Pseudo-steady-state assumption

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

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 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... 

We should mention that this and the succeeding batch-solids derivations are based on the pseudo steady-state assumption. This assumes that conditions change slowly enough with time for the system to be at steady state at any instant. Since a batch of solids can only be used if deactivation is not too rapid this assumption is reasonable. 

A pseudo-steady-state assumption was introduced for HONO, in view of its ready decomposition to N2... 

Prior to gelation, two types of radicals with solid- and liquid-like mobility are present they are possibly located in microgels (solid-like mobility) and in monomers (liquid-like mobility). The concentration of free radicals increases continuously, so that the pseudo steady-state assumption cannot be applied to model the reaction kinetics. 

FFB Model. Use of FFB units in the industry for catalyst characterization is primarily due to excellent temperature control and resulting isothermal reactor temperature. The operating conditions of the FFB activity unit used in our study are given in Table I. The material balance using a pseudo steady state assumption (at any given instant the vapor is in contact with catalyst of almost uniform activity) gives ... 

At the reaction interface 6 between the ore and iron layers, using the pseudo steady state assumption the dimensionless material balances may be represented by... 

Pseudo-Steady-State Assumption (PSSA). This assumption, limited to chain reactions with termination, states that the absolute value of the rate of change of free radicals with time is so small relative to the other terms in the differential equations for free radicals, that it can be set equal to zero. Aris (3) refers to this as an... 

P — Pseudo-steady-state assumption C = Continuous variable for chain length L = Laplace transforms Z = Z-transforms E = Eigenzeit transform M = Moments of distribution N = Numerical techniques... 

Because of the time-dependency of the metabolic fluxes, there are no direct methods for their analysis in-vivo. Nevertheless, intracellular fluxes can be quantified assuming that the intracellular concentration of metabolites is constant at all times (pseudo-steady state assumption). For a given metabolic network, the balance around each metabolite imposes a number of constraints on the system. In general, if there are fluxes and K metabolites, then the degree of freedom is F = J — K . Through the measurements of F fluxes, i.e., nutrient uptake, growth... 

Neilsen [40] and Reiss and LaMer [41] solved ttie same problem with its movii boundary and pseudo-steady state assumption, giving... 

This analysis constitutes a pseudo-steady state assumption that is, the flux is assumed to be constant (Le., equation 14.25 and 14.27) for a particular core radius of the sphere. This is a reasonably good approximation because the volume of vapor produced is very large compared to the volume of liquid evaporated by a factor of approximately 1000. This results in an equation for the drying time which is of the form t = t f Rc/Ro)-... 

The differential equation for the mass balance of gaseous reactant B reacting with a pellet of radius under the pseudo-steady-state assumption is... 

The material balance for the gaseous reactant A in the ash layer of the pellet under the pseudo-steady-state assumption is represented by ... 

This type of solution method is possible for reactions where deactivation is slow, and a pseudo steady-state assumption can be made when solving the mass balance equations. Thus, these equations are applicable to reactions where the activity loss is first-order in both the poison and the active sites, and where deactivation is slow compared to the main reaction. A similar type of approach was taken by Johnson et al. (5), for oxygen consumption and carbon content during catalyst regeneration and by Bohart and Adams (6), for chlorine consumption and absorbence capacity of charcoal. 

The empty-site requirement in Eq. (28) can be physically interpreted in one of two different ways either the adsorbed A and B have to rearrange prior to reaction, or they are bound to more than one adsorption site. For the latter case, the intermediate concentration is low, thus allowing a pseudo-steady-state assumption. Through the application of bifurcation analysis and catastrophe theory this model was found to predict a very rich bifurcation and dynamic behavior. For certain parameter values, sub- and supercritical Hopf bifurcations as well as homoclinic bifurcations were discovered with this simple model. The oscillation cycle predicted by such a model is sketched in Fig. 6c. This model was also used to analyze how white noise would affect the behavior of an oscillatory reaction system... 

Pore Mouth (or Shell Progressive) Poisoning This mechanism occurs when the poisoning of a pore surface begins at the mouth of the pore and moves gradmuly inward. This is a moving boundary problem, and the pseudo-steady-state assumption is made that the boundary moves slowly compared with diffusion of poison and reactants and reaction on the active surface. P is the fraction of the pore that is deactivated. The poison diffuses through the dead zone and deposits at the interface between the dead and active zones. The reactants diffuse across the dead zone without reaction, followed by diffusion-reaction in the active zone. 

To evaluate the gradient in Eq. (14-2) consider the diffusion of A through the layer of F. With the pseudo-steady-state assumption this process can be evaluated independently of the change in r. Consider a small element of thickness Ar at location r in the product layer (Fig. 14-3). The steady-state mass balance of A around this layer is... 

Ishikawa, H., Maeda, T., Hikita, H and Miyatake, K. (1988) The computerised derivation of rate equations for enzyme reactions on the basis of the pseudo-steady-state assumption and the rapid-equilibrium assumption. Biochem. J. 251, 175-181. 

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. 

A very important particular case of gas-solid reactions is that in which a component of the solid phase reacts with the reactant gas phase very rapidly and the particle size remains constant. This establishes a reaction interface between the two zones in one zone there is diffusion (but no reaction) to the interface, and in the other there is only reaction at the interface. This is usually called the shrinking core model. If again we make the pseudo-steady-state assumption, then from equation (7-84)... 

An apparent overall effectiveness factor of the catalyst is obtained by applying the pseudo-steady-state assumption to the mass balance equations within the catalyst, 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. 

Recently, Bhattacharya (1996) developed a simple, general framework for modeling PTC reactions in liquid-liquid systems. The main feature of this analysis is that it can model complex multistep reactions in both aqueous and organic phases and is thus applicable to both normal and inverse PTC reactions. It does not resort to the commonly made pseudo-steady-state assumption nor does it assume extractive equilibrium. 

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