In batch systems

Under some conditions, it is observed that complex oscillatory sequences develop even in batch systems, typically towards the end of the oscillatory phase of the reaction. Transient chaos —see section A3.14.3.3— appears to be established [18]. 

Sorption Rates in Batch Systems. Direct measurement of the uptake rate by gravimetric, volumetric, or pie2ometric methods is widely used as a means of measuring intraparticle diffusivities. Diffusive transport within a particle may be represented by the Fickian diffusion equation, which, in spherical coordinates, takes the form... 

Rate equations are used to describe interphase mass transfer in batch systems, packed beds, and other contacting devices for sorptive processes and are formulated in terms of fundamental transport properties of adsorbent and adsorbate. 

The following details establish reactor performance, considers the overall fractional yield, and predicts the concentration profiles with time of complex reactions in batch systems using the Runge-Kutta numerical method of analysis. 

Fig. 5.17. Free lipase in batch system Product inhibition plots (Left Hanes-woolf Right Curve fit).

Equations (1.1) to (1.3) are diflerent ways of expressing the overall mass balance for a flow system with variable inventory. In steady-state flow, the derivatives vanish, the total mass in the system is constant, and the overall mass balance simply states that input equals output. In batch systems, the flow terms are zero, the time derivative is zero, and the total mass in the system remains constant. We will return to the general form of Equation (1.3) when unsteady reactors are treated in Chapter 14. Until then, the overall mass balance merely serves as a consistency check on more detailed component balances that apply to individual substances. 

Autocatalysis can cause sustained oscillations in batch systems. This idea originally met with skepticism. Some chemists believed that sustained oscillations would violate the second law of thermodynamics, but this is not true. Oscillating batch systems certainly exist, although they must have some external energy source or else the oscillations will eventually subside. An important example of an oscillating system is the circadian rhythm in animals. A simple model of a chemical oscillator, called the Lotka-Volterra reaction, has the assumed mechanism ... 

Complex mycelium pellets of Trametes versicolor with activated carbon powder were investigated for decolorization of Acid Violet 7 [49]. The complex pellets showed the best dye removal. The dye was almost completely removed in 6 h. For complex pellets, maximum decolorization rate (Vmax mg/L h) and half velocity concentration (Ks mg/L) was calculated 130.5 and 345.0 in batch system, respectively. 

Although the main cost advantage arises in batch systems the same principle can be applied to continuously operated scrubbers. This allows operation at a lower standing hypochlorite concentration in the scrubber and thus leads to improved process safety... 

Because much of the sorption/desorption research been carried out in batch systems (see Sect. 4.1.1, where particles are suspended in a well-mixed aqueous solution), the present discussion is mainly for the phenomena occurring on the intraparticle scale (i.e., within individual solid soil grains or within aggregates that are stable in aqueous systems). 

In the past, the majority of high-pressure homogeneous catalytic reactions were conducted in batch systems, which may cause problems in scale-up for SCFs because of the higher pressures needed for achieving the supercritical state. Therefore, continuous processing has also been investigated in the last years. It would be preferable for industrial-scale SCF reactions, because it involves smaller and, hence, safer equipment [144-150]. In addition, capital costs are likely to be lower than in batch systems. 

Continuous flocculation in stirred tanks shows similar trends with regard to agitation/mixing conditions as for batch systems. The overall performance of continuous flocculation processes tends to be inferior to that which can be obtained in batch systems. This is attributed to a closer approach to equilibrium in the continuous case. In-line flocculation, by injection of polymer into turbulent flow in a pipe, is widely practised in industry and offers certain advantages. Multiple addition at several points along the length of a pipe is preferred to single-point addition. 

In this section, the basic theory required for the analysis and interpretation of adsorption and ion-exchange kinetics in batch systems is presented. For this analysis, we consider the transient adsorption of a single solute from a dilute solution in a constant volume, well-mixed batch system, or equivalently, adsorption of a pure gas. Moreover, uniform spherical particles and isothermal conditions are assumed. Finally, diffusion coefficients are considered to be constant. Heat transfer has not been taken into account in the following analysis, since adsorption and ion exchange are not chemical reactions and occur principally with little evolution or uptake of heat. Furthermore, in environmental applications,... 

As an example, we consider the oxidation of a series of monosubstituted anilines by Mn02 in batch systems. In this case, quite a good correlation between log kK (expressed relative to R of 4-chloroaniline) and E]/2(ArX ) is obtained (Fig. 14.20). The slope of -0.54 indicates that, similar to what we have postulated for the reduction of NACs by surface-bound Fe(II) (see Fig. 14.10 /), the overall reaction rate is determined not solely by the actual electron transfer but also by other steps such as precursor complex formation. Comparable results (slopes of between-0.5 and -0.6) were obtained for the reaction of Mn02 at pH 4 with a series of substituted anilines (Laha and Luthy, 1990), and with a series of substituted phenols at pH 4.4 (Stone, 1987). In all these cases, only initial pseudo-first-order rate constants determined with clean Mn02 were considered. In the presence of solutes such as Mn2+ that may adsorb to the oxide surface, much slower reaction rates and much... 

If the processes just described are assumed to characterize the transfer of mass and energy in a fixed-bed adsorber, the conservation principles may be applied to them to describe the temperature and concentration as a function of time and position. Presenting the equations for a fixed-bed geometry has the advantage of including also equations, as special cases, for transient adsorption in single particles or groups of particles in batch systems. 

Attarakih M, Bart HJ, Faqir NM. An approximate optimal moving-grid technique for the solution of discretized population balances in batch systems. Proceedings of ESCAPE 12 European Symposium on Computer-Aided Process Engineering, The Hague, 2002. 

For a series reaction network the most important variable is either time in batch systems or residence time in continuous flow systems. For the reaction system A - B - C the concentration profiles with respect to time in a batch reactor (or residence time in a PFR) are given in Figure 6. 

Smith EH. Uptake of heavy metals in batch systems by a recycled iron-bearing material. Water Res 1996 30 2424-2434. 

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