Adsorption wave equations

In view of the number of assumptions that have been made in deriving the adsorption wave equations and considering the fact that no experimental data are available for adsorption of gas in a flow system on materials other than charcoal, experimental work was carried out to evaluate the critical conditions for each variable in the equation. 

Equation 17.75 is important as it illustrates, for the equilibrium case, a principle that applies also to the non-equilibrium cases more commonly encountered. The principle concerns the way in which the shape of the adsorption wave changes as it moves along the bed. If an isotherm is concave to the fluid concentration axis it is termed favourable, and points of high concentration in the adsorption wave move more rapidly than points of low concentration. Since it is physically impossible for points of high concentration to overtake points of low concentration, the effect is for the adsorption zone to become narrower as it moves along the bed. It is, therefore, termed self-sharpening. 

The inter-pellet air velocity = 0.233 m/s. The velocity uc with which the adsorption wave moves through the column may be obtained from equation 17.79. 

The left-hand side of equation 17.123 is the velocity of a point of fixed concentration on the adsorption wave. For a linear isotherm and if longitudinal diffusion is neglected, all points of concentration will move at the same velocity. Changing the pressure will affect u and, to a lesser extent, Ka. 

The speed of the adsorption wave can be readily derived by introducing the linear isotherm assumption and the chain mle derivative of q with respect to t. The wave speed results because the assumptions turn Eq. (9.10) into a kinematic wave equation and the wave speed W is instantly recognized as ... 

If a sound theoretical adsorption wave expression could be applied to this problem, it would not only greatly reduce the number of experiments necessary to define completely the geometry of the bed, but could also be useful in the elucidation of the mechanism of adsorption of various gases on different types of adsorbents, from which information could be derived for their improvement or to indicate when the adsorbent has attained its maximum efficiency. Theoretical treatment of the problem has been made by various investigators (73,74,75). However, these workers did not have sufficient experimental data to support their views. The problem of adsorption by charcoal was treated by Wicke (76) and a number of useful differential equations derived. However, a real... 

Chronocoulometry can also be applied to the cases discussed in Section 14.3.2 where only adsorbed species are electroactive (57). In this situation the potential step causes only double-layer charging and the electrolysis of the adsorbed species. One can estimate Qdi by steps between a potential at the foot of the adsorption wave, and potentials, Ef, beyond the adsorption wave. If Q is not a function of E in the region of the wave, the following equation results (57) ... 

It has already been pointed out that the forms of solution to the adsorption wave problem are similar to those representing the elution of tracer in a residence-time distribution experiment. Closest to what we considered in Chapter 5 are solutions of Levenspiel and Bischolf [O. Levenspiel and K.B. Bischolf, Adv. Chem. Eng., 4, 95 (1963)] and Lapidus and Amundson [L. Lapidus and N.R. Amundson, J. Phys. Chem., 56, 984 (1952)]. With the rate equation written as in equation (9-8a), these are... 

A weak cw circularly polarized probe beam is applied to the sample and together with its reflection off the mirror creates a standing wave probe field which monitors and when the probe wavelength is matched to the pump wavelength. Solving Maxwell s Equations for the probe field inside the sample volume with a modulated adsorption constant and index of refraction, for the case where yields a coupled wave equation for the inci-... 

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. 

Equation 17.102 was derived on the assumption that concentration and thermal waves propagated at the same velocity. Amundson et al.<4y> showed that it was possible for the temperatures generated in the bed to propagate as a pure thermal wave leading the concentration wave. A simplified criterion for this to occur can be obtained from equations 17.75 and 17.101. Since there is no adsorption term associated with a pure thermal wave, and if changes within the bed voids are small, then ... 

Polanyi (27) as early as 1921 suggested that the activated state consists of free atoms which are adsorbed or bound by large affinities (heat of adsorption or diminution of the homogeneous activation energy) to the catalyst. A more quantitative treatment became possible after the theory of wave mechanics presented equations for the mutual interaction of different covalent bonds. London (28) showed that the total interaction energy of three atoms X, Y, and Z is... 

Mercury binding leads to an increase of mass of the gold layer which can be detected by electro-acoustic transducers based on quartz microbalance (QMB the abbreviation QCM = quartz crystal microbalance is also widely used), surface acoustic waves (SAW)—devices [20] or microcantilevers [21,22], Adsorption of mercury vapour increases resonance frequency of shear vibrations of piezoelectric quartz crystals (Fig. 12.2). This process can be described by Sauerbrey equation [23]. For typical AT-cut quartz, this equation is... 

Cyclic voltammetry of all the ferrocenyl dendrimers on a Pt anode showed all the ferrocenyl centers to be equivalent as only one wave was observed. It was possible to avoid adsorption even using CH2CI2 for the small ferrocenyl dendrimers, but the use of MeCN was necessary for the medium-sized ones (27-Fc, 54-Fc, and 81-Fc). Finally, adsorption could not be avoided even with MeCN for the 243-Fc dendrimer. From the intensity of the wave, the number of ferrocenyl units could be estimated using the Anson-Bard equation [75], and the numbers found were within 5 % of the branch numbers except in the case of the 243-Fc dendrimer, for which the experimental number was too high (250) because of the adsorp-... 

Price and Halley (PH) [136] and Halley, Johnson, Price and Schwalm (HJPS) [137] have described a different theory of electron overspill into the layer between the solvent and metal-ion cores at a metal-electrolyte interface in the absence of specific adsorption of ions. Previous authors avoided the use of Schrodinger s equation altogether by introducing trial functions for the electron density function n(x). In contrast Halley and co-workers (HQ [138-141] used the Kohn-Sham version [122] of the variational principle of Hohenberg and Kohn [121] in which n(x) was described in terms of wave functions obeying Hartree-like equations. An effective one-electron Schrodinger equation is solved... 

For example, the amount of adsorbed SRlOl on a water/DCE interface (F) is given by the Gibbs equation F= —(1/2.3RT) dy/d log[SR101]. The F value was then calculated to be 5.0 x 10 mol cm ([SR101] = 1.0 x 10 M). When the interfacial area is assumed to be 1 cm for simplicity, the number of SRIOI molecules adsorbed on the interface (5.0 x 10 mol) is 5000 times higher than that expected to be involved in the excited volume by the evanescent wave (1.0 x 10 mol) without adsorption. Almost the same results with those for SRIOI were obtained for other dye/water/oil systems. At [SRIOI] = 1 x 10 M, therefore, the fluorescence response observed under the TIR conditions is ascribed essentially to that from the interface. 

A solvent dissolution, a vapor adsorption, any kind of surface-active substance exchange between the surface and the adjacent subphase, or heating makes the surface tension locally vary, thus generating Marangoni stresses and convection. Then, gravitocapillary waves (wavelength X and amplitude q) excited and sustained by the Marangoni effect in the shallow water waves approximation can be described by the equation ... 

Commonly, the purge and blowdown steps are governed by Equations (14.60) and (14.61). To analyze other steps typically requires additional equations that reflect the feature that enables PSA (as well as other adsorption processes) to operate efficiently. The feed, rinse, and (possibly) pressurization steps involve uptake of the more strongly adsorbed component (while the blowdown and purge steps usually do not). This phenomenon results in a sharp concentration front, sometimes called a constant pattern profile or a shock wave. Usually, it is the velocity of this sharp front through the packed bed (of a specific length) that governs the duration of the particular step. 

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