Diffusion-dispersion time constant

A breakthrough curve with the nonretained compound was carried out to estimate the axial dispersion in the SMB column. A Peclet number of Pe = 000 was found by comparing experimental and simulated results from a model which includes axial dispersion in the interparticle fluid phase, accumulation in both interparticle and intraparticle fluid phases, and assuming that the average pore concentration is equal to the bulk fluid concentration this assumption is justified by the fact that the ratio of time constant for pore diffusion and space time in the column is of the order of 10. ... 

Diffusion and dispersion processes can be characterised by a time constant for the process, given by... 

The aimealing kinetics of the light-induced defects are shown in Fig. 6.29. Several hours at 130 °C are needed to anneal the defects completely, but only a few minutes at 200 C. The relaxation is nonexponential, and in the initial measurements of the decay the results were analyzed in terms of a distribution of time constants, Eq. (6.78) (Stutzmann, Jackson and Tsai 1986). The distribution is centered close to 1 eV with a width of about 0.2 eV. Subsequently it was found that the decay fits a stretched exponential, as is shown in Fig. 6.29. The parameters of the decay-the dispersion, p, and the temperature dependence of the decay time, t - are similar to those found for the thermal relaxation data and so are consistent with the same mechanism of hydrogen diffusion. The data are included in Fig. 6.23 which describes the general relation between x and D,. The annealing is therefore the process of relaxation to the equilibrium state with a low defect density. 

Figure 2 shows the spectral response functions (5,(r), Eq. 1) derived firom the spectra of Fig. 1. In order to adequately display data for these varied solvents, whose dynamics occur on very different time scales, we employ a logarithmic time axis. Such a representation is also useful because a number of solvents, especially the alcohols, show highly dispersive response functions. For example, one observes in methanol significant relaxation taking place over 3-4 decades in time. (Mdtiexponential fits to the methanol data yield roughly equal contributions from components with time constants of 0.2, 2, and 12 ps). Even in sinqrle, non-associated solvents such as acetonitrile, one seldom observes 5,(r) functions that decay exponentially in time. Most often, biexponential fits are required to describe the observed relaxation. This biexponential behavior does not reflect any clear separation between fast inertial dynamics and slower diffusive dynamics in most solvents. Rather, the observed spectral shift usually appears to sirrply be a continuous non-exponential process. That is not to say that ultrafast inertial relaxation does not occur in many solvents, just that there is no clear time scale separation observed. Of the 18 polar solvents observed to date, a number of them do show prominent fast components that are probably inertial in origin. For example, in the solvents water [16], formamide, acetoniuile, acetone, dimethylformamide, dimethylsulfoxide, and nitromethane [8], we find that more than half of the solvation response involves components with time constants of 00 fs.

Obviously, the assumption of no lateral exchange between the subchannels is, strictly spoken, only (approximately) valid for a relative pitch of 1 however, for higher relative pitches it might be valid if the ratio of the residence time in the reactor to the time constant for lateral difhision/dispersion is much smaller than 1. For lateral diffusion/dispersion over a distance equal to the pitch (s) or the width of the reactor (d ), these ratios or Fourier numbers are, respectively. 

The mechanism of the exchange reaction can thus be deduced. PPP fixim the crystals is dissolving and diffusing from the surfaces into the oil. At the same time, the dissolved C-PPP in the oil is diffusing toward the surface of the crystal and recrystallizing in the crystals. The total concentration of PPP plus -PPP in the oil remains constant (note that the solution is always saturated). However the amount of dissolved -PPP in the dispersion decreases constantly. If we assumed that the exchange rate is constant, the rate of loss of -PPP should be directly proportional to the -PPP concentration in solution. 

The rate of convective mass transfer relative to the rate of mass transfer via interpellet axial dispersion is eqnivalent to the ratio of the diffusion time constant relative to the residence time for convective mass transfer. The interpellet Damkohler nnmber for reactant A is... 

