Particle radius, calculation

Figure 3 - Calculated Dependence of the PbS Specific Volatilization Rate and the Temperature Gradient of the Gas Phase - PbS Surface on the Surface Temperature and Particles Radius (Calculations are Based on (4) and (6) for Autogenous Smelting)...

Fig. 4.19 Phase diagrams in coordinates spin concentration-particle radius calculated for material parameters of KTO, temperature T = 300 K (a, b), 5 K (c, d) screening length A =0.2 nm (b,d), A = 4 nm (a, c). SG spin glass state with diffuse boundary undetermined within our model, FSG ferro-spin glass phase, FM ferromagnetic phase, PM paramagnetic phase [48]...

The dependence of transverse correlation radius Rex on the particle radius calculated on the basis of Eq. (4.73) is reported in Fig. 4.36. At temperatures T correlation radius Rex diverges at critical radius Rer (T) as anticipated from Eq. (4.73), see curves 1-3. The divergence corresponds to the size-induced ferroelectric phase transition. At temperatures T > 7), transverse correlation radius monotonically increases with the particle radius due to R) increase, see curves 4 and 5. 

Stress concentration Kis defined (1) as local stress/mean stress in a particle and calculated according to if = 1 + 2 LR) length and R is the radius of the crack tip. 

There have been several theories proposed to explain the anomalous 3/4 power-law dependence of the contact radius on particle radius in what should be simple JKR systems. Maugis [60], proposed that the problem with using the JKR model, per se, is that the JKR model assumes small deformations in order to approximate the shape of the contact as a parabola. In his model, Maugis re-solved the JKR problem using the exact shape of the contact. According to his calculations, o should vary as / , where 2/3 < y < 1, depending on the ratio a/R. 

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

Limitations It is desirable to have an estimate for the smallest particle size that can be effectively influenced by DEP. To do this, we consider the force on a particle due to DEP and also due to the osmotic pressure. This latter diffusional force will randomize the particles and tend to destroy the control by DEP. Eigure 20-31 shows a plot of these two forces, calculated for practical and representative conditions, as a function of particle radius. As we can see, the smallest particles that can be effectively handled by DEP appear to be in range of 0.01 to 0.1 pm (100 to 1000 A). 

DLVO Theory Applied to MCC Sols. In order to interpret the stability of MCC sols, several stability calculations were made. First, by using a spherical particle radius of 706A, A = 2.6 kT, ijto = 14 mV and T = 296°K at a salt concentration of 0.9 x 10 3M (region II), a secondary minimum was obtained at 500 A, V x =... 

The viscoelastic behavior of concentrated (20% w/w)aqueous polystryene latex dispersions (particle radius 92nm), in the presence of physically adsorbed poly(vinyl alcohol), has been investigated as a function of surface coverage by the polymer using creep measurements. From the creep curves both the instantaneous shear modulus, G0, and residual viscosity, nQ, were calculated. 

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

According to gel-filtration experiments conducted by P. Wills and J. Dijk (personal communication), proteins L17, L25, L28, L29, and L30 are compact LI, L4, L5, L6, L13, L16, L19, and L24 are moderately elongated and L2, L3, L9, Lll, L15, L23, L27, L32, and L33 are quite extended. A discrepancy between these results and those mentioned earlier is protein L9 which appears to be globular from hydrodynamic measurements (Giri et al., 1979), but the Stokes radius calculated from gel-filtration experiments was found to be quite large, suggesting an elongated particle. 

The relaxation rates calculated from Eq. (15) are smaller than the measured ones at low field, while they are larger at high field. OST is thus obviously unable to match the experimental results. However, water protons actually diffuse around ferrihydrite and akaganeite particles and there is no reason to believe that the contribution to the rate from this diffusion would not be quadratic with the external field. This contribution is not observed, probably because the coefficient of the quadratic dependence with the field is smaller than predicted. This could be explained by an erroneous definition of the correlation length in OST, this length is the particle radius, whilst the right definition should be the mean distance between random defects of the crystal. This correlation time would then be significantly reduced, hence the contribution to the relaxation rate. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

FIGURE 14.27 Calculated fraction of 550-nm light scattered upward (/3) as a function of solar zenith angle (0) and particle radius (yum). The refractive index of the particles is 1.4 (adapted from Nemesure et al., 1995). 

In Fig. 14.3 we plot (14.2) and (14.3) as functions of large-particle radius. There are of course several restrictions to be kept in mind, including 2aa 1 underlying the derivation of (7.2), which is only approximately satisfied for radii less than about 3 jum. To convince ardent Mie calculators that these simple expressions are approximately correct, we include single-size Mie calculations at 0.1-jLim intervals. Except for the interference maxima and minima in the Mie calculations, which are unlikely to be observed in natural aerosols, the simple treatment is quite good. 

The radius of the particle is calculated from its volume if we assume the particle to be a sphere. 

Besides the particle radius, the fundamental quantities indicated above are Dv, DP, KD, v, and a. v can be calculated from kinetic theory (5), Lai and Freiling have calculated Dv (6), and Norman has estimated DP and KP (9,12). Only the problem of estimating a remains. 

The experimental diffusion parameters, D /r., at 30°C. are presented in Table II for all the coals. Clearly, no correlation exists between diffusion parameter and rank. If r<> is taken as the average particle radius for the 200 X 325 mesh samples, an upper limit to the values of diffusion coefficient, D, is obtained. The diffusion coefficient ranges from 1.92 X 10 9 sq. cm./sec. for Kelley coal to 1.41 X 10"8 sk. cm./sec. for the Dorrance anthracite. Our previous studies on the change of D /n with particle size suggested that n is not necessarily the particle radius (7) but is a smaller distance related to the average length of the micropores in the particles. That is, the calculated... 

The size of the nanoparticles of BaC03 obtained was about 20-30 nm. Their specific area was calculated to be from 6.82 x 107 to 4.55 x 107m2/kg. The particles with 20 nm sizes have specific area of 6.82 x 107 m2/kg while particles with 30 nm size 4.55 x 107 m2/kg. Therefore, the specific area decreased with the increase of particle radius (Fig. 3). 

A quantitative comparison of particle expansion determined by the three methods is given in Table I. The particle diamete of the standard acrylic latex was determined by PCS to be 1120 A. This value was used in the calculation of the increase in particle radius at maximum expansion in each case. The sedimentation method yielded the largest increase in radius, 302 A, followed by the viscometric value of 240 K. Possibly the shear involved in the latter method resulted in a partial collapse of the surface layer. The value determined by PCS was found to be approximately half that determined by sedimentation. Since the PCS determination is presumed to be free of particle interactions at a concentration of 5 X 10 4%, we must conclude that the other two methods (at 1% solids) exhibit such interactions. As a result, the charged particles settle slower (19) and yield a higher viscosity than in the absence of these (repulsive) interactions. 

The particle radius (r) of homogeneous suspended solute may be calculated with the use of Eq. (4.88) (Nichols and Bailey, 1949) ... 

---