Radius of Packed Particles

To clarify the effect of radius of packed particles on Hs, Equation 11.20 can be rearranged as follows, in case the liquid film mass transfer resistance is negligible  

When the effective diffusivity of solutes D g can be approximated by the diffusivity in water, D, multiplied by a constant that includes the effects of particle porosity and tortuosity of pores in particles. Equation 14.3 can be written as follows  

Hs for a given velocity will increase, while the intercept of the straight lines on the y-axis, which corresponds to the value of the first term of Equation 11.20, is constant for different solutes. The value of the intercept will depend on the radius of packed particles, but does not vary with the effective diffusivity of the solute. 

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As the first term of the right-hand side of Equation 11.20 is independent of fluid velocity and is proportional to the radius fg (m) of particles packed as the stationary phase under usual conditions in chromatography separation, Hs (m) will increase linearly with the interstitial velocity of the mobile phase u (m s ), as shown in Figure 11.9. With a decrease in the effective diffusivities of solutes (m s ), Hs for a given velocity will increases, while the intercept of the straight lines on the y-axis, which corresponds to the value of the first term of Equation 11.20, is constant for different solutes. The value ofthe intercept will depend on the radius of packed particles, but does not vary with the effective diffusivity of the solute. 

This method is used particularly for colloids. A colloidal dispersion is forced through a long column packed with nonporous beads with an approximate radius of 10pm. Particles of different particle size travel with different speeds around the beads and are thus collected in size fractions. 

To start with, for simplicity, let us assume that all particles are of equal size in the oxide powder. If the radius of individual particles is 2r and if we take rr particles and pack them in a cube of each side 2m, then the total volume of the cube will be 8r n. The acmal volume occupied by the particles is, and hence the void space between the particles... 

We note that the theoretical isotherms exhibit quite different shapes when the radius of elementary particles and the packing type are changed as shown in Figs. 2 and 3. Therefore in later works we could easily obtain suitable experimental isotherms by choosing proper values of and and the type of packing by a trial-and-error method, in order to satisfy the experimental observations. 

Nevertheless, much is known about the structure of adsorbed 6-casein, certainly more flian is known for any other food protein, and various techniques have been used to study the adsorbed protein. The first evidence from DLS showed that 6-casein adsorbed to a polystyrene latex caused an increase in the radius of the particle by 10 to 15 nm (84). Later studies using small-angle X-ray scattering confirmed this and showed, in addition, that the bulk of the mass of the protein was close to the interface, so the interfacial layer was not of uniform density throughout (85). Neutron-reflectance studies also showed that most of the mass of protein was close to the interface (86). Only a relatively small portion of the mass of the adsorbed protein extends from the tightly packed interface into the solution, but it is this part which determines the hydrodynamics of the particle and which is almost certainly the soiuce of the steric stabilization which the 6-casein affords to emulsion droplets (84). It is to be noted that all of the studies just described were performed on latex particles or on planar interfaces however, it has also been demonstrated that the inter-facial structiues of 6-casein adsorbed to emulsion dro plets resemble those of the model particles (39, 85). Although detailed control of emulsion droplets dining their... 

Colloidal suspensions can be classified as soft sphere systems because the repulsive intoactions occur at some characteristic distance from the particle surface. For electrostatic and stoic stabilization, this distance is the Debye length (1/ K) and the thickness of the adsorbed polymer layer, respectively. For stoically stabilized suspensions, the adsorbed polymer layer leads to an increase in the hydrodynamic radius of the particle. When the adsorbed layer is densely packed, the principles described above for hard sphere systems are applicable, provided that the volume fraction of particles/is replaced by an effective volume fraction /gy given by... 

The comparison of obtained data shows that at a given radius of aerosol particles of SAS solution, the rate adsorption constants of this aerosol at its adsorption from air are essentially higher than the rate adsorption constants from water bulk. This allows one to realize technically the flotation process using the close-pack ionized adsorption monolayer of surfactants, specially obtained at air bubble surface by aerosol dispersion and, thus, to control this process efficiently. 

The elution of such gels is an example not of size exclusion but rather of hydrodynamic fractionation (HDF). However, it must be remembered that merely being able to physically fit an insoluble material through the column interstices is not the only criterion for whether the GPC/HDF analysis of an insoluble material will be successful. A well-designed HDF packing and eluant combination will often elute up to the estimated radius in Eq. (5), but adsorption can drastically limit this upper analysis radius. For example, work in our laboratory using an 8-mm-bead-diameter Polymer Laboratories aqueous GPC column for HDF found that that column could not elute 204 nM pSty particles, even though Eq. (5) estimates a critical radius of —1.5 jam. 

It is well known in crystallography that, when spheres of equal radius are packed together, the closest type of packing is one in which each sphere has 12 other spheres in contact with it. In Sec. 24 it was mentioned that in water at room temperature each molecule has, on the average, only 4.4 other molecules in contact with it. If we wanted to place one or two additional H20 molecules in contact with any H20 molecule, there would be plenty of room to do this without seriously disturbing the neighbors that are already in contact with this molecule. Similarly, if this molecule is replaced by a solute particle of the same size, the same remark could be made about placing molecules in contact with the solute particle. 

A first-order chemical reaction occurs isothermally in a reactor packed with spherical catalyst pellets of radius R. If there is a resistance to mass transfer from the main fluid stream to the surface of the particle in addition to a resistance within the particle, show that the effectiveness factor for the pellet is given by ... 

The simplest form of RSM is constmcted by placing N identical sohd monospheres of radius Rsp in a single empty volume without any correlation between the positions of the spheres, so that some of the spheres remain isolated and others overlap. The porosity, e, of such chaotic packing is determined by Kolmogorov as the probability of finding an arbitrarily chosen point outside of the space of the particles, and is equal to... 

The presence of the pores adds two parameters - the pore volume fraction and the pore radius. The predicted Rp Increases as the pore radius decreases suggesting a preference tor small pore packings. However, for a small pore radius of 1.0 pm a single value of the separation factor corresponds to two values of the particle diameter (13). Such double-valued behavior Is of course undesirable In an analytic technique. 

Figure 6. Separation factor-particle diameter tehavior as a function of packing diameter for the pore-partitioning model. Parameters are the same as in Figure 3 with the exception of the interstitial capillary radius which was computed from the hed hydraulic radius (Equation 11 (7.) with void fraction = 0.358).

Table 1. Molecular Weight Equivalent to the Hydraulic Radius of a Column Packed with 2-, 5-, and 10-pm Particles...

Equation (19) shows that the minimum radius will increase as the square root of the extra column dispersion and as the square root of (a-1) but, increase inversely with the square root of the particle diameter. (However,it will be shown later that, that if the column is packed with particles of optimum diameter for the particular separation then the column radius will become linearly related to the function (a-1)). 

There is no radial velocity, and the axial velocity across the radius of the packed bed is uniform. Schwartz and Smith (1953) found that the velocity across the diameter of a packed bed is not uniform for radial aspect ratios (tube-to-particle diameter) less than about 30, due to the significant effect of the increased void space near the wall where the particles are locally ordered. This result has been verified by Hoiberg et al. (1971) for a packed bed reactor with radial aspect ratio about 50. They considered a radial velocity variation suggested by experimental observations with a sharp peak about 15% greater than the mean fluid velocity situated close to the wall. Simulations using their model showed results virtually identical to those obtained with a uniform velocity profile.3... 

When the radius of particles packed and/or the column height are changed, the number of theoretical plates of a large-scale column must be kept equal to that of a small column that is,... 

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