In the frequency response method, first applied to the study of zeolitic diffusion by Yasuda [29] and further developed by Rees and coworkers [2,30-33], the volume of a system containing a widely dispersed sample of adsorbent, under a known pressure of sorbate, is subjected to a periodic (usually sinusoidal) perturbation. If there is no mass transfer or if mass transfer is infinitely rapid so that gas-solid mass-transfer equilibrium is always maintained, the pressure in the system should follow the volume perturbation with no phase difference. The effect of a finite resistance to mass transfer is to cause a phase shift so that the pressure response lags behind the volume perturbation. Measuring the in-phase and out-of-phase responses over a range of frequencies yields the characteristic frequency response spectrum, which may be matched to the spectrum derived from the theoretical model in order to determine the time constant of the mass-transfer process. As with other methods the response may be influenced by heat-transfer resistance, so to obtain reliable results, it is essential to carry out sufficient experimental checks to eliminate such effects or to allow for them in the theoretical model. The form of the frequency response spectrum depends on the nature of the dominant mass-transfer resistance and can therefore be helpful in distinguishing between diffusion-controlled and surface-resistance-controlled processes. 

D, which has the same dimension unit as the molecular diffusion coefficient D. Usually is much larger than because it incorporates all effects that may cause deviation from plug flow, such as radial velocity differences, eddies, or vortices. The key parameter determining the width of the RTD is the ratio between the axial dispersion time and the space-time r, which corresponds to the mean residence time in the reactor t at constant fluid density. This ratio is often called Bodenstein number Bo). 

As was shown earlier, the presence of the CPE of fractal impedance produces a distribution of the time constants. In addition, other elements such as the Warburg (semi-infinite or finite-length) linear or nonlinear diffusion, porous electrodes, and others also produce a dispersion of time constants. Knowledge about the nature of such dispersion is important in the characterization of electrode processes and electrode materials. Such information can be obtained even without fitting the experimental impedances to the corresponding models, which might be still unknown. Several methods allow for the determination of the distribution of time constants [378, 379], and they will be briefly presented below. 

The most commonly used technique for determining 5 is photon correlation spectroscopy (PCS) [also known as quasi-elastic light scattering (QELS)]. PCS has become one of the standard tools of the trade for the colloid chemist. In this technique concentration fluctuations arising from the diffusive motion of the dispersion particles give rise to fluctuations in the dielectric constant of the medium are monitored photometrically. These fluctuations decay exponentially with a time constant related to the diffusion coefficient, Ds, of the scatterer, which can in turn be related to its hydrodynamic radius through the Stokes-Einstein equation ... 

In the case of quasi-elastic light scattering from a mono-dispersed polymer in solution, the time constant for the autocorrelation function appropriate to translational diffusion is given by ... 

The dispersion model with particle diffusion always assumes complete external contacting of particles by liquid which may not be the case in trickle flow. This means that the effective diffusional time constant is increased in trickle flow resulting in a reduced apparent effective diffusivity which is based on the total external surface area. Using this diffusivity in the expression for the Thiele modulus, and equating it to the modulus defined for trickle-bed operation by Dudukovic (130) results in the following estimate of the external catalyst contacting efficiency, UcE ... 

As the particles in a coUoidal dispersion diffuse, they coUide with one another. In the simplest case, every coUision between two particles results in the formation of one agglomerated particle,ie, there is no energy barrier to agglomeration. Applying Smoluchowski s theory to this system, the half-life, ie, the time for the number of particles to become halved, is expressed as foUows, where Tj is the viscosity of the medium, k Boltzmann s constant T temperature and A/q is the initial number of particles. 

A monolithic system is comprised of a polymer membrane with dmg dissolved or dispersed ia it. The dmg diffuses toward the region of lower activity causiag the release of the dmg. It is difficult to achieve constant release from a system like this because the activity of the dmg ia the polymer is constantly decreasiag as the dmg is gradually released. The cumulative amount of dmg released is proportional to the square root of time (88). Thus, the rate of dmg release constantly decreases with time. Again, the rate of dmg release is governed by the physical properties of the polymer, the physical properties of the dmg, the geometry of the device (89), and the total dmg loaded iato the device. 

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