Diffusion coefficient, effective axial

The mixing effect is a combination of turbulence and molecular diffusion. Both are first order transport processes therefore we define, in analogy to the diffusion coefficient, the axial dispersion or mixing coefficient 0 as follows... 

Table 1.6 Characteristic quantities to be considered for micro-reactor dimensioning and layout. Steps 1, 2, and 3 correspond to the dimensioning of the channel diameter, channel length and channel walls, respectively. Symbols appearing in these expressions not previously defined are the effective axial diffusion coefficient D, the density thermal conductivity specific heat Cp and total cross-sectional area S, of the wall material, the total process gas mass flow m, and the reactant concentration Cg [114].

Figure 8. Effective axial diffusivity coefficients for solids mixing, (From DeGroot, 1967.)...

In considering axial dispersion as a diffusive flow, we assume that Fick s first law applies, with the diffusion or effective diffusion coefficient (equation 8.5-4) replaced by an axial dispersion coefficient, D,. Thus, for unsteady-state behavior with respect to a species A (e.g., a tracer), the molar flux (NA) of A at an axial position x is... 

Lastly, we studied the effect of 7-stress on the effective time to steady state and the corresponding magnitude of the peak hydrogen concentration. We found that a negative T -stress (which is the case for axial pipeline cracks) reduces both the effective time to steady state and the peak hydrogen concentration relative to the case in which the T -stress effect is omitted in a boundary layer formulation under small scale yielding conditions. This reduction is due to the associated decrease of the hydrostatic stress ahead of the crack tip. It should be noted that the presented effective non-dimensional time to steady state r is independent of the hydrogen diffusion coefficient D 9. Therefore, the actual time to steady state is inversely proportional to the diffusion coefficient (r l/ ). 

Taylor (T2) and Westhaver (W5, W6, W7) have discussed the relationship between dispersion models. For laminar flow in round empty tubes, they showed that dispersion due to molecular diffusion and radial velocity variations may be represented by flow with a flat velocity profile equal to the actual mean velocity, u, and with an effective axial dispersion coefficient Djf = However, in the analysis, Taylor... 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

With turbulence, there is an effective axial dispersion coefficient 3, called Aris Taylor diffusion, which is driven by the turbulent eddies,... 

The longitudinal diffusion coefficient D has been formulated by the hole theory in Sect. 6.3.2. If the similarity ratio X in this theory is chosen to be 0.025 for the rod with the axial ratio 50, Eq. (58) with Eq. (56) gives the solid curve in Fig. 16a. Though it fits closely the simulation data, the chosen X is not definitive because the change in D(l is small and the definition of the effective axial ratio is ambiguous. Though not shown here, Eq. (53) for D, by the Green function method describes the simulation data equally well if P and C, are chosen to be 1000 and 1, respectively. 

It should be pointed out that for a low pressure gas the radial- and axial diffusion coefficients are about the same at low Reynolds numbers (Rediffusion effects may be important at velocities where the dispersion effects are controlled by molecular diffusion. For Re = 1 to 20, however, the axial diffusivity becomes about five times larger than the radial diffusivity [31]. Therefore, the radial diffusion flux could be neglected relative to the longitudinal flux. If these phenomena were also present in a high-pressure gas, it would be true that radial diffusion could be neglected. In dense- gas extraction, packed beds are operated at Re > 10, so it will be supposed that the Peclet number for axial dispersion only is important (Peax Per). 

Axial, film, and macropore Maxwell and Knudsen diffusion coefficients are estimated based on relatively standard formulations ( 6, 7, 8, 9), using estimates of physical properties shown in Table I to compute the diffusivity values for the systems studied. The effect of errors in the estimated values of these properties will be discussed later. 

G. I. Taylor s concept of the effective axial diffusion coefficient, which has proved so useful in combining variable axial advection with radial transfer into one parameter, works best when there is no exchange of a passive tracer with the pipe walls. An analogue of his method, which should be applicable when development lengths are large and there is exchange at the wall, has yet to be provided. It would be of great value. 

Unfortunately the theory, when used for data interpretation, is based on many simplifications that are not always valid. The success of the method depends on the significance of the axial dispersion and the interfacial mass transfer and on the accuracy of the description of these effects. A comparative study of different measuring technique by [25] has shown, for the chromatographic method, that experimental uncertainties may lead to significantly broader confidence limits for the diffusion coefficients than for measurements in single pellets in a diffusion cell. 

