Concentration waves

Zaikin A N and Zhabotinsky A M 1970 Concentration wave propagation in two-dimensional liquid-phase self-oscillating system Nature 225 535-7... 

Pneumatic systems use the wave motion to pressurize air in an oscillating water column (OWC). The pressurized air is then passed through an air turbine to generate electricity. In hydrauhc systems, wave motion is used to pressurize water or other fluids, which are subsequendy passed through a turbine or motor that drives a generator. Hydropower systems concentrate wave peaks and store the water dehvered in the waves in an elevated basin. The potential energy suppHed mns a low head hydro plant with seawater. 

Determination of Controlling Rate Factor The most important physical variables determining the controlhng dispersion factor are particle size and structure, flow rate, fluid- and solid-phase diffu-sivities, partition ratio, and fluid viscosity. When multiple resistances and axial dispersion can potentially affect the rate, the spreading of a concentration wave in a fixed bed can be represented approximately... 

The variable Ti defined by Eq. (16-127) or (16-129) is a throughput parameter, equal to unity (hence, the I subscript) at the time when the stoichiometric center of the concentration wave leaves the bed. This important group, in essence a dimensionless time variable, essentially determines the location of the stoichiometric center of the transition in the bed at any time. 

The solution to this model for a deep bed indicates an increase in velocity of the fluid-phase concentration wave during breakthrough. This is most dramatic for the rectangular isotherm—the instant the bed becomes saturated, the fluid-phase profile Jumps in velocity from that of the adsorption transition to that of the fluid, and a near shocklike breakthrough curve is obseived [Coppola and LeVan, Chem. Eng. Sci.,36, 967(1981)]. 

It turns out that packed beds much less than a hundred particles thick behave as if they were weU-stirred due to the entrance effect. Although it may seem odd that a packed bed can behave as if weU-stirred, it typically takes at least a 100-particle bed depth in order for a phig-flow concentration wave to develop. 

Pt2,V and Pt y have been investigated at 1393 K and 1224 K respectively and we have explored the [100] and [110] planes of the reciprocal lattice. The measured Intensities have been Interpreted in a Sparks and Borie approach with first order displacements parameters and using a model Including 29 a(/ ) for PfsV and 21 for PtsV. In figure 1 is displayed the intensity distribution due to SRO a q) in the [100] plane. As for PdjV, the diffuse intensity of Pt V is spread along the (100) axes with maxima at the (100) positions, whereas the ground state is built on (1 j 0) concentration wave ( >022 phase). 

We have measured the experimental SRO contribution in PtsV and Pt V alloys. The PtsV SRO displays maxima at (100) positions despite a ground state built with the (1 0) concentration wave. For Pt V, the maxima are not located at special points of the fee lattice. 

Fig. 5 illustrates a peculiar kinetic phenomenon which occurs when an initially disordered alloy is first annealed at temperature T corresponding to area b in Fig. 1 and then quenched to the final temperature T into the spinodal instability area d antiphase boundaries "replicate , generating approximately periodic patterns. This phenomenon reflects the presence of critical, fastest growing concentration waves under the spinodal instability (the Calm waves ). Lowering of the temperature to T < T results in lowering of the minority concentration minimum ("c-well ) within APB, while the expelled solute atoms build the c-bank adjacent to the well . Due to the... 

A similar treatment applies for the unstable regime of the phase diagram (v / < v /sp), where the mixture decays via spinodal decomposition.For the linearized theory of spinodal decomposition to hold, we must require that the mean square amplitude of the growing concentration waves is small in comparison with the distance from the spinodal curve. 

In eq. (1) the ETT order parameter z = s(p-Pc) measures, in a convenient direction, the chemical potential from that corresponding to the ETT. From the values given in Table I for the above s and q, we readily see that the occurrence of the ETTs discussed in this paper always implies an increase of the alloy free energy. Thus, CuPt random alloys, that just below and above the equiatomic concentration present both the relevant ETT s, are less stable than CuPd or AgPd and, thus more likely to be destabilised. Moreover, the proximity to both the critical concentrations implies large contributions to the BSE from the X and L points. Now, the concentration wave susceptibility, Xcc(q). as observed by Gyorffy and Stocks, is proportional to... 

