Steady-state concentrations, asymptotic solutions

Expressions for the steady-state concentration profile have also been derived for some more complex countercurrent systems. The extension to a (plug flow) system in which the equilibrium relationship is of Langmuir rather than linear form (constant separation factor) is given by Pratt. The solution for a linear plug flow system in which the mass transfer rate is controlled by intraparticle diffusion rather than by the linear rate law has been derived by Amundson and Kasten while the asymptotic behavior of a dispersed plug flow Langmuir system has been investigated by Rhee and Amundson. ... 

For membrane transport experiments, the relevant membrane is sandwiched between two solutions a donor typicaUy at constant dmg concentration, C = Cg, and a receiver at zero concentration, C = 0. The dmg concentration in the receiver is monitored as a function of time and the cumulative amount transported, has a linear asymptote with time where M is the area,/ is the steady-state flux, /is the time, and / is the time lag. 

The assumption of a steady-state profile in the oil laminates and small concentration drops in the water layers may be used to derive asymptotic solutions for the permeation problem. It may be shown that (2) for P P y and t[Pg.36]

For a chemical reaction system, the characteristics of the periodic solutions are uniquely determined by the kinetic constants as well as by the concentrations of the reactants and final products. Starting from the neighborhood of steady state as an initial condition, the system asymptotically attains a closed orbit or limit cycle. Therefore, for long times, the concentrations sustain periodic undamped oscillations. The characteristics of these oscillations are independent of the initial conditions, and the system always approaches the same asymptotic trajectory. Generally, the further a system is in the unstable region, the faster it approaches the limit cycle. 

The steady-state didiroic ratio of liquid crystalline solutions of TOLG (Fig. 3) increa s with external field strength and the asymptotic value is 4.5—4.7, regardless of the polymer concentration for comfdetely birefiringent solutions (23). It may be safe to s that all the polymer molecules are parallel or neady parallel within molecular a regates (31). Therefore, the value of 7 for the partide is tentatively assumed... 

The first 3D model of FFF was developed in Ref. 2. The 3D diffusion-convection equation was solved with the help of generalized dispersion theory, resulting in the equations for the cross-sectional average concentration of the solute and dispersion coefficients and K2, representing the normalized solute zone velocity and the velocity of the corresponding peak width growth, respectively. Unfortunately, only the steady-state asymptotic values of dispersion coefficients Ki oo) and K2 oo) were determined in Ref. 2, leading to the prediction of the solute peaks much wider than the experimental ones. 

There are simple algebraic solutions for the linear ideal model of chromatography for the two main coimter-current continuous separation processes. Simulated Moving Bed (SMB) and True Moving Bed (TMB) chromatography. Exphcit algebraic expressions are obtained for the concentration profiles of the raffinate and the extract in the columns and for their concentration histories in the two system effluents. The transition of the SMB process toward steady state can be studied in detail with these equations. A constant concentration pattern can be reached very early for both components in colimm III. In contrast, a periodic steady state can be reached only in an asymptotic sense in colunms II and IV and in the effluents. The algebraic solution allows the exact calculation of these limits. This result can be used to estimate a measure of the distance from steady state rmder nonideal conditions. 

Imagine the reactor is initially at this steady state and at t 0 we perturb the temperature and concentration by small amounts. We would like to know whether or not the system returns to the steady state after this initial condition perturbation. If so, we call the steady-state solution (asymptotically) stable. If not, we call the steady state unstable. Obviously we can solve numerically the nonlinear differential equations to answer this question, but then we answer the question on a case-by-case basis. By linearizing the nonlinear differential equations, we can gain further insight without resorting to full numerical solution. Consider the Taylor series expansion of the.nonlinear functions f, fz... 

In addition to finding the concentrations that make all the time derivatives in the rate equations vanish, it is useful to have another piece of information about such a time-independent or steady state. If the system starts at the steady state and is then subjected to a small perturbation, for example, injection or removal of a pinch of one of the reactants, we may ask whether the system will return to the original state or will evolve toward some other asymptotic behavior. The question we are asking here is whether or not the state of interest is stable. One of the basic tools of nonlinear chemical dynamics is stability analysis, which is the determination of how a given asymptotic solution to the rate equations describing a system will respond to an infinitesimal perturbation. 

One very important mathematical result facilitates the analysis of two-dimensional (i.e., two concentration variables) systems. The Poincare Bendixson theorem (Andronov et al., 1966 Strogatz, 1994) states that if a two-dimensional system is confined to a finite region of concentration space (e.g., because of stoichiometry and mass conservation), then it must ultimately reach a steady state or oscillate periodically. The system cannot wander through the concentration space indefinitely the only possible asymptotic solution, other than a steady state, is oscillations. This result is extremely powerful, but it holds only for two-dimensional systems. Thus, if we can show that a two-dimensional system has no stable steady states and that all concentrations are bounded—that is, the system cannot explode—then we have proved that the system has a stable periodic solution, whether or not we can find that solution explicitly. 

Equation 2.3g represents the ultimate steady-state value of the pollutant concentration Qs, which is attained after a long period of time. Note that this result is also obtained by setting the time derivative in Equation 2.3c equal to zero. The entire solution curve is shown in Figure 2.3a and demonstrates the asymptotic approach to steady-state conditions as time goes to infinity. 

The mixed solutions LYS-MR were prepared with using of lysozyme and MR after 24 h incubated separately at 25°C in darkness. After mixing of these solutions 1 1 they were stored 3 h at 25°C away from the light. The dynamic surface tension was measured over 60,000 s to guarantee steady-state of the adsorption layer. A t oo asymptotic extrapolation was used to find the steady-state surface tension values. Standard deviations were always less, than 0.5 mN/m, and duplicate measurements were made for each MR concentration. Finally, phosphate buffer were confirmed not to present surface activity by measuring separately the surface tension of a phosphate buffer solution, obtaining values practically equal to those of pure water. 

When a carbon fiber microelectrode modified with alkaline phosphatase was immersed in a solution containing 4-aminophenyl phosphate, a steady-state current (at an applied potential of +0.30V this potential was employed in order to ensure mass transport control) due to the oxidation of 4-aminophenol was reached within 3-4 sec. in cating a rapid response. Moreover, the magnitude of the steady state current was dependent on the solution concentration of 4-aminophenyl phosphate. The response was linear at the lower concentrations (Figure 5) and (asymptotically) reached a saturation limit (as expected) for substrate concentrations above 20mM. From the linear portion of the calibration curve, a sensitivity of 1.4x10 nA M and a limit of detection of 5x10"were determined. These values are, in fact, superior to those that we had previously obtained at conventionally sized electrodes. 

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