Small Effective Diameters

The simplest way of improving radial uniformity in a tubular reactor is to reduce the tube diameter since this will increase and at/R. A practical lower 

y is a distance coordinate analogous to r that is zero at the center of the slit H is analogous to R and is the slit half-height. Equation 8.53 assumes that the walls at the sides of the slit are so distant that they have no effect on the flow over most of the slit width. 

Equation 8.53 predicts that the fastest moving fluid will experience two-thirds of the mean residence time, 0.67F. This is a worthwhile improvement in uniformity compared to a mbe where the fastest fluid has only half the mean residence time. However, the major advantage of duct and slit devices is that they can be fabricated 

---

Let us focus our adention for the moment on a small volume in space, dr, and on particles in the volume with a given velocity v. Let us sit on such a particle and ask if it might collide in time t with another particle whose velocity is v, say. Taking the effective diameter of each particle to be a, as described above, we see... 

To be specific let us have in mind a picture of a porous catalyst pellet as an assembly of powder particles compacted into a rigid structure which is seamed by a system of pores, comprising the spaces between adjacent particles. Such a pore network would be expected to be thoroughly cross-linked on the scale of the powder particles. It is useful to have some quantitative idea of the sizes of various features of the catalyst structur< so let us take the powder particles to be of the order of 50p, in diameter. Then it is unlikely that the macropore effective diameters are much less than 10,000 X, while the mean free path at atmospheric pressure and ambient temperature, even for small molecules such as nitrogen, does not exceed... 

A commercial bacterial cellulose product (CeUulon) was recently introduced by Weyerhaeuser (12). The fiber is produced by an aerobic fermentation of glucose from com symp in an agitated fermentor (13,14). Because of a small particle diameter (10 P-m), it has a surface area 300 times greater than normal wood cellulose, and gives a smooth mouthfeel to formulations in which it is included. CeUulon has an unusual level of water binding and works with other viscosity builders to improve their effectiveness. It is anticipated that it wiU achieve GRAS status, and is neutral in sensory quaUty microcrystaUine ceUulose has similar attributes. 

Theoretical studies indicate that for dt < 2 nm the effect of strain energy exceeds that of the room temperature thermal energy, so that it is only at small nanotube diameters that the strain energy associated with nanotube... 

The function f(k ) is shown plotted against the thermodynamic capacity ratio in Figure 1. It is seen that for peaks having capacity ratios greater than about 2, the magnitude of (k ) has only a small effect on the optimum particle diameter because the efficiency required to effect the separation tends to a constant value for strongly retained peaks. From equation (1) it is seen that the optimum particle diameter varies as the square root of the solute diffusivity and the solvent viscosity. As, in... 

The results provided in the literature for stress with biological particle systems, whereby gas distributors with small hole diameters, i.e. with smaller bubble sizes, have a more negative effect on cells (see e.g. [4, 30,31]), are frequently not comparable, as in these studies there was differing stress during bubble formation at the gas distributor due to different hole velocities. 

In order to simplify the analysis, we will consider the capillary flow of a hquid L in a horizontal small tube (diameter much smaller than the capillary length), in order to avoid complications due to gravity effects (Fig. 14). 

As will be outlined below, the computation of compressible flow is significantly more challenging than the corresponding problem for incompressible flow. In order to reduce the computational effort, within a CED model a fluid medium should be treated as incompressible whenever possible. A rule of thumb often found in the literature and used as a criterion for the incompressibility assumption to be valid is based on the Mach number of the flow. The Mach number is defined as the ratio of the local flow velocity and the speed of sound. The rule states that if the Mach number is below 0.3 in the whole flow domain, the flow may be treated as incompressible [84], In practice, this rule has to be supplemented by a few additional criteria [3], Especially for micro flows it is important to consider also the total pressure drop as a criterion for incompressibility. In a long micro channel the Mach number may be well below 0.3, but owing to the small hydraulic diameter of the channel a large pressure drop may be obtained. A pressure drop of a few atmospheres for a gas flow clearly indicates that compressibility effects should be taken into account. 

With the addition of a pseudopotential interaction between electrons and metal ions, the density-functional approach has been used82 to calculate the effect of the solvent of the electrolyte phase on the potential difference across the surface of a liquid metal. The solvent is modeled as a repulsive barrier or as a region of dielectric constant greater than unity or both. Assuming no specific adsorption, the metal is supposed to be in contact with a monolayer of water, modeled as a region of 3-A thickness (diameter of a water molecule) in which the dielectric constant is 6 (high-frequency value, appropriate for nonorientable dipoles). Beyond this monolayer, the dielectric constant is assumed to take on the bulk liquid value of 78, although the calculations showed that the dielectric constant outside of the monolayer had only a small effect on the electronic profile. 

The work currently being conducted by Satyanarayan, Kumar, and Kuloor (S3) indicates that the effect of surface tension is more involved than hitherto appreciated. Some of their data are presented in Figs. 6. and 7. They find that at very small orifice diameters or at very large flow rates, the surface tension variation has negligible influence on the bubble volume. For higher orifice diameters, the influence is more pronounced at small flow rates, as is evident from Fig. 7. 

Fig. 6. Effect of surface tension on bubble volume for small orifice diameters under constant pressure conditions.

Photodiodes occur in many different varieties and are useful in both steady-state and time-resolved fluorescence studies. Photodiodes designed for use in steady-state or on microsecond time-scales are inexpensive and have effective areas up to a few square millimeters, and are capable of efficiently matching to simple focusing optics. However, as the temporal resolution increases so does the cost, and the effective area has to be reduced. For example, APDs with response times in the 50 psec region have effective diameters ofca. 10 /small active area of high-speed devices is currently the primary drawback in fluorescence studies. Also, photodiodes other... 

Detection If a small capillary diameter is desired for efficiency purposes, the detection part of the capillary can be adapted for better detection sensitivity. Examples are the bubble cell capillary and the Z-cell. In the bubble cell capillary, the capillary diameter is enlarged at the detection window so that better concentration sensitivity is obtained. If you implement a bubble cell capillary in your pharmaceutical analysis method, it is important to test different batches. Test also whether you need a bubble cell capillary or whether you can gain similar sensitivity increase with a proper injection procedure. Also, check the effect of the bubble cell on band broadening. An approximately three-times sensitivity enhancement is possible. 

To account for their data (Fig. 2.7), Mondain-Monval et al. hypothesized that these two forces simply add and that the repulsion between micelles and droplets increases the effective diameter of the droplets (or micelles) [22]. This force is derived by integrating the osmotic pressure Posm over the accessible zone for micelles of diameter 2r (r = 2.35 nm) from 6 = n to 9 = 7t -Oi, with 9i defined in Fig. 2.6. The distance at which the small micelles are excluded from the gap between the droplets is evidently influenced by the electrostatic micelle-droplet repulsion. To account for this repulsion, droplets (or micelles) may be considered as particles of effective radius (a + S) [or micelles of radius (r + 5)]. From... 

---