Residence time distribution molecular diffusion

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some rough correlations of the Peclet number are proposed by Wen (in Petho and Noble, eds.. Residence Time Distribution Theory in Chemical Tngineeiing, Verlag Chemie, 1982), including some for flmdized beds. Those for axial dispersion are ... 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

The pilot reactor is a tube in isothermal, laminar flow, and molecular diffusion is negligible. The larger reactor wiU have the same value for t and will remain in laminar flow. The residence time distribution will be unchanged by the scaleup. If diffusion in the small reactor did have an influence, it wiU lessen upon scaleup, and the residence time distribution will approach that for the diffusion-free case. This wiU hurt yield and selectivity. 

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. 

The dispersion model assumes that the residence-time distribution of a real tubular reactor can be regarded as the superimposition of the plug flow that is characteristic of the ideal tubular reactor and diffusionlike axial mixing, characterized by an axial dispersion coefficient which has the same dimensions as, but can be much larger than, the molecular diffusion coefficient. The following effects can contribute to the axial mixing ... 

Residence time distribution can be an important issue in the selection process. Microreactors usually operate at Reynolds numbers lower than 200. In this regime, laminar flow prevails and mass transfer is dominated by molecular diffusion. An injected substance in the channel will dissipate caused by the flow profile in the channel. Hence the input signal will be broadened until it reaches the exit of the channel (Figure 3.2). The extent of such a distribution depends on the channel design. In microchannels the mixing process can then be described by the Fourier number (no axial diffusion, dominating radial diffusion D ). A high Fourier Po number leads to a narrow residence time distribution ... 

Equation (4.10.117) expresses that a tubular reactor is formally represented by axial dispersion superimposed on plug flow. The dispersion coefHcient is then needed, for example, based on the measurement of the residence time distribution or based on correlations for D x- Note that D x considers axial convection, molecular diffusion, and the effect of the radial velocity profile. 

Mass Transport and Residence Time Distribution in Microchannels In microchannels, we have a strong laminar flow as the Re number is typically in the range 10-500 (Emig and Klemm, 2005). Thus, at first sight, one could come to the conclusion that we have no plug flow behavior and a lower conversion compared to an ideal PFR (Section 4.10.2.7). However, this is not the case if molecular diffusion in the radial direction is relatively fast, that is, the time for radial diffusion is much shorter than the average residence time (Section 4.10.6) ... 

Figure 4.10..80 Typical residence time distribution in a microreactor (with a liquid phase) and a laminar tube without influence of molecular diffusion. Comparison with Figure 4.10.54 shows that the Bo number for the microreactor is >80 and, thus, we have almost plug flow. Adapted from Emig and Klemm (2005).

Radial mixing by eddies and by diffusion, which reduces radial concentration gradients, and con uently reduces the residence time distribution on the molecular scale, particularly in turbulent flow. 

In fact the viscosity influences both the heat balance and the mass balance. It has been shown how the heat transfer coefficient is affected by the viscosity. But the energy dissipation by the stirrer is also strongly dependent on viscosity (see Section 11.4.4). Furthermore, viscosity affects the molecular diffusion, the mass transport, the mixing time, or the residence time distribution, and therefore the reaction rate. Since the reaction rates influence the chain length and particle sizes, they have a direct effect on the polymer properties. In turn they affect the viscosity and the shear forces - there is a feedback effect. Such complex interactions cannot be described by analytical equations, so empirical models must be used. Often... 

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if tlie Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m /sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

We have shown previously that non-Flory distributions often reflect the transport-limited removal of reactive olefins from catalyst pellets on Ru and Co catalysts (4,5,14,40,41,44). This proposal is consistent with the similar effects of bed residence time and of molecular size on chain growth probability and product functionality. It accounts for the observed effects of convective and diffusive rates of reactive olefins and for the non-Flory distribution of highly paraffinic hydrocarbons formed from synthesis gas on Co and Ru catalysts. 

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