Axial velocities of particles and effective diffusion coefficients of solids are constant throughout the reactor. 

The general approach for modelling catalyst deactivation is schematically organised in Figure 2. The central part are the mass balances of reactants, intermediates, and metal deposits. In these mass balances, coefficients are present to describe reaction kinetics (reaction rate constant), mass transfer (diffusion coefficient), and catalyst porous texture (accessible porosity and effective transport properties). The mass balances together with the initial and boundary conditions define the catalyst deactivation model. The boundary conditions are determined by the axial position in the reactor. Simulations result in metal deposition profiles in catalyst pellets and catalyst life-time predictions. 

The analysis of outlet peaks is based on the model of processes in the column. Today the Kubi n - Kucera model [14,15], which accounts for all the above-mentioned processes, as long as they can be described by linear (differential) equations, is used nearly exclusively. Several possibilities exist for obtaining rate parameters of intracolumn processes (axial dispersion coefficient, external mass transfer coefficient, effective diffusion coefficient, adsorption/desorption rate or equilibrium constants) from the column response peaks. The moment approach in which moments of the outlet peaks are matched to theoretical expressions developed for the system of model (partial) differential equations is widespread because of its simplicity [16]. The today s availability of computers makes matching of column response peaks to model equations the preferred analysis method. Such matching can be performed in the Laplace- [17] or Fourier-domain [18], or, preferably in the time-domain [19,20]. 

There are four unknown parameters in the theoretical impulse response for porous particles, h(t) the pellet diffusion time, tdif (which contains the effective diffusion coefficient of the pair T-C, Dtc, td.fs R p/D.f( , R is the radius of the pellet equivalent sphere), the mean residence time of the carrier-gas in the interparticle space, tc (tc = v/L with the carrier gas linear interstitial velocity, v, and column length, L), Peclet number, Pe (Pe = L.v/E, with E the effective axial dispersion coefficient) and the adsorption parameter, 5q (see below). Because matching with four unknown parameters would give highly correlated parameters, it is better to determine some parameters independently,... 

The velocity-independent term A characterises the contribution of eddy (radial) diffusion to band broadening and is a function of the size and the distribution of interparticle channels and of possible non-uniformiiies in the packed bed (coefficient A.) it is directly proportional to the mean diameter of the column packing particles, dp. The term B describes the effect of the molecular (longitudinal) diffusion in the axial direction and is directly proportional to the solute diffusion coefficient in the mobile phase, D, . The obstruction factor y takes into account the hindrance to the rate of diffusion by the particle skeleton. 

In the bubble column the velocity profile of recirculating liquid is shown in Fig. 27, where the momentum of the mixed gas and liquid phases diffuses radially, controlled by the turbulent kinematic viscosity Pf When I/l = 0 (essentially no liquid feed), there is still an intense recirculation flow inside the column. If a tracer solution is introduced at a given cross section of the column, the solution diffuses radially with the radial diffusion coefficient Er and axially with the axial diffusion coefficient E. At the same time the tracer solution is transported axially Iby the recirculating liquid flow. Thus, the tracer material disperses axially by virtue of both the axial diffusivity and the combined effect of radial diffusion and the radial velocity profile. 

As for axial diflfusivity included in Eq. (4-2), the situation is not much different from that for radial diflfusivity. Pozin et al. (P6) report that the mean value of axial diflfusivity is 2.5 times the value of Er, when radial and axial diffusion are assumed to be homogeneous and nonisotropic throughout the bubble column. Hence it is reasonable to assume for the local axial diffusion coefficient Ez = Ipr, with i of order unity. On the other hand, one has the relations that Ey = Vm/cL [see Eq. (4-4)] and = au so that E equals the product (aC/eJvf Accordingly, the cross-sectionally averaged axial diflfusivity E is expressed by the relation Ez = where the overbar shows an effective mean value. 

Increasing surfactant concentrations in the aeration cell has been found to decrease bubble diameter, bubble velocity, axial diffusion coefficient, but increase bubble s surface-to-volume ratio, and total bubble surface area in the system. The effect of a surface-active agent on the total surface area of the bubbles is also a function of its operating conditions. The surfactant s effect is pronounced in the case of a coarse gas diffuser where the chances of coalescence are great and the effectiveness of a surface-active solute in preventing coalescence increases with the length of its carbon chain. 

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