Fraction packing

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

The virial series in temis of the packing fraction q = upa /3 is then... 

In this expression, the packing fraction r] = Ttpa- /6, and the other two parameters are related to this by... 

Figure A3.6.13. Density dependence of die photolytic cage effect of iodine in compressed liquid n-pentane (circles), n-hexane (triangles), and n-heptane (squares) [38], The solid curves represent calculations using the diffusion model [37], the dotted and dashed curves are from static caging models using Camahan-Starling packing fractions and calculated radial distribution fiinctions, respectively [38],...

Since the total integer atomic mass (M) is given by the number of protons and neutrons, then M = P + N. Because of the masses of the electrons in an atom and a packing fraction of mass in each nucleus, the actual atomic mass is not an integer. 

For troubleshooting fractionators see the troubleshooting section in this book. For a general understanding of how trayed and packed fractionating columns work, illus-... 

The adsorption isotherms have been obtained according to the procedure of Ford and Glandt for a hard sphere fluid in a hard sphere matrix [23]. Let us denote the packing fraction of matrix species by rjf rj = -nNc Ma /6). 

TABLE 1 The Partitioning Coefficient, K, for Hard Spheres in a Matrix of Chains with m Monomers at Packing Fraction r] ... 

FIG. 5 Adsorption isotherms for a hard sphere fluid from the ROZ-PY and ROZ-HNC theory (solid and dashed lines, respectively) and GCMC simulations (symbols). Three pairs of curves from top to bottom correspond to matrix packing fraction = 0.052, 0.126, and 0.25, respectively. The matrix in simulations has been made of four beads (m = M = 4). 

FiG. 7 The same as in Fig. 6, for the matrix made of chains with four beads (m = Af = 4) at packing fraction 0.126. The nomenclature of lines and symbols and their location are similar to those of Fig. 6. 

This fitted the data well up to volume fractions of 0.55 and was so successful that theoretical considerations were tested against it. However, as the volume fraction increased further, particle-particle contacts increased until the suspension became immobile, giving three-dimensional contact throughout the system flow became impossible and the viscosity tended to infinity (Fig. 2). The point at which this occurs is the maximum packing fraction, w, which varies according to the shear rate and the different types of packings. An empirical equation that takes the above situation into account is given by [23] ... 

Murch, D. R, Height of Equivalent Theoretical Plate in Packed Fractional Columns. Ind. Eng. Chem., 45, 2616 (1953). 

Fig. 3a,b. The phase behaviour of a system of hard ellipsoids, both prolate and oblate, as a function of the ellipticity, alb, plotted against a the packing fraction, p b the scaled number density, p ... 

FIG. 4 Nomialized concentration distribution of a 0.1 molar 1 1 electrolyte in an uncharged cylindrical pore of radius five times the diameter of the ions. The dashed line, solid up-triangles, and solid down-triangles are the neutral solvent particles, cations, and anions, respectively, in an SPM model with 0.3 solvent packing fraction. The open symbols are for the cations and anions in the RPM model. 

FIG. 8 Salt exclusion as a function of surface charge in a cylindrical pore in equilibrium with a 0.1 molar electrolyte. The open circles are GCMC results for 1 1 RPM electrolyte in a pore ofR = 5d The circles with a centered cross are results for a 2 1 electrolyte in a pore of = 5d. The up-trian-gles are results for a 2 1 electrolyte in a pore ofR = lOd. The solid circles are results for a 1 1 SPM model with 0.3 solvent packing fraction in a pore of = 5d. The solid squares are the same results for a pore of R = Id. 

FIG. 14 Comparison of total pressure between two charged surfaces in 1 1, 2 1, and 2 2 electrolytes. The open squares, up-triangles, and down-triangles are results of 1 1, 2 1, and 2 2, respectively, for the SPM model of 0.3 packing fraction. The corresponding solid symbols are for the RPM model. 

FIG. 19 Extrapolated zero-field conductivity versus pore radius for 0.1 M 1 1 RPM and SPM electrolytes at two solvent packing fractions, 0.1 and 0.2. The conductivity in a poreR = 1.2[Pg.647]

Packed fractional distillation columns run in the batch mode are often used for low-pressure drop vacuum separation. With a trayed column, the liquid holdup on the trays contributes directly to the hydraulic head required to pass through the column, and with twenty theoretical stages that static pressure drop is very high, e.g., as much as 100-200 mm Hg. 

---