Simple wave

Methods for analysis of fixed-bed transitions are shown in Table 16-2. Local equilibrium theoiy is based solely of stoichiometric concerns and system nonlinearities. A transition becomes a simple wave (a gradual transition), a shock (an abrupt transition), or a combination of the two. In other methods, mass-transfer resistances are incorporated. 

Local equilibrium theory Shows wave character—simple waves and shocks Usually indicates best possible performance Better understanding Mass and heat transfer very rapid Dispersion usually neglected If nonisothermal, then adiabatic... 

FIG. 16-2 Limiting fixed-bed behavior simple wave for unfavorable isotherm (top), square-root spreading for linear isotherm (middle), and constant pattern for favorable isotherm (bottom). [From LeVan in Rodtigues et al. (eds.), Adsorption Science and Technology, Kluwer Academic Publishers, Dotdtecht, The Nethedands, 1989 reptinted withpeimission.]... 

For a favorable isotherm d n lldc f < 0), Eq. (I6-I3I) gives the impossible result that three concentrations can coexist at one point in the bed (see example below). The correct solution is a shock (or abrupt transition) and not a simple wave. Mathematical theoiy has been developed for this case to give weak solutions to consei vation laws. The form of the solution is... 

It is also possible to have a combined wave, which has both gradual and abrupt parts. The general rule for an isothermal, trace system is that in passing from the initial condition to the feed point in the isotherm plane, the slope of the path must not decrease, if it does, then a shock chord is taken for that part of the path. Referring to Fig. 16-19, for a transition from (0,0) to (1,1), the dashes indicate shock parts, which are connected by a simple wave part between points Pi and Pg. 

For a simple wave, apphcation of the method of characteristics (hodograph transformation) gives... 

Nontrace isothermal systems give the adsorption effect (i.e., significant change in fluid velocity because of loss or gain of solute). Criteria for the existence of simple waves, contact discontinmties, and shocks are changed somewhat [Peterson and Helfferich, J. Phy.s. Chem., 69, 1283 (1965) LeVan et al., AIChE J., 34, 996 (1988) Frey, AJChE J., 38, 1649(1992)]. 

Using the isotherm to calculate loadings in equilibrium with the feed gives rii = 3.87 mol/kg and ri2 = 1.94 mol/kg. An attempt to find a simple wave solution for this problem fails because of the favorable isotherms (see the next example for the general solution method). To obtain the two shocks, Eq. (16-136) is written... 

The solution gives all of the expected asymptotic behaviors for large N—the proportionate pattern spreading of the simple wave if R > 1, the constant pattern if R < 1, and square root spreading for R = 1. 

For high feed loads, the shape of the diffuse traihng profile and the location of the leading front can be predicted from local equihbrium theory (see Fixed Bed Transitions ). This is illustrated in Fig. 16-35 for Tp = 0.4. For the diffuse profile (a simple wave ), Eq. (16-131) gives ... 

If Cm -I- 3Cii > 0, a centered simple wave will be produced by impact loading, and a record of this waveform suffices to determine the entire uniaxial stress-strain relation over the range of strains encountered. Vitreous silica is a material responding in this manner, and its coefficients have been determined by Barker and Hollenbach [70B01] (see also [72G02]) on the basis of a simple-wave analysis. 

The simple wave produced by impacting vitreous silica has approximately the form of a linear ramp of velocity. When this ramp wave is used to load another elastic solid placed in contact with the vitreous silica, a measurement... 

We note that it is possible to combine the method with correlation factor with the method using superposition of configurations to obtain any accuracy desired by means of comparatively simple wave functions. For a very general class of functions g(r12), one can develop the quotient (r r2)lg(r12) according to Eq. III.2 into products of one-electron functions y>k(r), which leads to the expansion... 

If a simple wave is not possible on physical grounds, then it (or part of it) is replaced by a shock, given by... 

Whereas the profile in linear wave equations is usually arbitrary it is important to note that a nonlinear equation will normally describe a restricted class of profiles which ensure persistence of solitons as t — oo. Any theory of ordered structures starts from the assumption that there exist localized states of nonlinear fields and that these states are stable and robust. A one-dimensional soliton is an example of such a stable structure. Rather than identify elementary particles with simple wave packets, a much better assumption is therefore to regard them as solitons. Although no general formulations of stable two or higher dimensional soliton solutions in non-linear field models are known at present, the conceptual construct is sufficiently well founded to anticipate the future development of standing-wave soliton models of elementary particles. 

When no analyhcal soluhon can describe the process satisfactorily it may be possible, working from Eq. (9.18) (which describes the length of the wave) and either Eq. (9.11) or (9.13) (the expression for the velocity of the adsorption wave), to assemble a simple wave mechanics solution that approximates the length and movement of the mass transfer front in the bed. As with analytical solutions this method can deliver useful results that may approximate the wave shape inside the bed and thus can be used to describe the shape and duration of the breakthrough curve that occurs as the wave intercepts and crosses the end of the bed. Such methods are generally only applicable for one or at most two adsorbable components. 

Figure 8.1 shows the expectation values of the electric moment for the Li+—H and covalent structures. The graph gives the negative of the moment for Li —H+ for easy comparison. In addition, the moment for the three-term wave function involving all three of the other functions is given. Although such a simple wave... 

This space-time diagram resembles the analogous space-time diagram for the case of a linear detonation shock (shown in Ref 2, p 5), except that there is one difference between the two cases. In the linear case, there is a simple wave in the fan-like region, which means that each radial line is a characteristic line and for each radial line,... 

In the spherical case, however, there is no simple wave, but for each radial line within the fan-like region, there is ... 

Fig 4. Straight characteristics and particle paths in a simple wave. Piston path (P), straight characteristics (solid), and particle paths (dashed) are shown. 

---