Transport roots, velocity

In this section we shall be concerned with a molecular theory of the transport properties of gases. The molecules of a gas collide with each other frequently, and the velocity of a given molecule is usually changed by each collision that the molecule undergoes. However, when a one-component gas is in thermal and statistical equilibrium, there is a definite distribution of molecular velocities—the well-known Maxwellian distribution. Figure 1 shows how the molecular velocities are distributed in such a gas. This distribution is isotropic (the same in all directions) and can be characterized by a root-mean-square (rm speed u, which is given by... 

In an aquifer, the total Fickian transport coefficient of a chemical is the sum of the dispersion coefficient and the effective molecular diffusion coefficient. For use in the groundwater regime, the molecular diffusion coefficient of a chemical in free water must be corrected to account for tortuosity and porosity. Commonly, the free-water molecular diffusion coefficient is divided by an estimate of tortuosity (sometimes taken as the square root of two) and multiplied by porosity to estimate an effective molecular diffusion coefficient in groundwater. Millington (1959) and Millington and Quirk (1961) provide a review of several approaches to the estimation of effective molecular diffusion coefficients in porous media. Note that mixing by molecular diffusion of chemicals dissolved in pore waters always occurs, even if mechanical dispersion becomes zero as a consequence of no seepage velocity. 

It is the fluctuating element of the velocity in a turbulent flow that drives the dispersion process. The foundation for determining the rate of dispersion was set out in papers by G. 1. Taylor, who first noted the ability of eddy motion in the atmosphere to diffuse matter in a manner analogous to molecular diffusion (though over much larger length scales) (Taylor 1915), and later identified the existence of a direct relation between the standard deviation in the displacement of a parcel of fluid (and thus any affinely transported particles) and the standard deviation of the velocity (which represents the root-mean-square value of the velocity fluctuations) (Taylor 1923). Roberts (1924) used the molecular diffusion analogy to derive concentration profiles... 

The transport equations describing the instantaneous behavior of turbulent liquid flow are three Navier-Stokes equations (transport of momentum corresponding to the three spatial coordinates r, z, in a cylindrical polar coordinate system) and a continuity equation. The instantaneous velocity components and the pressure can be replaced by the sum of a time-averaged mean component and a root-mean-square fluctuation component according to Reynolds. The resulting Reynolds equations and the continuity equation are summarized below ... 

I think it is fair to say that the merits and demerits of DPD are still debated. In my opinion, the DPD technique does have a problem with the hydrodynamics, which relaxes in the same time and distance scale as the dissolved particles. In reality, because of the near incompressibility of the solvent, the hydrodynamics relaxes essentially instantaneously on that particle s timescale of structural evolution. One other problem of the technique, as pointed out by Marsh and Yeomans, is that the temperature of the system depends on the value of the time step (as the dissipative force is inversely proportional to the square root of the time step). In an interesting article, Lowe looked at DPD from the perspective of another thermostatting procedure, but which conserves momentum and enhances the viscosity. Besold et al. examined the various integration schemes used in DPD and found differences in the response fimctions and transport coefficients. These artefacts can be largely suppressed by using velocity-Verlet-based schemes in which the velocity dependence of the dissipative forces is taken into account. 

Were it the case that auxin transport takes place within a plant part—a stem, a petiole, a root—by travelling in a solution of constant concentration (density) in a continuous stream of constant velocity (speed), then, the efficiency of the... 

Measurements and calculations of velocities, densities, and intensities of transport to characterize hormone translocation usually imply that these quantities be constant and that the hormone moves in a stream. They do not, however, allow for degradation and/or immobilization, i.e., leakage of molecules out of the stream, to take place. Yet such phenomena do take place, and do vary with time and distance from the hormone source. Variations in the density of mobile auxin have been demonstrated even within short transport sections (e.g., Kaldewey 1963 in Fritillaria axes Newman 1965, 1970 in Avena coleoptiles Kaldewey and Kraus 1972 in Gossypium seedlings Kaldewey etal. 1974 in Pisum internodes Kaldewey 1976 in Tulipa axes). The commonly observed decline of mobile auxin as a function of distance from the auxin source indicates that not all auxin molecules move with the same velocity. The same conclusion may be drawn from the tpyical initial gradual increase of hormone flux into basal receivers which occurs before linearity of the time course is reached (e.g., Hertel 1962, Hertel and Leopold 1963, de la Fuente and Leopold 1973 in Helianthus hypocotyls McCready and Jacobs 1963 a, b, in petioles and Smith and Jacobs 1968 in hypocotyls of Phaseolus de la Fuente and Leopold 1966 in Coleus internodes Thornton and Thimann 1967 in Avena coleoptiles Greenwood and Goldsmith 1970 in Pinus embryonic hypocotyls Wilkins and Cane 1970, Wilkins etal. 1972 and Shaw and Wilkins 1974 in Zea roots Kaldewey et al. 1974 in Pisum internodes Tsurumi and Ohwaki 1978 in Vida roots). 

Consequently, the time-dependent fluid front position in a surface-directed microfluidic device tends to be dependent on system geometry, intrinsic fluid properties, fluid-substrate interactions and the square root of time. This dependency has been derived theoretically and observed for a number of capillary-driven microfluidic systems such as the v-groove geometry [5, 6]. Noting that the ratio yZ/u. is a characteristic capillary velocity, Uq, enables the rearrangement UcLcosOt) /. This relation perhaps leads to a more tangible description of fluid specific capillary transport. 

There are two distinct regimes (1) the high U regime, which covers most practical flow rates experienced in microfluidic fuel cells and (2) the low U regime for low flow rates in channels with small hydrauhc diameter. In the high U regime, the transport limited current is proportional to the cubic root of the mean velocity [20] ... 

How wide a range of linear velocity must be studied The following table shows the results of a calculation of the effect of linear velocity on the fractional conversion from an ideal PFR. The reaction considered is A R, which was assumed to be irreversible and first order in A, with A r = 0. The reactor was assumed to be isothermal. The mass-transfer coefficient was assumed to be proportional to the square root of linear velocity, a relationship that is reasonably typical for flow-through packed beds. Finally, to provide a starting point for the calculation, rjkylc/kc was taken to be 1.0 when the outlet fractional conversion of A was 0.50. When rjkylc/kc = 1, the resistance to extmial transport is equal to the resistance to reaction inside the catalyst particle. At this condition, /ly arbitrarily was assigned a value of 1. 

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