Molecular diffusion, definition

The importance of dilfusion in a tubular reactor is determined by a dimensionless parameter, SiAt/S = QIaLKuB ), which is the molecular diffusivity of component A scaled by the tube size and flow rate. If SiAtlB is small, then the elfects of dilfusion will be small, although the definition of small will depend on the specific reaction mechanism. Merrill and Hamrin studied the elfects of dilfusion on first-order reactions and concluded that molecular diffusion can be ignored in reactor design calculations if... 

Molecular diffusion (or self-diffusion) is the process by which molecules show a net migration, most commonly from areas of high to low concentration, as a result of their thermal vibration, or Brownian motion. The majority of reactive transport models are designed to simulate the distribution of reactions in groundwater flows and, as such, the accounting for molecular diffusion is lumped with hydrodynamic dispersion, in the definition of the dispersivity. 

We might properly refer to this value as the apparent Peclet number, because by many formal definitions the Peclet number accounts for the relative importance of advection and molecular diffusion, without mention of hydrodynamic dispersion. 

Bischoff and Levenspiel (B14) present some calculations using existing experimental data to check the above predictions about the radial coefficients. For turbulent flow in empty tubes, the data of Lynn et al. (L20) were numerically averaged across the tube, and fair agreement found with the data of Fig. 12. The same was done for the packed-bed data of Dorweiler and Fahien (D20) using velocity profile data of Schwartz and Smith (Sll), and then comparing with Fig. 11. Unfortunately, the scatter in the data precluded an accurate check of the predictions. In order to prove the relationships conclusively, more precise experimental work would be needed. Probably the best type of system for this would be one in laminar flow, since the radial and axial coefficients for the general dispersion model are definitely known each is the molecular diffusivity. 

Molecular diffusion deals with the relative motion of one kind of atom or molecule against a set of reference molecules. As explained in the introduction to this chapter (remember the trip in the dining car through the Swiss Alps), the reference system itself may move relative to some chosen coordinates. We called such directed motion advection. If one really looks very closely and wants to use crystal-clear definitions, it turns out that there is more than one way to choose the reference system. Each choice leads to a different separation between diffusion and advection, resulting in different diffusion coefficients. 

It is not unreasonable to use the left-hand side of this equation as the definition of the effective diffusion constant K, the more so as it will be shown that any distribution tends to normality. With this definition K is the sum of the molecular diffusion coefficient, D, and the apparent diffusion coefficient k = oP-U2I 48D, which was discovered by Taylor in his first paper (Taylor 1953, equation (25)). Equation (26), however, is true without any restriction on the value of p, or on the distribution of solute. The constant 1/48 is a function of the profile of flow, and for so-called piston flow with x — 0 this constant is zero and K = D as it should. 

Another difficulty is related to the mass transfer by convection, as, by definition, the films are stagnant and hence, there should be no mass transport mechanism, except for molecular diffusion in the direction normal to the interface (Kenig, 2000). Nevertheless, convection in films is directly accounted for in correlations. Moreover, in case of reactive systems, the film thickness should depend on the reaction rate, which is beyond the two-film theory consideration. 

Microbial mats are communities in surface water ecosystems where bacteria and bacterial processes dominate. In microbial mats dissolved nutrients and metabolites are transformed by one-dimensional (vertical) molecular diffusion. The distinction between microbial mats and biofilms is not sharp. By definition (Fenchel et al, 1998), microbial mats are typically stratified vertically with respect to different functional types of bacteria. Microbial mats are thicker (often several millimeters) than biofilms. In microbial mats various types of filamentous prokaryotes are the most conspicuous part and they are responsible for the mechanical coherence of the mat. The mechanical stability of microbial mats is reinforced by the bacterial excretion of mucous polymers, producing a gelatinous matrix. 

To illustrate the importance of the definition of the deterministic and stochastic velocity components 5, and let us suppose a puff of species of known concentration distribution c(x, o) at time to- In the absence of chemical reaction and other sources, and assuming molecular diffusion to be negligible, the concentration distribution at some later time is described by the following advection equation ... 

