Pseudo- steady-state

The overall value in of the economic potential increases significantly with the increased value of the product, but each case shows the same maximum and at the same point in time. The changes in mass of B and D are apparently linear in time over most of the run up to 10,000 time units. Why should this be Linear time dependence indicates a constancy of slope. However, this is a fully transient, that is, time-dependent, system. How can we have a constant slope, that is, a constant rate of change, in the concentrations for a fully transient system To understand this we must reintroduce the concept of the pseudo-steady state. 

A situation that can arise in the well-stirred fed-batch reactor is one in which the rate of consumption of the added component is balanced exactly by its rate of addition. In this case the rate of change of the mass of A in the reactor is effectively zero as long as there is sufficient B present for reaction to take place. This leads to a period of operation that can be considered to be a steady state, but we refer to it is a pseudo-steady state because at the same time the rate of change of B is real and constant. We will go back to the equations of change to understand what this means  

If the rate of change of the concentration of A is zero, then the following simplifications apply  

If the feed flow rate and concentration of A are constant, then we would find that the concentrations of B and D are linear in time and with oppositely signed slopes. We use stst to designate the steady-state time-dependent concentrations of B and D. 

General spell1 Possible spelling error new symbol name qaf is similar to existing symbol qAf . 

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Figure B 1.16.9 shows background-free, pseudo-steady-state CIDNP spectra of the photoreaction of triethylamine with (a) anthroquinone as sensitizer and (b) and (c) xanthone as sensitizer. Details of the pseudo-steady-state CIDNP method are given elsewhere [22]. In trace (a), no signals from the p protons of products 1 (recombination) or 2 (escape) are observed, indicating that the products observed result from the radical ion pair. Traces (b) and (c) illustrate a usefiil feature of pulsed CIDNP net and multiplet effects may be separated on the basis of their radiofrequency (RF) pulse tip angle dependence [21]. Net effects are shown in trace (b) while multiplet effects can...

Figure Bl.16.9. Background-free, pseudo-steady-state CIDNP spectra observed in the photoreaction of triethylamine with different sensitizers ((a), antliraquinone (b), xanthone, CIDNP net effect (c), xanthone, CIDNP multiplet effect, amplitudes multiplied by 1.75 relative to the centre trace) in acetonitrile-d3. The stmctiiral formulae of the most important products bearing polarizations (1, regenerated starting material 2, N,N-diethylvinylamine 3, combination product of amine and sensitizer) are given at the top R denotes the sensitizer moiety. The polarized resonances of these products are assigned in the spectra. Reprinted from [21].

Mechanism. The thermal cracking of hydrocarbons proceeds via a free-radical mechanism (20). Siace that discovery, many reaction schemes have been proposed for various hydrocarbon feeds (21—24). Siace radicals are neutral species with a short life, their concentrations under reaction conditions are extremely small. Therefore, the iategration of continuity equations involving radical and molecular species requires special iategration algorithms (25). An approximate method known as pseudo steady-state approximation has been used ia chemical kinetics for many years (26,27). The errors associated with various approximations ia predicting the product distribution have been given (28). 

On the basis of the pseudo-steady state assumption, the net rate of disappearanee is zero, therefore... 

Briggs and Haldane [8] proposed a general mathematieal deseription of enzymatie kinetie reaetion. Their model is based on the assumption that after a short initial startup period, the eoneentration of the enzyme-substrate eomplex is in a pseudo-steady state (PSS). Eor a eonstant volume bateh reaetor operated at eonstant temperature T, and pH, the rate expressions and material balanees on S, E, ES, and P are... 

Thischapterhasbeendevotedtocontinuousreactorsandtheiranalyses.Wehaveexamined the powerful idealizations of the CSTR and PER. Pseudo-steady states and steady states... 

The inlet monomer concentration was varied sinusoidally to determine the effect of these changes on Dp, the time-averaged polydispersity, when compared with the steady-state case. For the unsteady state CSTR, the pseudo steady-state assumption for active centres was used to simplify computations. In both of the mechanisms considered, D increases with respect to the steady-state value (for constant conversion and number average chain length y ) as the frequency of the oscillation in the monomer feed concentration is decreased. The maximum deviation in D thus occurs as lo 0. However, it was predicted that the value of D could only be increased by 10-325S with respect to the steady state depending on reaction mechanism and the amplitude of the oscillating feed. Laurence and Vasudevan (12) considered a reaction with combination termination and no chain transfer. 

The reactor is operated in the semibatch mode with component A being sparged into the stirred tank. Unreacted A and the reaction products leave through the gas phase so that the mass of liquid remains constant. To the extent that these assumptions are true and the catalyst does not deactivate, a pseudo-steady-state can be achieved. Find (flg)o j. Assume that Henry s law is valid throughout the composition range and ignore any changes in the gas density. 

Measurements Using Liquid-Phase Reactions. Liquid-phase reactions, and the oxidation of sodium sulfite to sodium sulfate in particular, are sometimes used to determine kiAi. As for the transient method, the system is batch with respect to the liquid phase. Pure oxygen is sparged into the vessel. A pseudo-steady-state results. There is no gas outlet, and the inlet flow rate is adjusted so that the vessel pressure remains constant. Under these circumstances, the inlet flow rate equals the mass transfer rate. Equations (11.5) and (11.12) are combined to give a particularly simple result ... 

