Diffusion characteristic time

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

Fig. 6. A typical correlation function obtained for IFABP in 20 mM phosphate buffer at pH 7.3 at room temperature. The experiment was performed using a ConfoCor 2 LSM combination instrument (Carl Zeiss-Evotec, Jena, Germany) and the correlation function data (G(t)) were fitted to the form G(r) = G(0)/(1 + x/rD), where Td is the characteristic diffusion time. An additional exponential component improves the fit. 

In order to compare a number of different zeolite preparations we have found it convenient to determine not the diffusivity of o-xylene per se, but to characterize the samples by measuring the time (tQ 3) it takes to sorb 30% of the quantity sorbed at infinite time. The characteristic diffusion time, t0 3, is a direct measure of the critical mass transfer property r2/D ... 

Tj-hem TD = to2/4Dt the chemical relaxation time is much larger than the characteristic diffusion time so that there is no chemical exchange during diffusion through the excitation volume. The autocorrelation function is then given by... 

Characteristic diffusion time Characteristic hydrodynamic time... 

The maximum value of the diffusivity occurs when zJzi — 0.5 and has a magnitude 0.21w.Zj. For typical meteorological conditions this corresponds to a diffusivity of 0(100 m sec" ) and a characteristic diffusion time defined by of 0(5zi/w ). Yamada (1977), for example, has observed dififusivities of 0(100 m sec" ) when simulating the Wangara day 34 field experiment. Above the surface layer the observational evidence is inadequate to verify more than an order of magnitude estimate of the diffusivity. Clearly there is a need for more field data to establish the shape of the profile in the upper portions of the mixed layer. 

The above models describe a simplified situation of stationary fixed chain ends. On the other hand, the characteristic rearrangement times of the chain carrying functional groups are smaller than the duration of the chemical reaction. Actually, in the rubbery state the network sites are characterized by a low but finite molecular mobility, i.e. R in Eq. (20) and, hence, the effective bimolecular rate constant is a function of the relaxation time of the network sites. On the other hand, the movement of the free chain end is limited and depends on the crosslinking density 82 84). An approach to the solution of this problem has been outlined elsewhere by use of computer-assisted modelling 851 Analytical estimation of the diffusion factor contribution to the reaction rate constant of the functional groups indicates that K 1/x, where t is the characteristic diffusion time of the terminal functional groups 86. ... 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

The characteristic diffusion time for any UME geometry where the transition from semi-infinite linear diffusion (transient) to hemispherical or spherical diffusion (steady state) occurs may be given as... 

A prominent trade-off in fixed-bed reactor design concerns the catalyst particle size. What is the basis for the choice of a certain particle size When the catalyst performance is to be optimized, the application of the Thiele model helps to provide an answer (Figure 7). The Thiele modulus accounts for the competition between the chemical reaction and the limitation of transport of reactants by diffusion in a porous catalyst particle. It is defined as the square root of the ratio of the characteristic diffusion time fo = L /D and the characteristic reaction time (r. For a... 

The fluorescence fluctuations measured by FCS can be analyzed in several ways. The most common technique, autocorrelation analysis, provides information about characteristic diffusion time of fluorescent molecules through the observation volume. It also reports on the average number of molecules present in the observation volume, and thus the concentration of fluorescent moleculesn (14, 49, 56, 57). Other types of FCS analysis can be used to analyze molecular brightness and the oligomeric state of the fluorescent molecule. Finally, cross-correlation FCS monitors fluctuations jointly from molecules labeled with two or more different fluorophores. This technique provides a powerful approach to assay for intermolecular interactions, because molecules that are bound either directly or indirectly to one another should diffuse as a single unit (8, 59). 

Other dimensionless groups similar to the Deborah number are sometimes used for special cases. For example, in a steady shearing flow of a polymeric fluid at a shear rate y, the Weissenberg number is defined as Wi = yr. This group takes its name from the discoverer of some unusual effects produced by normal stress differences that exist in polymeric fluids when Wi 1, as discussed in Section 1.4.3. Use of the term Weissenberg number is usually restricted to steady flows, especially shear flows. For suspensions, the Peclet number is defined as the shear rate times a characteristic diffusion time to [see Eq. (6-12) and Section 6.2.2]. 

Here, Qa and Qb account for the different optical properties of the fluo-rophores that distinguish A and B as well as the laser intensity and other instrumental factors. K = kab/kba, and td = w jAD is the characteristic diffusion time for a Gaussian excitation intensity profile with exp(—2) radius w, and S is the area of the laser spot. It is readily shown that for equilibrium systems Gab (t) = Gba (t) due to the fact that kabCB = kbaC [25]. The NESS fluxes can be obtained from the initial slope of the correlation functions. 

Another criterion for predicting if the transport in polymeric gels is controlled by diffusion (Fickian) or by relaxation, is to determine the diffusional Deborah number De), which is a ratio between the characteristic polymer relaxation time of the polymer (2) when it is subject to a swelling stress and a characteristic diffusion time (6), defined as the coefficient between the square of the sample thickness (h) and the coefficient of water diffusion in the polymeric gel... 

The incorrectness of the steady-state approach was noted by the authors of Ref. 3 and can be explained as follows. There are two characteristic diffusion times in an FFF channel = wV4D and 1 2 = b l4D, where D is the diffusion coefficient of solute molecules. As the ratio b/w > 40, then > 1600. Experimental values of retention time are usually equal to several but are never as large as IbOOt i- Thus, the steady-state values of and K2 corresponding to t tj)2 are never reached during the experiment. (For the channel with w = 200 /zm, b = l cm, and solute with D = 10 cm /s,... 

A crude first estimate of the characteristic diffusion time Tesc can be made by neglecting propagation losses for a primary nucleus P and assuming that Qs = 0 for a secondary nucleus S. If we also neglect collision losses by the secondary nucleus after it is produced, then the solution of Eq. 1 is... 

The mixing mechanism in this laminar flow system must be diffusion because the mixing must occur normal to the laminar flow streamlines. Laminar flow prevails in systems with these small dimensions. Thus, the characteristic mixing time is the characteristic diffusion time, defined for two colaminar streams mixing in a cylindrical channel as ... 

Tj, bulk temperature in the reaction fluid (K) tjj characteristic diffusion time (s)... 

Table 16.2 is based on the properties of water. Suppose that the fluid is air at 1 atm. Calculate the Reynolds number and the characteristic diffusion time for the various duct sizes. The velocity in the 2-m pipe remains 1 m s . ... 

The ratio of the characteristic relaxation time to the characteristic diffusion time is the so-called Deborah number (De). The smaller De is, the more fluidic the material appears. 

Ihble 14-10. Amplification factor and characteristic diffusion time in the lactate recycling system as a function of the enzyme loading. 

With this diffusion coefficient, assuming a 100-/rm thick membrane, the characteristic diffusion time is 10 days t = L 2j2D). 

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