase separation. This was based on a lead by Wilson (24) who proposed a local volume fraction concept or domain theory for explaining deviations from ideality. More recently, Prausnitz (27) reported on further work involving anomalous viscosities in high solid polymer solutions. High viscosities may occur when solvent mixtures are used which have values near the extreme ends of the solu-... 

The local mole fraction is converted to the local volume fraction as follows ... 

In general, D can be a function of local concentration as discussed in the context of Fig. 20.3-6. As noted by Stem, a convenient form of Fick s Law can be written as shown in Eq. (20-4.1) in terms of the local volume fraction of the penetrant in the polymar, u, and the mutual diffusion confllcienl D.I J... 

For the moment, let us assume that the local volume fraction is the only slow variable of interest in the system, < >(f, t). Since the overall composition of the blend is conserved, we have a continuity equation... 

If the fluctuation in the local volume fraction of component 2 in a binary mixture is denoted by dcp the fluctuation in entropy density is... 

The quantities /1,2 and A, signify interaction energies between molecules. The quantity x,2 has the meaning of a local composition. Furthermore, a local volume fraction of the component 1 can be formulated as ... 

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