Negligible Dispersion

Experimental fixed bed reactors are commonly heated electrically or cooled, and radial temperature gradients within the packed bed may occur. The criterion for exclusion of the influence of radial heat dispersion (negligible radial temperature profile) is ... 

These equations are the steady-state version of Eqs. 9.6 through 9.15 with the assumptions of negligible axial dispersion, negligible external mass transfer resistance, isothermal pellets, and constant v and physical properties. Note that the external heat transfer resistance is still present as given by Eq. 9.36. Needless to... 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Chemical Reaction Measurements. Experimental studies of incineration kinetics have been described (37—39), where the waste species is generally introduced as a gas in a large excess of oxidant so that the oxidant concentration is constant, and the heat of reaction is negligible compared to the heat flux required to maintain the reacting mixture at temperature. The reaction is conducted in an externally heated reactor so that the temperature can be controlled to a known value and both oxidant concentration and temperature can be easily varied. The experimental reactor is generally a long tube of small diameter so that the residence time is well defined and axial dispersion may be neglected as a source of variation. Off-gas analysis is used to track both the disappearance of the feed material and the appearance and disappearance of any products of incomplete combustion. 

Several modes of waste management are available. The simplest is to dilute and disperse. This practice is adequate for the release of small amounts of radioactive material to the atmosphere or to a large body of water. Noble gases and slightly contaminated water from reactor operation are eligible for such treatment. A second technique is to hold the material for decay. This is appHcable to radionucHdes of short half-life such as the medical isotope technetium-9 9m = 6 h), the concentration of which becomes negligible in a week s holding period. The third and most common approach to waste... 

Stainless steel flat six-blade turbine. Tank had four baffles. Correlation recommended for ( ) < 0.06 [Ref. 156] a = 6( )/<, where d p is Sauter mean diameter when 33% mass transfer has occurred. dp = particle or drop diameter <3 = iuterfacial tension, N/m ( )= volume fraction dispersed phase a = iuterfacial volume, 1/m and k OiDf implies rigid drops. Negligible drop coalescence. Average absolute deviation—19.71%. Graphical comparison given by Ref. 153. ... 

Both phases are siibstantiaUy in plug flow. Dispersion measurements of the hquid phase usuaUy report Peclet numbers, Uid /D, less than 0.2. With the usual smaU particles, the waU effect is negligible in commercial vessels of a meter or so in diameter, but may be appreciable in lab units of 50 mm (1.97 in) diameter. Laboratory and commercial units usuaUy are operated at the same space velocity, LHSy but for practical reasons the lengths of lab units may be only 0.1 those of commercial units. 

Figure 3.2.1 illustrates the mixing in packed beds (Wilhelm 1962). As Reynolds number approaches the industrial range Rep > 100, the Peclet numbers approach a constant value. This means that dispersion is influenced by turbulence and the effect of molecular diffusion is negligible. 

Having established that a finite volume of sample causes peak dispersion and that it is highly desirable to limit that dispersion to a level that does not impair the performance of the column, the maximum sample volume that can be tolerated can be evaluated by employing the principle of the summation of variances. Let a volume (Vi) be injected onto a column. This sample volume (Vi) will be dispersed on the front of the column in the form of a rectangular distribution. The eluted peak will have an overall variance that consists of that produced by the column and other parts of the mobile phase conduit system plus that due to the dispersion from the finite sample volume. For convenience, the dispersion contributed by parts of the mobile phase system, other than the column (except for that from the finite sample volume), will be considered negligible. In most well-designed chromatographic systems, this will be true, particularly for well-packed GC and LC columns. However, for open tubular columns in GC, and possibly microbore columns in LC, where peak volumes can be extremely small, this may not necessarily be true, and other extra-column dispersion sources may need to be taken into account. It is now possible to apply the principle of the summation of variances to the effect of sample volume. 

In a packed column, however, the situation is quite different and more complicated. Only point contact is made between particles and, consequently, the film of stationary phase is largely discontinuous. It follows that, as solute transfer between particles can only take place at the points of contact, diffusion will be severely impeded. In practice the throttling effect of the limited contact area between particles renders the dispersion due to diffusion in the stationary phase insignificant. This is true even in packed LC columns where the solute diffusivity in both phases are of the same order of magnitude. The negligible effect of dispersion due to diffusion in the stationary phase is also supported by experimental evidence which will be included later in the chapter. 

PR valve risers in flammable service should also be sized such that exit velocities are at least 30 m/s under all foreseable contingencies (except fire) which would cause the valve to release. On the basis of experimental work and plant experience, this minimum velocity, in conjunction with the riser elevation requirements, has been shown to ensure effective dispersion. Entrainment of air and dilution result in a limited flammable zone, with a negligible probability of this zone reaching any equipment which could constimte an ignition source. 

