Transversal diffusive time

The representative case considered in Taylor (1993) is his case (B), where the longitudinal transport time L/uq is much bigger than the transversal diffusive time fl /D. The problem of a diffusive transport of a solute was studied experimentally and analytically. Two basically different cases were subjected to experimental verification in Taylor s paper ... 

Here, p is the local (transverse) Peclet number, which is the ratio of transverse diffusion time to the convection time. Per is the radial Peclet number (ratio of transverse diffusion time to a convection time based on pipe radius). We assume that p <4 1 while Per is of order unity. (Remark The parameter Pe /p — ux)L/Dm is also known as the axial Peclet number. Also note that for any finite Per or tube diameter, the axial Peclet number tends to infinity as p tends to zero.) When such scale separation exists, we can average the governing equation over the transverse length scale using the L-S technique and obtain averaged model in terms of axial length and time scales. 

Here, C( , z, t) is the scaled solute concentration in the fluid phase, Cw the solute concentration at the wall, 6 the normalized adsorbed concentration (O<0< 1), K the adsorption equilibrium constant, p the transverse Peclet number, T represents the adsorption capacity (ratio of adsorption sites per unit tube volume to the reference solute concentration), and Da is the local Damkohler number (ratio of transverse diffusion time to the characteristic adsorption time). We shall assume that p 4Cl while T and Da are order-one parameters. (In physical terms, this implies that transverse molecular diffusion and adsorption processes are much faster compared to the convection.)... 

For the case of p — 0 (which corresponds to adsorption, desorption, and transverse diffusion time scales going to zero), we have... 

Extrapolation, to liquid density, of thermalization time in gaseous water (also approximate) by Christophorou et al. (1975), based on drift velocity and transverse diffusion coefficient measurement, which gives tth 2.0 x 10-1+ s... 

In porous media the flow of water and the transport of solutes is complex and three-dimensional on all scales (Fig. 25.1). A one-dimensional description needs an empirical correction that takes account of the three-dimensional structure of the flow. Due to the different length and irregular shape of the individual pore channels, the flow time between two (macroscopically separated) locations varies from one channel to another. As discussed for rivers (Section 24.2), this causes dispersion, the so-called interpore dispersion. In addition, the nonuniform velocity distribution within individual channels is responsible for intrapore dispersion. Finally, molecular diffusion along the direction of the main flow also contributes to the longitudinal dispersion/ diffusion process. For simplicity, transversal diffusion (as discussed for rivers) is not considered here. The discussion is limited to the one-dimensional linear case for which simple calculations without sophisticated computer programs are possible. 

Microfluidic chemical processes are based on transverse molecular diffusion to the microchannel. In microspace, because the diffusion distance is short, rapid chemical processes are expected there. In order to clarify the time required for chemical processes in microspace, diffusion time f in one... 

Using a (radius of the pipe) and L (length of the pipe) as the characteristic lengths in the transverse and axial directions, respectively, CR as the reference concentration, and Dm R as the reference molecular diffusivity, we obtain four time scales in the system associated with convection (ic), local/transverse diffusion (tD), axial diffusion tz), and reaction (tR),... 

We have applied the discrete-exchange model to these data. An exchange between Na ion under a particular slow-motion condition and in the extreme narrowing limit is assumed. Transverse relaxation time and diffusion coefficient are written as follows ... 

Physical parameters Molecular 0.1-1 nm Dipole-dipole interaction Second moment, fourth moment of lineshape Incoherent magnetization transfer characteristic times for cross-polarization and exchange Mesoscopic lnm-0.1 p,m Longitudinal relaxation time Ti Transverse relaxation time T2 Relaxation time Tip in the rotating frame Solid-echo decay time T2e Spin-diffusion constant Microscopic 0.1-lOp.m Molecular self-diffusion constant D Macroscopic 10 p,m and larger Spin density... 

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