Velocity profile residence time distribution

For the design (Sauter diameter, droplet velocity, pressure profile, residence time distribution, flooding load, etc.) see [6.71, 6.73-6.75]. Extractor selection and operating parameters have to be determined exper-... 

The above derivation assumes straight streamlines and a monotonic velocity profile that depends on only one spatial variable, r. These assumptions substantially ease the derivation but are not necessary. Anal5dical expressions for the residence time distributions have been derived for noncircular ducts,... 

Figure 3.42 Evolution of a pulse at the entrance of a micro channel for different diffusion coefficients. Calculated concentration profile (left) and cumulative residence time distribution curve (channel 300 pm x 300 pm x 20 mm flow velocity 1 m s f = 10 s) [27],...

With the development of modern computation techniques, more and more numerical simulations occur in the literature to predict the velocity profiles, pressure distribution, and the temperature distribution inside the extruder. Rotem and Shinnar [31] obtained numerical solutions for one-dimensional isothermal power law fluid flows. Griffith [25], Zamodits and Pearson [32], and Fenner [26] derived numerical solutions for two-dimensional fully developed, nonisothermal, and non-Newtonian flow in an infinitely wide rectangular screw channel. Karwe and Jaluria [33] completed a numerical solution for non-Newtonian fluids in a curved channel. The characteristic curves of the screw and residence time distributions were obtained. 

Preliminary residence time distribution studies should be conducted on the reactor to test this assumption. Although in many cases it may be desirable to increase the radial aspect ratio (possibly by crushing the catalyst), this may be difficult with highly exothermic solid-catalyzed reactions that can lead to excessive temperature excursions near the center of the bed. Carberry (1976) recommends reducing the radial aspect ratio to minimize these temperature gradients. If the velocity profile in the reactor is significantly nonuniform, the mathematical model developed here allows predictive equations such as those by Fahien and Stankovic (1979) to be easily incorporated. 

In macroscopic reactors, knowledge of the velocity profile in the channel cross-section is a necessary and sufficient prerequisite to describe the material transport. In microscopic dimensions down to a few micrometers, diffusion also has to be considered. In fact, without the influence of diffusion, extremely broad residence time distributions would be found because of the laminar flow conditions. Superposition of convection and diffusion is called dispersion. Taylor [91] was among the first to notice this strong dominating effect in laminar flow. It is possible to transfer his deduction to rectangular channels. A complete fluid dynamic description has been given of the flow, including effects such as the influence of the wall, the aspect ratio and a chemical wall reaction on the concentration field in the cross-section [37]. 

It is important to study the bubble rise velocity and its radial profile in a gas-liquid system as these are closely related to the hydrodynamics, and mass and heat transfer [25]. Bubble rise velocity and its radial profile have also significant influences on gas and liquid residence time distributions. A suitable bubble rise velocity and radial profile can improve production efficiency. Bubble rise velocities in a... 

The residence time distribution of particles is related to the properties of the particles and the gas flow, including the size distribution, and the velocity of gas flow and its profile. In practically applicable impinging stream devices, the particles being processed usually have relatively narrower size distribution the diameter of the tube to particles size ratio, d Jd,h is normally very large ( 15) while the gas velocity is high... 

The maximum relative deviation is 36%. The parabolic velocity profile in the micro channels will not influence the residence time distribution, otherwise the deviation would be much larger. The reason for this is the fast balancing of the concentration in a small channel by dispersion. 

The SSE is an important and practical LCFR. We discussed the flow fields in SSEs in Section 6.3 and showed that the helical shape of the screw channel induces a cross-channel velocity profile that leads to a rather narrow residence time distribution (RTD) with crosschannel mixing such that a small axial increment that moves down-channel can be viewed as a reasonably mixed differential batch reactor. In addition, this configuration provides self-wiping between barrel and screw flight surfaces, which reduces material holdback to an acceptable minimum, thus rendering it an almost ideal TFR. 

An important advantage of the use of EOF to pump liquids in a micro-channel network is that the velocity over the microchannel cross section is constant, in contrast to pressure-driven (Poisseuille) flow, which exhibits a parabolic velocity profile. EOF-based microreactors therefore are nearly ideal plug-flow reactors, with corresponding narrow residence time distribution, which improves reaction selectivity. 

