Molecular diffusion coefficient coefficients

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some rough correlations of the Peclet number are proposed by Wen (in Petho and Noble, eds.. Residence Time Distribution Theory in Chemical Tngineeiing, Verlag Chemie, 1982), including some for flmdized beds. Those for axial dispersion are ... 

Dispersion The movement of aggregates of molecules under the influence of a gradient of concentration, temperature, etc. The effect is represented by Pick s law with a dispersion coefficient substituted for molecular diffusion coefficient. The rate of transfer = -Dg(dC/3z). 

D = molecular diffusion coefficient, sq ft/hr Em = Murphree vapor plate efficiency, %... 

The molecular diffusion coefficients in hquid phase can be estimated from the correlations of WiUce and Chang [47] for organic solutions and Hayduk and Minhas [48] for aqueous solutions, respectively. An extensive comparison of the available correlations is provided by Wild and Charpentier [49]. 

Tybsre m is the total mass of analyte collected, D the molecular diffusion coefficient, A the area of the diffusion channel, L the diffusion path length, C the analyte concentration in the air, and Tt the sampling time. In deriving equation (8.7) it was assumed t. that the sorbent is effective sink for the analyte and,... 

Fig. 2.7.5 Two-dimensional D—T2 map for Berea sandstone saturated with a mixture of water and mineral oil. Figures on the top and the right-hand side show the projections of f(D, T2) along the diffusion and relaxation dimensions, respectively. In these projections, the contributions from oil and water are marked. The sum is shown as a black line. In the 2D map, the white dashed line indicates the molecular diffusion coefficient of water,...

Fabrication processing of these materials is highly complex, particularly for materials created to have interfaces in morphology or a microstructure [4—5], for example in co-fired multi-layer ceramics. In addition, there is both a scientific and a practical interest in studying the influence of a particular pore microstructure on the motional behavior of fluids imbibed into these materials [6-9]. This is due to the fact that the actual use of functionalized ceramics in industrial and biomedical applications often involves the movement of one or more fluids through the material. Research in this area is therefore bi-directional one must characterize both how the spatial microstructure (e.g., pore size, surface chemistry, surface area, connectivity) of the material evolves during processing, and how this microstructure affects the motional properties (e.g., molecular diffusion, adsorption coefficients, thermodynamic constants) of fluids contained within it. 

NMR signals are highly sensitive to the unusual behavior of pore fluids because of the characteristic effect of pore confinement on surface adsorption and molecular motion. Increased surface adsorption leads to modifications of the spin-lattice (T,) and spin-spin (T2) relaxation times, enhances NMR signal intensities and produces distinct chemical shifts for gaseous versus adsorbed phases [17-22]. Changes in molecular motions due to molecular collision frequencies and altered adsorbate residence times again modify the relaxation times [26], and also result in a time-dependence of the NMR measured molecular diffusion coefficient [26-27]. 

The level of vapor movement in the unsaturated zone is much less important than transport in liquid form. However, this might not be true if the water content of the soil is very low or if there is a strong temperature gradient. The movement of vapor through the unsaturated zone is a function of temperature, humidity gradients, and molecular diffusion coefficients for water vapor in the soil. 

For a free-falling spherical particle of radius R moving with velocity u relative to a fluid of density p and viscosity p, and in which the molecular diffusion coefficient (for species A) is DA, the Ranz-Marshall correlation relates the Sherwood number (Sh), which incorporates kAg, to the Schmidt number (Sc) and the Reynolds number (Re) ... 

The gas A must transfer from the gas phase to the liquid phase. Equation (1) describes the specific (per m2) molar flow (JA) of A through the gas-liquid interface. Considering only limitations in the liquid phase, this molar flow notably depends on the liquid molecular diffusion coefficient DAL (m2 s ). Based on the liquid state theories, DA L can be calculated using the Stokes-Einstein expression, and many correlations have been developed in order to estimate the liquid diffusion coefficients. The best-known example is the Wilke and Chang (W-C) relationship, but many others have been established and compared (Table 45.4) [28-33]. 

Each of the mass transfer coefficients klA and k2A can be interpreted as a molecular diffusion coefficient, D, divided by a film thickness, z, for the gas phase and the water phase, respectively, i.e., k=D/z. However, this interpretation has no meaning in practice because of the lack of knowledge on the thickness of the two films. 

A number of approaches have been suggested for the determination of the molecular diffusion coefficient, D, of a component in water (Othmer and Thakar, 1953 Scheibel, 1954 Wilke and Chang, 1955 Hayduk andLaudie, 1974 Thibodeaux, 1996). Based on these five references, the diffusion coefficient ratio />/Jl2s / Dlq2 was found to vary within the interval 0.78-0.86 with an arithmetic mean value equal to 0.84. This value can be inserted in Equation (4.22) as a first estimate to determine Km. Equation (4.22) and the empirical expressions for KLC>2 outlined in Table 4.7 are the basis for the determination of the mass transfer coefficient for H2S, KL i S, and thereby, the emission of H2S from the wastewater into the sewer atmosphere. Further details relevant in this respect are dealt with in Section 4.4. 

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

The sub-grid-scale turbulent Schmidt number has a value of Scsgs 0.4 (Pitsch and Steiner 2000), and controls the magnitude of the SGS turbulent diffusion. Note that due to the filtering process, the filtered scalar field will be considerably smoother than the original field. For high-Schmidt-number scalars, the molecular diffusion coefficient (T) will be much smaller than the SGS diffusivity, and can thus usually be neglected. 

In this case, if the boundary and initial conditions allow it, either ej or c can be used to define the mixture fraction. The number of conserved scalar transport equations that must be solved then reduces to one. In general, depending on the initial conditions, it may be possible to reduce the number of conserved scalar transport equations that must be solved to min(Mi, M2) where M = K - Nr and M2 = number of feed streams - 1. In many practical applications of turbulent reacting flows, M =E and M2 = 1, and one can assume that the molecular-diffusion coefficients are equal thus, only one conserved scalar transport equation (i.e., the mixture fraction) is required to describe the flow. 

At high Reynolds numbers, it is usually possible to assume that the mean scalar fields (e.g., (cc are independent of molecular-scale quantities such as the molecular-diffusion coefficients. In this case, it is usually safe to assume that all scalars have the same molecular diffusivity T. The conserved-scalar transport equation then simplifies to37... 

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