Pseudo-steady state transfer

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. 

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 ... 

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). 

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). 

In most processes, steps 1, 3, 5 and 6 are in pseudo steady state and the mass transfer is governed by diifusion through the gas-liquid layers (steps 2 and 4). An additional step can appear if one deals with aggregates of cells (pellets), but we will not examine this case. 

Let us consider the burning of an ideal spherical particle in static gas. The oxidant diffuses to the surface of the particle to react with the carbon C + CO2, while the latter diffuses out from the surface of the particle. The combustion heat is transferred to the surrounding gas partially by convection and partially by radiation. The following assumptions were made in the modeling (1) The process is at a pseudo steady state. (2) The temperature the highest at the surface, and continuously drops down outwards from the surface of the particle and the concentration of oxidant is highest in the bulk... 

GPC has become especially useful in polymerization reaction engineering since it permits comparatively rapid and precise determination of MWD. As shown in Figure 5, Xw obtainable from the MWD in a batch reactor can be used to determine the initiation rate constant as well as kp2/kt assuming the pseudo-steady state, the absence of chain transfer, and termination by combination. Similar relationships with CSTR s are available. 

The differences between the TBR and the MR originate from the differences in catalyst geometry, which affect catalyst load, internal and external mass transfer resistance, contact areas, as well as pressure drop. These effects have been analyzed by Edvinsson and Cybulski [ 14,26] via computer simulations based on relatively simple mathematical models of the MR and TBR. They considered catalytic consecutive hydrogenation reactions carried out in a plug-flow reactor with cocurrent downflow of both phases, operated isothermally in a pseudo-steady state all fluctuations were modeled by a corresponding time average ... 

Dutta et al. [32] modified the pseudo-steady-state advancing reaction front model of Stroeve and Varanasi [30] by considering the polydispersity of the emulsion globules and the external phase mass transfer resistance. They also included the outer membrane film resistance in their model [5]. Their results were in good agreement with experimental data for phenol extraction. 

Asai et al. (1994) have developed a reaction model for the oxidation of benzyl alcohol using hypochlorite ion in the presence of a PT catalyst. Based on the film theory, they develop analytic expressions for the mass-transfer rate of QY across the interface and for the inter-facial concentration of QY. Recently, Bhattacharya (1996) has developed a simple and general framework for modeling PTC reactions in liquid-liquid systems. The uniqueness of this approach stems from the fact that it can model complex multistep reactions in both aqueous and organic phases, and thus could model both normal and inverse PTC reactions. The model does not resort to the commonly made pseudo-steady-state assumption, nor does it assume extractive equilibrium. This unified framework was validated with experimental data from a number of previous articles for both PTC and IPTC systems. 

Sysicm is,at pseudo steady state with regard to the panicle sizc i.e., panicle growth is slow compared to oxygen transfer. 

Figure 12.5 depicts schematically the gas- and aqueous-phase concentrations of A in and around a droplet. The aqueous-phase concentrations have been scaled by HART, to remove the difference in the units of the two concentrations. This scaling implies that the two concentration profiles should meet at the interface if the system satisfies at that point Henry s law. In the ideal case, described by (12.45), the concentration profile after the scaling should be constant for any r. However, in the general case the gas-phase mass transfer resistance results in a drop of the concentration from cA(oo) to cA(Rp) at the air-droplet interface. The interface resistance to mass transfer may also cause deviations from Henry s law equilibrium indicated in Figure 12.5 by a discontinuity. Finally, aqueous-phase transport limitations may result in a profile of the concentration of A in the aqueous phase from [A(/ ,)J at the droplet surface to [A(0)] at the center. All these mass transfer limitations, even if the system can reach a pseudo-steady state, result in reductions of the concentration of A inside the droplet, and slow down the aqueous-phase chemical reactions. 

The dynamics for a slow PT reaction and a mass transfer controlled instantaneous reaction were studied. Wu [63] and Wu and Meng [69] indicated that the pseudo-steady-state LLPTC model could describe the complicated nature of the LLPTC reaction. The rate equation from the report of Wu [63] is expressed as... 