FIG. 1 The mean force potential acting between colloidal species, /3fV (r), in adsorbed colloidal dispersion. In parts (a) and (b) the matrix density is taken as negligibly small, = 10 and = 0.193, respectively. In both parts, the evolution of the mean force potential with solvent density is shown p = 0.2, 0.3 and 0.4 (solid, dashed, and dotted lines, respectively). In part (c) the evolution of the PMF on matrix density is presented. The solvent density is held constant, p =0.3 the matrix density is Pmcr = 0.193, 0.386, and 0.772 (dotted, dashed, and solid lines, respectively). The diameter of the matrix species is = 7.055. The density of colloids is Pcg] = 10 , with Uc = 5, in all the cases in question. 

Optimum flowrates, resulting in high productivity and low eluent consumption, are estimated first for an ideal system , which means that kinetic and hydrodynamic dispersive effects are assumed to be negligible [46]. This procedure has recently been improved [57]. 

This type of liquid is characterized by direction independent, relatively weak dispersion forces decreasing with r-6, when r is the distance between neighbouring molecules. A simple model for this type of liquid, which accounts for many properties, was given by Luck 1 2> it is represented by a slightly blurred lattice-like structure, containing hole defects which increase with temperature and a concentration equal to the vapor concentration. Solute molecules are trapped within the holes of the liquid thus reducing their vapor pressure when the latter is negligible. 

In this equation, AG°CS is taken to be negligible for p- and y-cyclodextrin systems and to be constant, if there is any, for the a-cyclodextrin system. The AG W term is virtually independent of the kind of guest molecules, though it is dependent on the size of the cyclodextrin cavity. The AG dw term is divided into two terms, AG°,ec and AGs°ter, which correspond to polar (dipole-dipole or dipole-induced dipole) interactions and London dispersion forces, respectively. The former is mainly governed by the electronic factor, the latter by the steric factor, of a guest molecule. Thus, Eq. 2 is converted to Eq. 3 for the complexation of a particular cyclodextrin with a homogeneous series of guest molecules ... 

Silver chloride is light-sensitive decomposition occurs into silver and chlorine, and the silver remains colloidally dispersed in the silver chloride and thereby imparts a purple colour to it. The decomposition by light is only superficial, and is negligible unless the precipitate is exposed to direct sunlight and is stirred frequently. Hence the determination must be carried out in as subdued a light as possible, and when the solution containing the precipitate is set aside, it should be placed in the dark (e.g. in a locker), or the vessel containing it should be covered with thick brown paper. 

The proposed technique will be used here to illustrate the case of interfacial heat and multicomponent mass transfer in a perfectly mixed gas-liquid disperser. Since in this case the holding time is also the average residence time, the gas and liquid phases spend the same time on the average. If xc = zd = f, then for small values of t, the local residence times tc and td of adjacent elements of the continuous and dispersed phases are nearly of the same order of magnitude, and hence these two elements remain in the disperser for nearly equal times. One may conclude from this that the local relative velocity between them is negligibly small, at least for small average residence times. Gal-Or and Walatka (G9) have recently shown that this is justified especially in dispersions of high <6 values and relatively small bubbles in actual practice where surfactants are present. Under this domain, Eqs. (66), (68), (69) show that as the bubble size decreases, the quantity of surfactants necessary to make a bubble behave like a solid particle becomes smaller. Under these circumstances (pd + y) - oo and Eq. (69) reduces to... 

For example, for equal volumes of gas and liquid ( =0.5), Eq. (266) predicts that the Stokes velocity (which is already very small for relatively fine dispersions) should be reduced further by a factor of 38 due to hindering effects of its neighbor bubbles in the ensemble. Hence in the domain of high values and relatively fine dispersions, one can assume that the particles are completely entrained by the continuous-phase eddies, resulting in a negligible convective transfer, although this does not preclude the existence of finite relative velocities between the eddies themselves. 

Chemical methods. A known volume of the gas is passed over a suitable absorbent, the increase in mass of which is measured. The efficiency of the process can be checked by arranging a number of vessels containing absorbent in series and ascertaining that the increase in mass in the last of these is negligible. The method is very accurate but is laborious. Satisfactory absorbents for water vapour are phosphorus pentoxide dispersed in pumice, and concentrated sulphuric acid. 

Evaporation at 20°C is negligible a harmful concentration of airborne particles can, however, be reached quickly when dispersed. 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

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