Adsorption Wave Speed

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  

Here K is the slope of the linear isotherm. Eor non-trace systems or for rather high loadings the true slope K may be replaced by the slope of the operating line or chord of the isotherm, which quantity is simply Aq/Ac. The equivalent expression in molar loadings and gas phase mole fractions is  

When the quantity K is large there is no loss of accuracy to simply write the wave 

The latter relationship is readily derived from a mass balance on the contaminant entering the column. This wave speed is referred to as the stoichiometric wave speed. It applies strictly only to one component. It will be recognized that manipulation of Eq. (9.11) with similar substitutions yields the wave speed result for bulk separations where the derivative of v with respect to the spatial variable does not drop out and where two or more components may adsorb. The result has a very similar form but of course depends on not one isotherm slope but on two or more and very significantly the wave speed depends on the relative mole fractions of the adsorbing species and the initial loadings in the column. 

Before moving on to true mass transport issues it is worthwhile to point out that the quantity vep(Y - Yp) is in fact the water load per unit of time (whichever units of time v is expressed in.) Further that with an a-priori design process we may not know the required cross sectional area for flow and hence it may be more convenient to multiply the above relationship by and thus obtain the adsorbable contaminant input rate  

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The forgoing derivation and wave speed values will be found very helpful in discussing design methodologies for all adsorption systems. 

In the literature we can now find several papers which establish a widely accepted scenario of the benefits and effects of an ultrasound field in an electrochemical process [13-15]. Most of this work has been focused on low frequency and high power ultrasound fields. Its propagation in a fluid such as water is quite complex, where the acoustic streaming and especially the cavitation are the two most important phenomena. In addition, other effects derived from the cavitation such as microjetting and shock waves have been related with other benefits reported for this coupling. For example, shock waves induced in the liquid cause not only an enhanced convective movement of material but also a possible surface damage. Micro jets of liquid, with speeds of up to 100 ms-1, result from the asymmetric collapse of cavitation bubbles at the solid surface [16] and contribute to the enhancement of the mass transport of material to the solid surface of the electrode. Therefore, depassivation [17], reaction mechanism modification [18], surface activation [19], adsorption phenomena decrease [20] and the mass transport enhancement [21] are effects derived from the presence of an ultrasound field on electrode processes. We have only listed the main phenomena referring to the reader to the specific reviews [22, 23] and reference therein. 

The solution to either Eq. (9.9) or (9.10) has not one wave but two or more one for the adsorbed phase and a second in the gas phase. In many cases these waves move coincidentally but that assumption should not be invoked in all cases. The speed of these waves can be derived from the original pde together with the form of the adsorption equilibrium model. 

Changes in resonant freqnency measnred as a fnnction of time can be linked to the adsorption of a molecnlar layer on a qnartz snbstrate and is related to changes in the thickness of the layer. In the case of pnre adsorption, the decrease in resonant freqnency is connected with an increase in the layer thickness, as follows A/ = where v is the speed of the transverse wave in... 

The development of the wave-shape pattern is described as follows. The electric field activates and drives the surfactant molecules, which are adsorbed on the gel and deform it. As the adsorption progresses, the deformation occurs in such a way that the surface normal of the gel approaches parallel to the equipotential surface of the electric field. Fig. 7.22 illustrates the geometry of the gel and the electric field. Horizontal lines are the equipotential surfaces of the electric field. Arrows on the gel surface are normal vectors of the gel. Prom equation (2.7), the effect of the electric field to the gel disappears when the surface normal of the gel and the equipotential surface of the electric field become parallel (Fig. 7.22(a)). The angle of the tip of the gel 4> reaches maximum when the deformation speed near the root and one near the tip balance (Fig. 7.22(b)). The gel deformation works to deactivate the adsorption reaction and causes oscillatory motion. 

During the next half of the square wave cycle, the fluid from the top reservoir is pushed downward into the packed bed, which is suddenly cooled to Tcoia the colder fluid is now pushed into the bottom reservoir with the same speed. Since the adsorbent particles are cold, their adsorption capacity is much higher, thus purifying the fluid of solute i. The fluid coming into the bottom reservoir is substantially purified of species i. This cycle is ended when the volume of fluid introduced into the bottom reservoir during the second half of the cycle is equal to that removed from the same reservoir during the first half of the cycle. A new cycle is then initiated. Note that the top reservoir now has a solute concentration more than that in the initial... 

Figure 8.2 Three views of the photodissociation of ICN. Left the ground and excited state potential in the quasi-diatomic view where CN is regarded as an atom, plotted vs. the I—CN distance R. Middle the time evolution of the excited wave-packet, shown as probability ijr (fl,Af) vs. the l-C distance for different values of the delay time At. The initial wave function f R,At= 0) is quite localized and hence has a large spread in momentum. As the delay before probing increases, the wave-packet is moving out at a mean speed of about 2-10 cm s and broadens. Right experimental LIF signal profiles of CN vs. the delay time between the pump and probe pulses, for a series of frequencies. The topmost curve is at the resonance CN adsorption. Lower curves are for probing as the products are closer and closer in, so the peaks occur at earlier times [adapted from M. J. Rosker, M. Dantus, and A. H. Zewail, Sc/ence 241, 1200 (1988) and G. Roberts and A. H. Zewail, J. Phys. Chem. 95,7973 (1991) see also Bernstein and Zewail (1988),...

When R = 1 the thermal wave progresses through the bed more or less at the same speed as the mass transfer zone. Hence, virtually all the heat released on adsorption can be expected to be retained in the MTZ and the isothermal assumption should not be made unless either the heat of adsorption is low and/or the concentration of the adsorbable component is low. When R is very much less than unity the thermal wave lags behind the MTZ and hence the heat of adsorption can be retained in the equilibrium portion of the bed (that is, from the entrance up to Le shown in Figure 5.6 (b)). Retention of the heat of adsorption in this way is beneficial to the subsequent desorption step (Garg and Ausikaitis 1983). When R is very much greater than unity the heat is easily removed from the MTZ and it is safe to invoke the isothermal assumption. Further discussion on the crossover ratio is given in Section 7.5.3. 

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