Molecular diffusion tracer

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if tlie Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m /sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

Measurements of diffusion of tracer polymers in ordered block copolymer fluids is another potentially informative activity, since molecular diffusion is one of the most basic dynamic characteristics of a molecule. Balsara, et al. have measured the retardation of diffusion due to ordering in the diffusion of polystyrene tracer homopolymers in polystyrene-polyisoprene matrices of various domain sizes [167]. Measurement of the tracer diffusion of block copolymer molecules will also be important. Several interesting issues are directly addressable via measurements... 

In laminar flow, a similar mixing process occurs when the liquid is sheared between two rotating cylinders. During each revolution, the thickness of the fluid element is reduced, and molecular diffusion takes over when the elements are sufficiently thin. This type of mixing is shown schematically in Figure 7.3 in which the tracer is pictured as being introduced perpendicular to the direction of motion. 

Since in most situations the perturbation quantities (V and c() are not explicitly resolved, it is not possible to evaluate the turbulent flux term directly. Instead, it must be related to the distribution of averaged quantities - a process referred to as parameterization. A common assumption is to relate the turbulent flux vector to the gradient of the averaged tracer distribution, which is analogous with the molecular diffusion expression. Equation (35). 

At a close level of scrutiny, real systems behave differently than predicted by the axial dispersion model but the model is useful for many purposes. Values for Pe can be determined experimentally using transient experiments with nonreac-tive tracers. See Chapter 15. A correlation for D that combines experimental and theoretical results is shown in Figure 9.6. The dimensionless number, udt/D, depends on the Reynolds number and on molecular diffusivity as measured by the Schmidt number, Sc = but the dependence on Sc is weak for... 

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

Figures 13.15 and 13.16 show the findings for flow in pipes. This model represents turbulent flow, but only represents streamline flow in pipes when the pipe is long enough to achieve radial uniformity of a pulse of tracer. For liquids this may require a rather long pipe, and Fig. 13.16 shows these results. Note that molecular diffusion strongly affects the rate of dispersion in laminar flow. At low flow rate it promotes dispersion at higher flow rate it has the opposite effect.

Table 24.5 Characteristic Data of Chemical Tracers Used for the Gas Exchange Experiment Nondimensional Henry s Law Constant, Ki3/vi, Molecular Diffusion Coefficient in Air, Dia, and Water, Diw, Octanol-Water Partition Coefficient, Kiow, All valid for 4°C (Data from Cirpka et al., 1993)...

Some of the methods for measuring molecular diffusion coefficients, together with a few recent references, are (a) diaphragm cell [60,61] (b) boundary layer interferometry [59] (c) shearing plate interferometry [58] (d) chromatographic peak broadening [60] (e) nuclear magnetic resonance and electron spin resonance [62, 63] (f) electrolyte conductance [64] (g) isotopic tracers [65] and (h) laminar jets [66]. 

Dmoi = molecular diffusivity of the reactant (or tracer) in the fluid (Djnoi 1CT5 cm2/sec, for liquids, 1 cm2/sec for gases)... 

The characteristics of the experimental aquifer were independently determined from appropriate flowthrough column experiments or obtained directly from the literature. The dry bulk density of the sand ph= 1.61 kg/1, and the aquifer porosity 0=0.415 were evaluated by gravimetric procedures. The dimensionless retardation factor, R= 1.31, of the aqueous-phase TCE was determined from a column flowthrough experiment. The tortuosity coefficient for the aquifer sand was considered to be x =1.43 [75]. The molecular diffusion coefficient for the aqueous-phase TCE is D=0.0303 cm2/h [76]. The pool radius is r=3.8 cm. Bromide ion in the form of the moderately soluble potassium bromide salt was the tracer of choice [77 ] for the tracer experiment conducted in order to determine the longitudinal and transverse aquifer dispersivities a =0.259 cm and a-,— 0.019 cm, respectively. The experimental pool contained approximately 12 ml of certified ACS grade (Fisher Scientific) TCE with solubility of Cs=1100 mg/1 [78]. 

Tracer breakthrough profiles within the matrix regime confirmed that the dissolved gases and Br were slowly moving into the bedrock matrix and at different rates. Concentration profiles of the three tracers 6 m from the source and 0.8 m into the matrix relative to the fracture zone are shown in Figure 1.5(a). The movement of He and Ne into and from the matrix was more rapid than Br, which is consistent with the larger molecular diffusion coefficients for the dissolved gases relative to Br. These results support the notion that matrix diffusion contributed to the overall physical nonequilibrium process that controls solute transport in bedrock at this site (Maloszewski and Zuber, 1990, 1993). 

The effective diffusion coefficients. Die, obtained from the parameter tdjf, include contributions from the Knudsen diffusion mechanism and from the molecular diffusion mechanism. Because of the very low tracer concentrations the Bosanquet formula (3) is applicable. 

Here is the molecular diffusion coefficient of the pair C-T and K. 3 (2/3)V(8RgT/7tMT) is the Knudsen constant for the tracer T, Rg is the gas constant, T temperature, and Mt the tracer molecular weight. v t and vi/ are parameters characterizing the porous medium (transport parameters). stands for the integral mean radius of pores through which the... 

Here, the putative length scale represents some kind of mean displacement, or some Lagrangian decorrelation scale that is evidenced by a granularity in the tracer distribution (see Armi and Stommel, 1983 Jenkins, 1987 Joyce and Jenkins, 1993). The uf term is usually characterized as a turbulent diffusion coefficient k, because of its functional similarity to the molecular diffusion coefficient ... 

A very short pulse injection of tracer is made into a column. When the center of mass has traveled 2 m, the standard deviation is 0.1 m. The molecular diffusion coefficient is 10-5 cm2/sec. 

Molecular diffusion coefficients in water are usually determined in the laboratory by using a tracer in agar or gel to insure that the solution is totally free of turbulent motion. Values for gases and ions in pure water are presented in units of cm s in Table 9.1. The molecular diffusion coefficients and their temperature dependence for gases were calculated from the Eyring equation. 

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