Axial diffusion distribution

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. 

The axial voidage distribution resulting from mixing of dissimilar particles, as shown in Fig. 26, will now be examined in the light of the dynamics of these particles at the interfacial boundary between the properly juxtaposed phases of a binary particle mixture (Lou, 1964 Kwauk, 1973). Both the lighter particles at the top and the heavier particles at the bottom share the same tendency of invading the region occupied by the other. This behavior conforms to the concept of random walk, for which Fick s law can be adapted to describe the macroscopic diffusion flux of the smaller particles 2 ... 

We now consider the case of steady back-mixing experiment to measure the total axial diffusivity A small amount of fresh liquid is continuously introduced into the column cocurrently or countercurrently with the gas flow. A still smaller amount of tracer liquid is steadily introduced at a downstream section. Under such circumstances the concentration distribution of the tracer at a given cross section is steady, and the axial concentration gradient dc/dz is constant radially. As a consequence, Eq. (4-3) simplifies to... 

Distributed parameter, nonlinear, partial differential equations were soloed to describe oxygen transport from maternal to fetal bloody which flows in microscopic channels within the human placenta. Steady-state solutions were obtained to show the effects of variations in several physiologically important parameters. Results reported previously indicate that maternal contractions during labor are accompanied by a partially reduced or a possible total occlusion of maternal blood flow rate in some or all portions of the placenta. Using the mathematical modely an unsteady-state study analyzed the effect of a time-dependent maternal blood flow rate on placental oxygen transport during labor. Parameter studies included severity of contractions and periodicity of flow. The effects of axial diffusion on placental transport under the conditions of reduced maternal blood flow were investigated. 

As time goes on, the action of radial diffusion continues to inhibit axial dispersion by diffusion and convection and makes the mixed zone more uniform, as shown at time tj. Finally, at still larger times a quasi equilibrium is established. Here, convection, radial diffusion, and axial diffusion all contribute to the dispersion, with the net effect appearing as if the fluid were in plug flow, whereas in fact the velocity is radially distributed. With a further increase in time, the effect is only to increase the length of the mixed zone. 

Clearly there is a region between Pe 7 and Pe > 7, with 4L/a > Pe, where both radial and axial diffusion are important. Aris (1956), in a mathematically elegant paper, showed that the governing equation for the mean concentration distribution averaged over the tube cross section can be written in the form of the Taylor dispersion equation, with... 

The Taylor-Aris result can be shown in a somewhat simpler mathematical way by starting with the complete convective diffusion equation (Eq. 4.6.7), including the axial diffusion term. The procedure is essentially the same as Taylor s. Equation (4.6.7) is integrated over the tube cross section, since what is of interest is the average concentration, and the radial concentration distribution is given by Eq. (4.6.21). The replacement of dddx by dctdx is still made. The analysis follows through as before. In addition to requiring we must... 

If there is only one chemical reaction on the internal catalytic surface, then vai = — 1 and subscript j is not required for all quantities that are specific to the yth chemical reaction. When the mass transfer Peclet number which accounts for interpellet axial dispersion in packed beds is large, residence-time distribution effects are insignificant and axial diffusion can be neglected in the plug-flow mass balance given by equation (22-11). Under these conditions, reactor performance can be predicted from a simplified one-dimensional model. The differential design equation is... 

The model is axially distributed which permits the development of concentration gradients in the axial direction in each of the regions. This is accomplished by dividing the capillary-tissue unit into a number of axial segments (A seg)- The number of segments used is under the control of the modeler. The actual length of the capillary-tissue unit need not be known unless axial diffusion is included (see Model Assumptions). 

Concluding, the axial diffusion and nonuniform flow in FAIMS gaps (i) reduce the mean of residence times tjes by 30%, which increases sensitivity and decreases resolution of planar FAIMS and (ii) spread around the mean with hardly any effect on those metrics. The latter applies to operation at fixed or slow-changing Ec but not when Eq varies so fast that A c during compares with the peak widths in FAIMS spectra, in which case the width of distribution is important (4.3.11). Defining the physical limitations on E scan speed is cmcial to acceleration of FAIMS analyses that is topical to many existing and prospective applications. 

The mass stream caused by axial dispersion (sometimes called axial diffusion) in the fluid phase or the distribution of residence times is given by... 

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 ... 

This model divides the slurry bed reactor into several CSTR. The model can be regarded as a quasi axial diffusion model, and the idea of the model was that the interfacial resistance between the gas and liquid can be omitted fluid particle was considered weU-distributed the exit of pre-CSTR is the entrance of the latter stage CSTR. The model is shown in Eqs. (24) and (25) ... 

Fig. 5.26. Note that we qualify the preceding statement by including both ends of the medium. The immediate vicinity of the face 2 = 0 is also a special region of this medium and, from a practical standpoint, can also introduce certain complications. Some of these will be discussed in the next chapter. Suffice it to say for the present that the flux distribution near the face z = 0 involves the higher harmonics. Thus both ends of the medium are poorly described by the simple relation (5.246) and are to be avoided in the experiment when possible. The particular importance of a region such as AS to the diffusion-length experiment is due to the especially simple form of the axial flux distribution in this range. We have noted that in this range the flux distribution is exponential, and the exponent includes the constant k oo which is a function of the diffusion length L. Thus a knowledge of fcsoo yields the value of L.

The anisotropic mass diffusivity D, x is calculated using Eq. (3.37) the axial mass diffusivity is given in Fig. 5.38. Note that the wavy shape of u c and contours is as the result of existing intense wavy axial velocity distribution along radial direction as seen in Fig. 5.37. 

Abstract In this chapter, an exothermic catalytic reaction process is simulated by using computational mass transfer (CMT) models as presented in Chap. 3. The difference between the simulation in this chapter from those in Chaps. 4,5, and 6 is that chemical reaction is involved. The source term in the species conservation equation represents not only the mass transferred from one phase to the other, but also the mass created or depleted by a chemical reaction. Thus, the application of the CMT model is extended to simulating the chemical reactor. The simulation is carried out on a wall-cooled catalytic reactor for the synthesis of vinyl acetate from acetic acid and acetylene by using both c — Sc model and Reynolds mass flux model. The simulated axial concentration and temperature distributions are in agreement with the experimental measurement. As the distribution of lx shows dissimilarity with Dj and the Sci or Pri are thus varying throughout the reactor. The anisotropic axial and radial turbulent mass transfer diffusivities are predicted where the wavy shape of axial diffusivity D, along the radial direction indicates the important influence of catalysis porosity distribution on the performance of a reactor. 

Note The extemal-age distribution function for a laminar flow tubular reactor with no radial or axial diffusion is... 

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

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