Pseudo steady-state approximation

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). 

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

Therefore, we need to find approximate methods for simultaneous reaction systems that will permit finding analytical solutions for reactants and products in simple and usable form. We use two approximations that were developed by chemists to simplify simultaneous reaction systems (1) the equilibrium step approximation and (2) the pseudo-steady-state approximation... 

The first example we wiU use as an application of the equilibrium step approximation and pseudo-steady-state approximation is the reaction... 

Let us next apply the pseudo-steady-state approximation to see if it agrees with the experimentally observed rate. We write the mass balance on CN2O2 ... 

Thus we see that the pseudo-steady-state approximation gives orders of the reaction as the thermodynamic equilibrium approximation, the only difference being the definition of the rate constant... 

Thus both approximations predict rate expressions that agree with experimental data as long as the second term in the denominator of the pseudo-steady-state approximation is small. 

To show that the above rate expression can yield this rate expression, let us apply the pseudo-steady-state approximation on [CH3] and [CH3CO] and see what it predicts. Mass balances on all species yield. 

We will use the pseudo-steady-state approximation on [CH3] and on [CH3CO]. These... 

Applying the pseudo-steady-state approximation to A we obtain dCA ... 

Textbooks state that the pseudo-steady-state approximation will be valid if the concentration of a species is small. However, one then proceeds by setting its time derivative equal to zero (]/t/f = 0) in the batch reactor equation, not by setting the concentration (CH3CO ) equal to zero. This logic is not obvious from the batch reactor equations because setting the derivative of a concentration equal to zero is not the same as setting its concentration equal to zero. 

However, when we examine the CSTR mass balance, we see that the pseudo-Steady-state approximation is indeed that the concentration be small or that [CH3CO ]/t = 0. Thus by examining the CSTR version of the mass-balance equations, we are led to the pseudo-steady-state approximation naturally. This is expressly because the CSTR mass-balance equations are developed assuming steady state so that the pseudo-steady-state approximation in fact implies simply that an intermediate species is in steady state and its concentration is small. 

As for the quasi (pseudo)-steady-state case, the basic assumption in deriving kinetic equations is the well-known Bodenshtein hypothesis according to which the rates of formation and consumption of intermediates are equal. In fact. Chapman was first who proposed this hypothesis (see in more detail in the book by Yablonskii et al., 1991). The approach based on this idea, the Quasi-Steady-State Approximation (QSSA), is a common method for eliminating intermediates from the kinetic models of complex catalytic reactions and corresponding transformation of these models. As well known, in the literature on chemical problems, another name of this approach, the Pseudo-Steady-State Approximation (PSSA) is used. However, the term "Quasi-Steady-State Approximation" is more popular. According to the Internet, the number of references on the QSSA is more than 70,000 in comparison with about 22,000, number of references on PSSA. 

Consider a straight tubular runner of length L. A melt following the power-law model is injected at constant pressure into the runner. The melt front progresses along the runner until it reaches the gate located at its end. Calculate the melt front position, Z(f), and the instantaneous flow rate, Q t), as a function of time. Assume an incompressible fluid and an isothermal and fully developed flow, and make use of the pseudo-steady-state approximation. For a polymer melt with K = 2.18 x 10 N s"/m and n = 0.39, calculate Z(t) and Q(t)... 

The enolate ions are unstable intermediates, hence the pseudo steady state approximation can be applied to these intermediates, resulting in a kinetic model in which only stable components figure. It also can be proven (ref.5) that such a model will be mathematically equivalent to the one as follows from the network presented in figure 1. 

When the hydrate growth rate (dm/dt) is measured by the rate of gas consumption (drii/dt) the pseudo-steady-state approximation is made. That is, at any instant the rate of gas consumption by the hydrate is assumed equal to the rate of gas consumption from the gas phase. Frequently, experimenters monitor the amount of gas needed to keep the pressure constant in the hydrate vessel so that the driving force remains constant. In such cases, the rate of gas consumption from a separate supply reservoir is measured. 

To illustrate this approximation, let us consider a pressure flow in which the driving-force pressure drop varies with time. We set dp/dt and d /dt in the equations of continuity and motion, respectively, equal to zero and proceed to solve the problem as if it were a steady-state one, that is, we assume AP to be constant and not a function of time. The solution is of the form v = v(x, A P t), geometry, etc.). Because AP was taken to be a constant, v is also a constant with time. The pseudo-steady-state approximation pretends that the foregoing solution holds for any level of AP and that the functional dependence of v on time is v(x,-, t) = v(x,-, AP(t), geometry, etc.). The pseudo-steady state approximation is not valid if the values of A(pv)/At (At being the characteristic time of fluctuation of AP) obtained using this approximation contribute to an appreciable fraction of the mean value of the applied AP. 

Release kinetic expression for a tablet (radius rG and thickness /0) can be developed by using Higuchi s pseudo-steady-state approximation and mass balance. With macroscopic observation of the moving boundary of a dispersed drug tablet, Equation (6.76) can be formulated by ... 

It is perhaps easiest to explain the pseudo-steady-state approximation by way of an example. Consider the simple reaction A — B + C, whose elementary steps consist of the activation of A by collision with a background molecule M (in the atmosphere M is typically N2 and 02) to produce an energetic A molecule denoted by A, followed by the decomposition of A to give B and C. Thus, we write the mechanism as... 

There are three terms here because there are two ways to lose A and one way to form it. The reactive intermediate in this system of reactions is A. The pseudo-steady-state approximation states that the rate of formation of A is equal to its rate of loss in other words, [A ] does not change over time. Thus,... 

This analysis does not account for the heat required to heat the liquid filled core to a new temperature which is nearly equal to the liquid surface temperature. This amount of heat is small compared to the heat of evaporation. Again the pseudo-steady state approximation has been used for similar reasons. A summary of the derived equations for the drying time when transport in the pores is the rate determining step are given in Table 14.2. 

Some work has already been done on the simulation of transient behavior of moving bed coal gasifiers. However, the analysis is not based on the use of a truly dynamic model but instead uses a steady state gasifier model plus a pseudo steady state approximation. For this type of approach, the time response of the gasifier to reactor input changes appears as a continuous sequence of new steady states. 

Although the pseudo steady state approximation provides a useful tool for estimating some aspects of gasifier dynamics, it does not provide the means to examine the full range of dynamic behavior that one would expect to find for a gasifier. Therefore, a different approach has been taken here in that a nonlinear... 

One further note, the University of Delaware gasifier model used in the pseudo steady state approximation assumes that the gas and solids temperatures are the same within the reactor. That assumption removes an important dynamic feedback effect between the countercurrent flowing gas and solids streams. This is particularly important when the burning zone moves up and down within the reactor in an oscillatory manner in response to a step change in operating conditions. 

Pseudo-steady-state approximation for the unstable intermediate i.e., the concentration of these does not change during reaction... 

Ref. [8] gives a more rigorous derivation based on a pseudo steady state approximation. 

Pseudo-steady-state approximation for the unstable intermediate ... 

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