Multicomponent velocities

Adsorption Chromatography. The principle of gas-sohd or Hquid-sohd chromatography may be easily understood from equation 35. In a linear multicomponent system (several sorbates at low concentration in an inert carrier) the wave velocity for each component depends on its adsorption equihbrium constant. Thus, if a pulse of the mixed sorbate is injected at the column inlet, the different species separate into bands which travel through the column at their characteristic velocities, and at the oudet of the column a sequence of peaks corresponding to the different species is detected. 

It is seen from Figure 15 that the analysis time ranges from about 10,000 seconds (a little less than 3 hr) to about 30 milliseconds. The latter, high speed separation, is achieved on a column about 2 mm long, 12 microns in diameter, operated at a gas velocity of about 800 cm/second. Such speed of elution for a multicomponent mixture is of the same order as that of a scanning mass spectrometer. 

The proposed technique will be used here to illustrate the case of interfacial heat and multicomponent mass transfer in a perfectly mixed gas-liquid disperser. Since in this case the holding time is also the average residence time, the gas and liquid phases spend the same time on the average. If xc = zd = f, then for small values of t, the local residence times tc and td of adjacent elements of the continuous and dispersed phases are nearly of the same order of magnitude, and hence these two elements remain in the disperser for nearly equal times. One may conclude from this that the local relative velocity between them is negligibly small, at least for small average residence times. Gal-Or and Walatka (G9) have recently shown that this is justified especially in dispersions of high <6 values and relatively small bubbles in actual practice where surfactants are present. Under this domain, Eqs. (66), (68), (69) show that as the bubble size decreases, the quantity of surfactants necessary to make a bubble behave like a solid particle becomes smaller. Under these circumstances (pd + y) - oo and Eq. (69) reduces to... 

For a multicomponent system, the bulk flow velocity up is given by ... 

According to Maxwell s law, the partial pressure gradient in a gas which is diffusing in a two-component mixture is proportional to the product of the molar concentrations of the two components multiplied by its mass transfer velocity relative to that of the second component. Show how this relationship can be adapted to apply to the absorption of a soluble gas from a multicomponent mixture in which the other gases are insoluble and obtain an effective diffusivity for the multicomponent system in terms of the binary diffusion coefficients. 

The parameter D is known as the axial dispersion coefficient, and the dimensionless number, Pe = uL/D, is the axial Peclet number. It is different than the Peclet number used in Section 9.1. Also, recall that the tube diameter is denoted by df. At high Reynolds numbers, D depends solely on fluctuating velocities in the axial direction. These fluctuating axial velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for D is used for each component in a multicomponent system. 

Consider a fluid of molecules Interacting with pair additive, centrally symmetric forces In the presence of an external field and assume that the colllslonal contribution to the equation of motion for the singlet distribution function Is given by Enskog s theory. In a multicomponent fluid, the distribution function fi(r,Vj,t) of a particle of type 1 at position r, with velocity Vj at time t obeys the equation of change (Z)... 

Yabusaki, S. B., C. I. Steefel and B. D. Wood, 1998, Multidimensional, multicomponent subsurface reactive transport in non-uniform velocity fields code verification using an advective reactive streamtube approach. Journal of Contaminant Hydrology 30,299-331. 

In the foregoing discussion the diffusive mass fluxes are written in terms of the diffusion velocities, which in turn are determined from gradients of the concentration, temperature, and pressure fields. Such explicit evaluation of the diffusion velocities requires the evaluation of the multicomponent diffusion coefficients from the binary diffusion coefficients. 

In the formulation and solution of conservation equations, we tend to prefer the direct evaluation of the diffusion velocities as discussed in the previous section. However, it is worthwhile to note that the Stefan-Maxwell equations provide a viable alternative. At each point in a flow field one could solve the system of equations (Eq. 3.105) to determine the diffusion-velocity vector. Solution of this linear system is equivalent to determining the ordinary multicomponent diffusion coefficients, which, in this formulation, do not need to be evaluated. 

The objective of this problem is to explore the multicomponent diffusive species transport in a chemically reacting flow. Figure 3.18 illustrates the temperature, velocity, and mole-fraction profiles within a laminar, premixed flat flame. These profiles are also represented in an accompanying spreadsheet (premixed h2. air-flame. xls). 

Species fluxes calculated by either the multicomponent (Section 12.7.2) or the mixture-averaged (discussed subsequently in Section 12.7.4) formulations are obtained from the diffusion velocities V, which in turn depend explicitly on the concentration gradients of the species (as well as temperature and pressure gradients). Solving for the fluxes requires calculating either all j-k pairs of multicomponent diffusion coefficients Dy, or for the mixture-averaged diffusion coefficient D m for every species k. 

The Stefan-Maxwell equations (12.170 and 12.171) form a system of linear equations that are solved for the K diffusion velocities V. The diffusion velocities obtained from the Stefan-Maxwell approach and by evaluation of the multicomponent Eq. 12.166 are identical. 

Evaluate the four multicomponent diffusion velocities k using Eq. 12.166. Verify that the sum of the diffusive mass fluxes is zero,... 

Sedimentation velocity. Tire relative molecular mass Mr can also be measured from observation of the velocity of movement of the boundary (or boundaries for multicomponent systems) between solution and solvent from which the macromolecules have sedi-... 

We begin with the simplest case. A vacancy flux j° (driven, for example, by inhomogeneous particle radiation) flows across a multicomponent crystal (k = 1,2,.., n) and the component fluxes are restricted to one sublattice. We assume no other coupling between the fluxes except the lattice site conservation, which means that we neglect cross terms in the formulation of SE fluxes. (An example of coupling by cross terms is analyzed in Section 8.4.) The steady state condition requires then that the velocities of all the components are the same, independent of which frame of reference has been chosen, that is,... 

The rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

Optimization of the coiled-tube heat exchanger is quite complex. There are numerous variables, such as tube and shell flow velocities, tube diameter, tube pitch, and layer spacing. Other considerations include single-phase and two-phase flow, condensation on either the tube or shell side, and boiling or evaporation on either the tube or shell side. Additional complications come into play when multicomponent streams are present, as in natural gas liquefaction, since mass transfer accompanies the heat transfer in the two-phase region. 

---