As illustrated in Fig. 1, the activated carbon displays the highest conversion and selectivity among all the catalysts during the initial reaction period, however, its catalytic activity continues to decrease during the reaction, which is probably caused by coke deposition in the micropores. By contrast, the reaction over the CNF composites treated in air and HN03 can reach a pseudo-steady state after about 200 min. Similiar transient state is also observed on the CNFs and the untreated composite. Table 3 collects the kinetic results after 300 min on stream over catalysts tested for the ODE, in which the activity is referred to the BET surface area. The air-treated composite gives the highest conversion and styrene selectivity at steady state. 

In PAMPA measurements each well is usually a one-point-in-time (single-timepoint) sample. By contrast, in the conventional multitimepoint Caco-2 assay, the acceptor solution is frequently replaced with fresh buffer solution so that the solution in contact with the membrane contains no more than a few percent of the total sample concentration at any time. This condition can be called a physically maintained sink. Under pseudo-steady state (when a practically linear solute concentration gradient is established in the membrane phase see Chapter 2), lipophilic molecules will distribute into the cell monolayer in accordance with the effective membrane-buffer partition coefficient, even when the acceptor solution contains nearly zero sample concentration (due to the physical sink). If the physical sink is maintained indefinitely, then eventually, all of the sample will be depleted from both the donor and membrane compartments, as the flux approaches zero (Chapter 2). In conventional Caco-2 data analysis, a very simple equation [Eq. (7.10) or (7.11)] is used to calculate the permeability coefficient. But when combinatorial (i.e., lipophilic) compounds are screened, this equation is often invalid, since a considerable portion of the molecules partitions into the membrane phase during the multitimepoint measurements. 

The mathematical solution to moving boundary problem involves setting up a pseudo-steady-state model. The pseudo-steady-state assumption is valid as long as the boundary moves ponderously slowly compared with the time required to reach steady state. Thus, we are assuming that the boundary between the salt solution and the solid salt moves slowly in the tablet compared to the diffusion... 

The design q>roblem can be approached at various levels of sophistication using different mathematical models of the packed bed. In cases of industrial interest, it is not possible to obtain closed form analytical solutions for any but the simplest of models under isothermal operating conditions. However, numerical procedures can be employed to predict effluent compositions on the basis of the various models. In the subsections that follow, we shall consider first the fundamental equations that must be obeyed by all packed bed reactors under various energy transfer constraints, and then discuss some of the simplest models of reactor behavior. These discussions are limited to pseudo steady-state operating conditions (i.e., the catalyst activity is presumed to be essentially constant for times that are long compared to the fluid residence time in the reactor). 

A 4 in. diameter open can has a 1/4 in. diameter hole in the bottom. The can is immersed bottom down in a pool of water, to a point where the bottom is 6 in. below the water surface and is held there while the water flows through the hole into the can. How long will it take for the water in the can to rise to the same level as that outside the can Neglect friction, and assume a pseudo steady state, i.e., time changes are so slow that at any instant the steady state Bernoulli equation applies. 

The extension of the same mechanistic reasoning to the corresponding thermal process (carried out in the dark) is not generally rigorous. Most commonly, the adiabatic electron-transfer step (kET) is significantly slower than the fast back electron transfer and follow-up reactions (fcf) described in Section 7, and the pseudo-steady-state concentration is too low for the ion-radical pair to be directly observed (equation 99). 

If elementary rate laws are assumed for each step, and if Nc. is essentially constant over a short time, a (pseudo-) steady-state rate law can be developed ... 

Pseudorandom vectors, 26 1002 Pseudo steady-state approximation, 10 599 Pseudovitamins, 17 651, 25 807 Pseudo-volumetric chemical reaction, 25 278... 

If the ideas are used that an infinite sea of liquid surrounds each gas bubble and a pseudo-steady state exists, then the molar flux can be described by the familiar expression... 

The Randles-SevCik equation usually gives a more accurate value since all measurements are performed under pseudo-steady-state conditions. In practice, a Randles-SevCik plot of Ip against is drawn for a redox couple of known... 

Discussion. Fixed bed cracking reactors as well as commercial moving bed reactors operate under steady state or pseudo-steady state conditions ( ). Observed selectivity (eg., ratio of yield of branched to n-paraffin) in a steady state catalytic reactor is independent of space velocity (1, 17). The selectivity depends on intrinsic rate constants and diffusivities of the reacting species which depend on temperature. Thus, the selectivity observations reported here are applicable to commercial FCC units operating at space velocities different from that employed in this study. 

For ease of solution, it is assumed that the spherical shape of the pellet is maintained throughout reaction and that the densities of the solid product and solid reactant are equal. Adopting the pseudo-steady state hypothesis implies that the intrinsic chemical reaction rate is very much greater than diffusional processes in the product layer and consequently the reaction is confined to a gradually receding interface between reactant core and product ash. Under these circumstances, the problem can be formulated in terms of pseudo-steady state diffusion through the product layer. The conservation equation for this zone will simply reflect that (in the pseudo-steady state) there will be no net change in diffusive flux so... 

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