Particle radius, expressions

Particle sizes are measured in microns, p, A micron is 1/1000 millimeter or 1/25,400 inch. A millimicron, m,U, is 1/1000 of a micron, or 1/1,000,000 millimeter. Usually particle size is designated as the average diameter in microns, although some literature reports particle radius. Particle concentration is often expressed as grains/cubic feet of gas volume. One grain is 1/7000 of a pound. 

The rate expression for reaction of a cylindrical particle (radius r) is [469]... 

A diffusion-limited reaction proceeding in spherical particles (radius r) obeys a rate expression obtained by combining eqn. (10) with the contracting volume relation [eqn. (7), n = 3], viz. 

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 ). 

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. 

This formula generalizes the conclusion reached in Tables 1.2 and 1.3. It shows clearly that for a fixed amount of material, the surface area is inversely proportional to the radius for uniform, spherical particles. At the same time, the formula reminds us that some lower limit for Rs must be imposed since the relationship is undefined for Rs = 0. If SI units were used consistently, Asp would be expressed in m2 kg-1 however, m2 g 1 are the most commonly used units for this quantity. In the event of nonuniform or nonspherical particles, alternate expressions for Equation (2) have to be used. The following example considers the case of cylindrical particles. 

In order to test this model, we measured the quantum yield of the electron transfer to methylviologen as a function of the particle radius of CdS nanoparticle.13) The dependence of the electron transfer yield on the particle size well proved the applicability of the 2D ladder model to this system. For a low excitation limit of gV< 1, the quantum yield is independent of the light intensity, as expressed by... 

Kawaguchi et al. [12] have modified Paine s model for the dispersion copolymerization of amphiphilic PEO macromonomers. The authors have modeled the variation of the particle size and its distribution with reaction conditions. For example, the expressions for the critical conversion (x Xthe particle radius (r), and the surface area (S) occupied by a PEO chain are as follows ... 

The corresponding expression for mj(tn) can be derived according to the particular reaction. In general terms the rate of mass loss, -dmj/dt, is a function of particle radius rj, kinetic rate R(Tj, Tp, P), and gas phase compositions y s. This can be... 

Equation (3.58) and Equation (3.61) are the Hixson and Crowell cube-root and the Higuchi and Hiestand two-thirds-root expressions, respectively. The cube-root and the two-thirds-root expressions are approximate solutions to the diffusional boundary layer model. The cube-root expression is valid for a system where the thickness of the diffusional boundary layer is much less than the particle radius whereas the two-thirds-root expression is useful when the thickness of the boundary layer is much larger than the particle radius. In general, Equation (3.57) is more accurate when the thickness of the boundary layer and the particle size are comparable. 

In the above expressions, the subscripts p and w denote the reflections from the particle and the boundary wall, respectively. The superscript (z ) represents the z th reflection from the particle surface or the solid boundary. Note that Eqs. (58) and (59) are actually power series in k, which is the ratio of the particle radius a to the distance between the particle center and the boundary d. The advantage of this method is that one needs to consider boundary conditions associated with only one surface at a time. After solving for the electric field and the fluid velocity for each reflection, the electrophoretic velocity can be calculated and expressed by... 

When the double layer remains at equilibrium but has a finite thickness, the interactions between the particles do exist. The numerical results obtained by Shugai et al. [57] showed that the pair-interaction contribution to the mean electrophoretic mobility of a suspension can be significant for values of the scaled particle radius Ka < 10. Ennis and White [56] used the reflection results to obtain a complicated expression for a as a function of m. Interested readers should be referred to their paper for the general formula. For large Ka, the asymptotic expression for a is... 

As we discussed in Sect. 3.1.1, Hansen et al. [15] made significant improvements to the concept of the radical capture efficiency proposed by Nomura et al. [ 14]. Taking this concept into consideration, they examined the effect of radical desorption on micellar particle formation in emulsion polymerization [ 65 ]. Assuming that radical entry is proportional to the x power of the micelle radius and the polymer particle radius, they proposed the following general expression for the rate of particle formation ... 

For colloidal materials the small size of the semiconductor particles severely restricts the magnitude of the electric field that a particle can support. Albery and Bartlett first considered the potential distribution within spherical semiconductor particles [144]. For large particles, an expression equivalent to that for planar electrodes given in Eq. 6 was derived. For small semiconductor particles the total band bending within the semiconductor, Fb, is limited by the radius r, Eq. 23 ... 

Figure 7 Dependence of light-scattering efficiency of ammonium sulfate aerosol on the amount of material in the particle and RH. The scattering efficiency is expressed as a scattering coefficient cr p, here at wavelength 550 nm, per amount of sulfate. The auxiliary abscissa scale gives the particle radius (Nemesure et al., 1995) (reproduced by permission of American Geophysical Union from J. Geophys. Res.

The above expression has limited applicability, however, since it holds only when the gap D between particle surfaces is small compared to the particle radius, but still large compared to the Debye length. Fortunately, the superposition approximation can be generalized by linearly adding the potentials for two isolated spheres. This gives (Russel et al. 1989)-------------------------------------------------------------------------------------------... 

Figure 1 shows measured Peak shifts (Peak shift is defined as the distance from the particle edge to the point of maximum Vanadium concentration, expressed as a fraction of the particle radius) for various catalysts vs. fractional distance from the top to the reactor. Qualitatively, the results are in full agreement with the theory that predicts ... 

Dynamic viscosity also is called the coefficient of viscosity. Note that dynamic viscosity divided by density gives the kinematic viscosity which is used in Eq. [2-20]. For most gases, the dynamic viscosity increases with increasing temperature. At 18°C, the dynamic viscosity of air is 1.83 X 10 4 g/(cm sec), or poise. If particle radius is expressed in micrometers, values for g and pij are substituted, and Ap is set to 1 g/cm3, Stokes law reduces to a convenient rule of thumb for air ... 

For a diffusion-limited reaction proceeding in spherical particles (radius a) the rate expression is obtained by combining the parabolic law, equation (3.7), with the contracting volume equation (3.3) (with = 3) to give ... 

The effect of particle size and dissolution rate has been known since the pio-neeringworkofNoyes and Whitney (1897), and Hixson and Crowell (1931) subsequently derived a highly useful equation that expresses the rate of dissolution based on the cube root of the weight of the particles. When the Hixson-Crowell model is applied to micronized particles, for which the thickness of the aqueous diffusion layer around the dissolving particles is comparable to or larger than the radius of the particle, the change in particle radius with time is given by ... 

Electrically charged particles in aqueous media are surroimded by ions of opposite charge (counterions) and electrolyte ions, namely, the electrical double layer. The quantity He represents the energy of repulsion caused by the interaction of the electrical double layers. The expression for He depends on the ratio between the particle radius and the thickness of the electrical double layer, k, called the Debye length. For K.a > 5 (Quemada and Berli, 2002) ... 

The force and torque exerted on a solid particle were obtained in the form of a power series with respect to RJl, where is the particle radius and I is the distance from the center of the particle to the wall. Lorentz derived an asymptotic expression for the motion of a sphere along the normal... 

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