The laminar-flow reactor with segregation and negligible molecular diffusion of species has a residence-time distribution which is the direct result of the velocity profile in the direction of flow of elements within the reactor. To derive the mixing model of this reactor, let us start with the definition of the velocity profile. 

The coefficient of kp in (26-14) is written in terms of the concentration dependence of cpA via the definition of the binary molecular diffusivity in equation (25-76a) ... 

Diffusion phenomena can also be characterized by a time scale, the diffusion time, tp. The exact definition of t will be given later. The diffusion time is a measure of the time available for molecular diffusion phenomena to take place before mixing of the liquid phase makes the concentration uniform. Therefore, t decreases as the mixing or turbulence of the liquid phase is increased. 

Defining the proportionality constant as the molecular diffusivity D, the definition for D 3 becomes... 

One last useful definition is the tortuosity t. The tortuosity relates the effective diffusivity in the pores - effective to 6 molecular diffusivity in free solution, Dnjoiecuiar... 

I. Definition of mass-transfer coefficient. Since our understanding of turbulent flow is incomplete, we attempt to write the equations for turbulent diffusion in a manner similar to that for molecular diffusion. For turbulent mass transfer for constant c, Eq. (6.1-6) is... 

Still air can be viewed in two ways. One somewhat catholic definition would include air velocity below a level detectable by the optomotor reaction, thus embracing discrete plume structmes similar to those generated in air speeds which can be monitored by the optomotor reaction. A second and more obvious definition would encompass chemical stimuli generated in still air. Dispersion in completely still air would be dominated by molecular diffusion, rather that turbulent diffusion in an airflow, but Mankin et al. (1980) has pointed out that in most still air situations turbulent diffusion will overwhelm molecular diffusion (see Chapter 3). The spatial distributions of chemical stimulus in these two cases will obviously be quite different. 

A considerable amount of work has been published on optimizing the experimental conditions for minimum analysis time under various constraints [52]. One complication arises from the definition of reduced mobile-phase velocity. The actual mobile-phase velocity depends largely on the molecular-diffusion coefficient of the analyte. Thus, very small particles can be used for the analysis of high molecular mass compounds, which have low values. The actual flow rate required will remain compatible with pressure constraints despite the resulting high pneumatic or hydraulic resistance. Detailed results obviously depend greatly on the mode of chromatography used. 

Equations 15.9 and 15.10 are empirical with respect to the definition of the mass transfer coefficients, but the form of the equations is based on molecular diffusion theory. Applying the theory to a multi-component mixture where each component has a distinct diffusivity is impracti-cally complex and must rely on diffusivity data for all the components in the mixture. To derive usable equations from the diffusion theory, certain simplifying assumptions must be made. The basis for the derivation of Equations 15.9 and 15.10 is to assume that mass transfer takes place either as equimolar counter diffusion or as unimolar diffusion under dilute conditions. 

In equimolar counterdiffusion, knowledge of the mass-transfer coefficient (or fc or and the values of partial pressures (or concentrations or mole fractions) at the two ends of the diffusion path (z = 0 and z = Sg) were sufficient to determine Naz (or Ngz). In all other cases of molecular diffusion, the flux ratio Nr needs to be known [Pg.104]

An ideal gas has by definition no intermolecular structure. Also, real gases at ordinary pressure conditions have little to do with intermolecular interactions. In the gaseous state, molecules are to a good approximation isolated entities traveling in space at high speed with sparse and near elastic collisions. At the other extreme, a perfect crystal has a periodic and symmetric intermolecular structure, as shown in Section 5.1. The structure is dictated by intermolecular forces, and molecules can only perform small oscillations around their equilibrium positions. As discussed in Chapter 13, in between these two extremes matter has many more ways of aggregation the present chapter deals with proper liquids, defined here as bodies whose molecules are in permanent but dynamic contact, with extensive freedom of conformational rearrangement and of rotational and translational diffusion. This relatively unrestricted molecular motion has a macroscopic counterpart in viscous flow, a typical property of liquids. Molecular diffusion in liquids occurs approximately on the timescale of nanoseconds (10 to 10 s), to be compared with the timescale of molecular or lattice vibrations, to 10 s. 

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