CO concentration at the outlet of each zone was continuously measured using a CO analyzer (Shimadzu CGT-7000). To evaluate the performance of the reactors, the conversion of CO for the PBR (Xco) with 4g of catalyst and the time-average conversion of CO for the SCMBR (Tea) with 2g of catalyst in each zone were calculated and compared. It should be noted that the CO concentration wave used for Eq. (1) was obtained whrai the system is at cyclic steady state (after 30 min of operation). 

Fig. 4 Concentration waves of CO observed at the reactor outlet after reaching cyclic steady state...

Figure 3. Time-averaged reaction rate as a function of period. Key (concentration wave amplitude) Q, 0.24 , 0.12 and-----------, steady-state rate.

The initial concentration distribution is therefore simply translated at the velocity of the liquid steady flow and full equilibrium between the liquid and its matrix require that the amount of element transported by the concentration wave is constant. In more realistic cases, either the flow is non-steady due to abrupt changes in fluid advection rate or porosity, or solid-liquid equilibrium is not achieved. These cases may lead to non-linear terms in the chromatographic equation (9.4.35) and unstable behavior. The rather complicated theory of these processes is beyond the scope of the present book. 

If the velocities of the thermal and concentration waves are equal, then from equations 17.75 and 17.101 ... 

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 ... 

For a bed initially free of adsorbate, the thermal wave propagates more quickly than the concentration wave if ... 

It has been assumed that the gas and solid have the same temperature at any point, and that the fluid concentration is constant throughout a pellet at a value equal to that immediately outside the pellet. Within the limits of these assumptions, the thermal wave velocity up is independent of temperature. As discussed in Section 17.8.4, the velocity of the thermal wave relative to that of the concentration wave can be positive, as it normally is in liquids, negative or zero. 

Figure 17.37 shows a thermal wave plotted as a dotted line of distance against time. The velocity uc of the concentration wave depends on where it is in relation to the thermal wave, as can be seen by comparison with the full line in the Figure 17.37. 

The net movement upwards of a concentration wave is greater in the direct mode. Fewer cycles are needed to achieve a given separation. Nevertheless, the recuperative mode is probably the more convenient method to use on a commercial scale. Indeed,... 

When the steady state becomes unstable, the system moves away from it and often undergoes sustained oscillations around the unstable steady state. In the phase space defined by the system s variables, sustained oscillations generally correspond to the evolution toward a limit cycle (Fig. 1). Evolution toward a limit cycle is not the only possible behavior when a steady state becomes unstable in a spatially homogeneous system. The system may evolve toward another stable steady state— when such a state exists. The most common case of multiple steady states, referred to as bistability, is of two stable steady states separated by an unstable one. This phenomenon is thought to play a role in differentiation [30]. When spatial inhomogeneities develop, instabilities may lead to the emergence of spatial or spatiotemporal dissipative stmctures [15]. These can take the form of propagating concentration waves, which are closely related to oscillations. 

In this small experimental space four variables (A, B, C, D) were optimized. The concentration waves of components A and B are placed along the X axis, while that of the components C and D along the Y axis. As far as there are only two variables along each axis there are only four combinations for the visualization of the experimental hologram. Two selected holograms are shown in Figure 1. 

Table VII shows that, for the methanation reactor model, the dynamic response of the gas temperatures and CO and C02 concentrations should be much faster (by two orders of magnitude) than the response of the catalyst and thermal well temperatures. This prediction is verified in the dynamic responses shown in Figs. 18 and 19 and the previous analysis of the thermal and concentration wave velocities.

In this chapter we shall treat some particular instances of the system (3.1.15) and the related phenomena. Thus in 3.2, we shall concentrate upon binary ion-exchange and discuss the relevant single nonlinear diffusion equation. It will be seen that in a certain range of parameters this equation reduces to the porous medium equation with diffusivity proportional to concentration. Furthermore, it turns out that in another parameter range the binary ion-exchange is described by the fast diffusion equation with diffusivity inversely proportional to concentration. It will be shown that in the latter case some monotonic travelling concentration waves may arise. 

The third contribution to the chemical potential is due to strain. If A and B atoms (ions) have different size, clustering results in elastic lattice distortions. By making a Fourier transformation, one can decompose the concentration profile into harmonic plane waves [D. DeFontaine (1975)]. The elastic energy contributions of these concentration waves are additive in the Unear elastic regime and yield Ea. Therefore, we may write... 

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