Dimensionless particle volume fraction

Hereafter, instead of the number concentrahon c, the dimensionless particle volume fraction < ) cV is used. Note that by definition... 

Figure 14 Dimensionless real modulus G7 against the dimensionless particle volume fraction c- Reproduced with permission from T. Hao, Y. Chen, Z. Xu, Y. Xu and Y. Huang, Chin. J. Polym. Sci., 12(1994)97...

Fig. 3. The variation of the dimensionless equilibrium doublet concentration with the radius of particles of unit density, in air at 1 atm and 298 K, for a Hamakcr constant of 10 12 erg and for different initial particle volume fractions.

Here [>7] is the intrinsic viscosity, which for suspensions is the dilute limit of the viscosity increment per unit particle volume fraction, divided by the solvent viscosity. Thus, it is a dimensionless quantity defined as... 

FIGURE 4.64 Qualitative presentation of basic relations in rheology of suspensions (a) rate of strain, Y, vs. applied stress, x (see Equation 4.326) (b) average viscosity of a suspension, q, vs. rate of strain, Y (c) dimensionless parameter x (Equation 4.330) vs. particle volume fraction c[). 

Fig. 2. Solid, dash and dash-dot lines represent solutions for v= 0, 0.5u and u respectively. Variation of (a) non-dimensional velocity, u =u/(dg)l/2, (b) particle volume fraction and (c) non-dimensional granular temperature, T =T/dg, with dimensionless flow height.

Recommended Values of Aerosol Particle Volume Fraction in Air VylVK, dimensionless) and Volume Fraction of Organic Matter in Aerosols dimensionless) for Different Environmental Scenarios... 

Finally, the particle volume fraction, 4), is a critical dimensionless parameter. For monodisperse suspensions of spheres of radius a, = n(47t/3)a with n the number of particles per volume. As noted in the introduction, with gravitational effects eliminated when the difference in phase densities vanishes, Ap = 0, a suspension with any volume fraction 0 < ( ) < ( )max be formed and studied, a feature unique to suspensions. When interest is in the properties approaching the maximum packing limit, ( )niax/ it is common to describe the properties of the suspension in terms of a deviation from the limiting value i.e., in terms of 1 -< )/( )max-... 

Fig. 3. Zenz plot. Correlation of bed voidage, S, the volume fraction of the fluidized bed that is occupied by gas, for values of S from 1.0 to 0.5, and dimensionless velocity and particle properties, where p — p )/(3 PgU ). The horizontal lines represent the different values of e = 0.5...

Al = Cross-secdonal area allocated to lighL phase, sq ft Ap = Area of particle projected on plane normal to direction of flow or motion, sq ft A, = Cross-sectional area at top of vessel occupied by continuous hydrocarbon phase, sq ft ACFS = Actual flow al conditions, cu ft/sec bi = Constant given in table c = Volume fraction solids C = Overall drag coefficient, dimensionless D = Diameter of vessel, ft Db = See Dp, min Dc = Cyclone diameter, ft Dc = Cyclone gas exit duct diameter, ft Dh = Hydraulic diameter, ft = 4 (flow area for phase in question/wetted perimeter) also, DH in decanter design represents diameter for heavy phase, ft... 

In this section alone, in order to avoid confusion the intrinsic viscosity [rf] represents that of a particle and is dimensionless and the intrinsic viscosity [rj] represents that of a polymer and has units of gem-3. From these expressions we know that as we increase the volume fraction above each of these values, the product demarcates the structural transition. For particles there are many situations where it is relatively easy to determine the volume fraction, but for polymers the situation is more complex and it is most convenient to use a mass concentration. We can rewrite this relationship in terms of the mass of added particles in gem-3 ... 

Other dimensionless groups that compare the thickness of the adsorbed polymer layer to the radius of the particle or the radius of gyration of the polymer to the particle radius in polymer/colloid mixtures can also be easily defined. We are mostly concerned with the volume fraction and the Peclet number Pe in our discussions in this chapter. However, the other dimensionless groups may appear in the equations for intrinsic viscosity of dispersions when the dominant effects are electroviscous or sterically induced. 

From the above expressions it can be seen that reduced and intrinsic viscosities have the unit of reciprocal concentration. When one considers particle shape and solvation, however, concentration is generally expressed in terms of the volume fraction of the particles (i.e. volume of particles/total volume) and the corresponding reduced and intrinsic viscosities are, therefore, dimensionless. 

The first term wt = (l+2rc)1/2 of the power series w defined in Eq. (6-10) plays a special role within the interaction model in that it represents a perfect gas phase. If Vo, pc, Tc and R represent the molar volume of a compound, the critical pressure and critical temperature of the system and the gas constant, then the product pcV0 is reduced to llw of the product RTC due to the interaction between the particles in the system. Taking into account an empty (free) volume fraction in the critical state, the critical molar volume is written as Vc = V(). Consequently, a dimensionless critical... 

A grafted layer of polymer of thickness L increases the effective size of a colloidal particle. In general, dispersions of these particles in good solvents behave as non-Newtonian fluids with low and high shear limiting relative viscosities (fj0 and rj ), and a dimensionless critical stress (a3aJkT) that depends on the effective volume fraction = (1 + L/a)3. The viscosities diverge at volume fractions m0 < for mo < fan < 4>moo> the dispersions yield and flow as pseudoplastic solids. 

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. ... 

To solve the SCF equations, we make use of the discretisation scheme of Scheutjens and Fleer [69], It is understood that here we cannot give full details on the SCF machinery. For this we refer to the literature [67,70-72]. However, pertinent issues and approximations will be mentioned in passing. The radial coordinate system is implemented using spherical lattice layers r = 1,..., tm, where layers r = 1,..., 5 are reserved for the solid particle. The number of sites per layer is a quadratic function of the layer number, L r) o= i. The mean-field approximation is applied within each layer, which means that we only collect the fraction of lattice sites occupied by segments. These dimensionless concentrations are referred to as volume fraction (p r). We assume that the system is fully incompressible, which means that in each layer the volume fraction of solvent = 1 — (r) — volume fractions are the segment potentials u r). The segment potentials can be computed from the volume fractions as briefly mentioned below. 

The data of Mewis et al. (39) and d Haene (40) for poly(methyl methacrylate) spheres stabilized by poly( 12-hydroxy stearic acid) and dispersi in decalin correlate reasonably well with results for hard spheres for low to moderate volume fractions, although the critical stress is somewhat smaller. For highly concentrated dispersions, however, packing constraints cause some interpenetration of the layers at rest and viscous forces at high shear rates drive the particles even closer together. Consequently, the effective layer thickness decreases with increasing 0 and Pe, the dimensionless shear rate. 

As dimensionless concentration variable (f> is used throughout. In case of hard colloidal particles the quantity volume fraction. For polymers and penetrable hard spheres (j> refers to the relative concentration with respect to overlap (see (1.24))... 

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