Equations describing velocity profiles can be used, among other applications, to study the effect of different rheological models on the distribution of velocities and to understand the concept of residence-time distribution across the cross-section of a pipe or a channel. 

A parabolic gas velocity profile. As a result of the velocity profile a residence time distribution arises i.e. longer residence time near the wall (less dilution with inert gas) results in higher gas fractions. Parabolic gas velocity profiles are observed by a number of authors in comparable reactor configurations. ... 

The reactor is in laminar flow. The velocity profile gives a distribution of residence times and thus different extents of reaction for various portions of the fluid. Molecular diffusion is negligible, and thermal diffusion must also be negligible if the reaction is nonisothermai. The performance of the reactor is generally worse than that of piston flow with the same value of I but better than that of a CSTR. 

Calculate the residence-time distribution (RTD) for a tubular reactor undergoing steady, laminar flow (Hagen-Poiseuille flow). The velocity profile for Hagen-Poiseuille flow is 2, p. 51]... 

The laminar-flow reactor with segregation and negligible molecular diffusion of species has a residence-time distribution which is the direct result of the velocity profile in the direction of flow of elements within the reactor. To derive the mixing model of this reactor, let us start with the definition of the velocity profile. 

The way we have presented the one-dimensional dispersion model so far has been as a modification of the plug-flow model. Hence, u is treated as uniform across the tubular cross section. In fact, the general form of the model can be applied in numerous instances where this is not so. In such situations the dispersion coefficient D becomes a more complicated parameter describing the net effect of a number of different phenomena. This is nicely illustrated by the early work of Taylor [G.I. Taylor, Proc. Roy. Soc. (London), A219, 186 (1953) A223, 446 (1954) A224, 473 (1954)], a classical essay in fluid mechanics, on the combined contributions of the velocity profile and molecular diffusion to the residence-time distribution for laminar flow in a tube. 

With a packed-bed reactor, the velocity profile is complex and changing with distance, as the fluid flows around and between the particles. However, when the bed depth is many times the particle diameter (L/dp > 40), the residence time distribution of the fluid is quite narrow, and plug flow can be assumed. 

A complete analytical examination of the role of distribution of the flow velocity over the radius of a tube is obviously impossible. A formulated problem for a complete description of the flow of rheokinetic liquid seems to be quite difficult and it is clear that the first steps in investigating a two-dimensional flow were based on very simple assumptions. In a number of works [43,44], the authors took a fixed parabolic profile which is incorrect in principle for the flow of polymerizing media and leads to important mistakes. This is demonstrated very well in Ref. [45] where the possibility for styrene polymerization in a tubular reactor has been estimated it hse been shown that, if a real distribution of flow velocities and residence times over the radius is taken into account, the answer must be negative, in Ref. [44] however, a positive answer is obtained for an a priori parabolic profile. 

It is known that microchannel reactors, due to their small dimension and well defined structure have many advantages compared to conventional fixed bed reactors. The main ones are an efficient temperature control and well defined flow patterns. As the channel diameters are in the order of micrometers, microreactors operate under laminar flow conditions resulting in a parabolic velocity profile. But, due to the short radial diffiision times the radial concentration profile is flat, resulting in a narrow residence time distribution of the reactant. The latter characteristic is of crucial importance in the actual sbufy. Only reactors with an uniform residence time can be used to get meaningful kinetics information under periodic operation at short cycle periods [9]. 

This section will analyze how the velocity profiles, axial mixing, and residence time distribution are related. It will be shown why simple conveying screws have poor axial mixing capability. New mixer geometries that are specifically designed to improve backmixing will be discussed. 

Knowledge of the residence time distribution (RTD) of an extruder provides valuable information about the details of the conveying process in the machine. The RTD is directly determined by the velocity profiles in the machine. Thus, if the velocity profiles are known, the RTD can be calculated. Various workers have made theoretical calculations of the RTD in singie screw extruders [48-50]. Obviously, theoretical calculations of the RTD require knowledge of the velocity profiles in the machine. Thus, the predicted RTD is only as accurate as the velocity profiles that form the basis of the calculations. In single screw extruders, the velocity profiles can be determined reasonably well, although usually a substantial number of simplifying assumptions are made. In other screw extruders, e.g., twin screw extruders, calculation of velocity profiles is rather complex and thus prediction of the RTD more difficult. 

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