A similar model that specifically considers the poison deposition in a catalyst pellet was presented by Olson [5] and Carberry and Gorring [6], Here the poison is assumed to deposit in the catalyst as a moving boundary of a poisoned shell surrounding an unpoisoned core, as in an adsorption situation. These types of models are also often used for noncatalytic heterogeneous reactions, which was discussed in detail in Chapter 4. The pseudo-steady-state assumption is made that the boundary moves rather slowly compared to the poison diffusion or reaction rates. Then, steady-state diffusion results can be used for the shell, and the total mass transfer resistance consists of the usual external interfacial, pore diffusion, and boundary chemical reaction steps in series. 

Under the pseudo-steady-state equilibrium, the number of moles of metal ion diffuse per second through the boundary layers is constant. A model developed by Danes [30,31] was one of the initial models developed based on the steady-state mass transfer of metal ion in HFSLM. Simple assumptions... 

Approximate solution of Plank for freezing. Plank (P2) has derived an approximate solution for the time of freezing which is often sufficient for engineering purposes. The assumptions in the derivation are as follows. Initially, all the food is at the freezing temperature but is unfrozen. The thermal conductivity, of the frozen part is constant. All the material freezes at the freezing point, with a constant latent heat. The heat transfer by conduction in the frozen layer occurs slowly enough so that it is under pseudo-steady-state conditions. 

Tronconi and co-workers (98,116) have validated against experiment a more complex, heterogeneous, transient model, accovmting also for diffusion and reaction of NO and NH3 inside the porous walls of extruded honeycomb SCR catalysts. The model equations are presented in Table 5 x and z are the intraporous and axial coordinate, respectively is the ammonia adsorption capacity of the catalyst 6 is the NH3 surface coverage is the effective intraporous diffiisiv-ity s is the monolith wall half-thickness i is the gas velocity in the monolith channels are gas-solid mass transfer coefficients and dh is the hydraulic diameter of the monolith channels. Notably, a pseudo-steady-state assumption... 

R. S. Barlow, Analysis of the Adsorption Process and Desiccant Systems-A Pseudo Steady State Model for Coupled Heat and Mass Transfer, Solar Energy Research Institute, Golden, CO, Report No. SERI/TR-631-1330, 1982. 

Diffusion-Type Mass Transfer Models for Type 1 FacUitation. The state-of-the-art model for Type 1 facilitation is the advancing front model (2,7,8), In this model, the solute is assumed to react instantaneously and irreversibly with the internal reagent at a reaction surface which advances into the globule as the reagent is consumed. A perturbation solution to the resulting nonlinear equations is obtained. In general, the zero-order or pseudo-steady-state solution alone often gives an adequate representation of the diffusion process. 

Here is an effective thermal conductivity of the material bed being dried and Aj, is the cross-sectional area of the bed. The rate at which the dry-wet interface location coordinate z changes is very low. We may therefore assume a pseudo steady state condition. This heat flux will evaporate water correspondingly, the dry-wet interface location will change and the volume of the wet bed, zA, will decrease. The rate of heat transfer from the plate bottom must equal the rate of evaporation of the local moisture in the bed ... 

The above analysis/description of solvent flux and macrosolute rejection/retention/ttansmission far an ultra-flllration membreme was carried out in the context of a pseudo steady state analysis in a batch cell (Figure 6.3.26 (a)). Back diffusion of the macrosolute from the feed solution-membrane interface to the bulk solution takes place by simple difflision against the small bulk flow parallel to the force direction. The resulting mass-transfer coefficients for macrosolutes will be quite small the solvent flux levels achievable will be quite low. For practically useful ultrafiltration rates, the mass-transfer coefficient is increased via different flow configurations with respect to the force. 

The external mass transfer resistance has been neglected such that N is set to Nb- This equation can be integrated with the pseudo steady state assumption to give ... 

---