Steady states first-order approximation

We conclude the present discussion of the one-dimensional problems in transport theory with some remarks about the diffusion approximation and the relation of the partial currents introduced in Chap. 5 to the directed, or angular, flux. For this purpose we require the steady-state first-order approximations to the Eqs. (7.84). These are obtained by retaining only terms for at = 0, 1 thus, the diffusion theory is based on the requirement that terms in ot > 2 are negligibly small in comparison... 

These equations hold if an Ignition Curve test consists of measuring conversion (X) as the unique function of temperature (T). This is done by a series of short, steady-state experiments at various temperature levels. Since this is done in a tubular, isothermal reactor at very low concentration of pollutant, the first order kinetic applies. In this case, results should be listed as pairs of corresponding X and T values. (The first order approximation was not needed in the previous ethylene oxide example, because reaction rates were measured directly as the total function of temperature, whereas all other concentrations changed with the temperature.) The example is from Appendix A, in Berty (1997). In the Ignition Curve measurement a graph is made to plot the temperature needed for the conversion achieved. 

Figure 6 Approximations of the thermodynamic branch steady-state multiplicity case (see Figure 1). Solid line is the first-term hypergeometric approximation. Circles correspond to the higher-order hypergeometric approximation (m = 3). Dashed line is the first-order approximation in the vicinity of thermodynamic equilibrium. Dash-dots correspond to the second-order approximation in the vicinity of thermodynamic equilibrium.

The steady-state reaction rate and relaxation time are determined by these two constants. In that case their effects are coupled. For the steady state we get in first-order approximation instead of Equation (13) ... 

The steady-state approximation on the base of the rule (13) is a linear function of the restored-and-cut cycles rate-limiting constants. It is the first-order approximation. 

Note that the above approximation is a first order approximation. If we were to use a central difference, we would increase the order, but contrary to what is expected, this choice will adversely affect the accuracy and stability of the solution due to the fact the information is forced to travel in a direction that is not supported by the physics of the problem. How convective problems are dealt will be discussed in more detail later in this chapter. The following sections will present steady state, transient and moving boundary problems with examples and applications. 

For hydrodynamic electrodes, in order to solve the convective-diffusion equation analytically for the steady-state limiting current, it is necessary to use a first-order approximation of the convection function(s) (such as the Leveque approximation for the channel). These approximate expressions for the steady-state mass transport limited currents were introduced in Section 4 (see Table 5). 

Both physical and technological properties of copolymers are influenced by the sequence distribution in the macromolecular chains. The mathematical relationships governing the distribution, first developed by Alfrey and Goldfinger (7), are based upon kinetic and statistical considerations implying three fundamental assumptions a) steady state copolymerization, b) terminal effect only (i.e. influence of the last, but not of the penultimate unit of a growing chain on the addition of the next monomeric unit), and c) constant monomer feed. Under these assumptions, which may be defined as a first order approximation, the copolymers are described by two quantities, the ratio / of the molar fractions of the two monomers and the product of reactivity ratios... 

For the electron-only transport conditions considered here, under weak illumination conditions and negligible dark conductivity, the Fourier coefficient of the first-order space-charge field at steady-state can be approximated to... 

Among the Cg hydrocarbons, the yield of n-hexane was very small as compared to that of 2,3-dimethylbutane. It was concluded that the concentration of n-propyl radicals in this system is small, and the formation of /i-butane by radical recombination (k ,) may be neglected in a first-order approximation. The fraction of triplet methylene then corresponds to 2C2H6-I-isobutane from triplet CHj. The latter quantity may be derived from a steady-state treatment by use of the known relative rate coefficients of radical recombinations, viz. 

Although LNT is valid close to equilibrium, it is a good step for understanding the behaviour of systems beyond equilibrium as a first approximation. We will try to examine theory and experiment for typical steady states in order to assess to what extent... 

As shown in the example in this chapter, inverse response is caused by two competing processes - the faster of which takes the process first in a direction opposite to the steady state. We can approximate this as two first-order processes with gains of opposite sign, so that the combined effect is given by... 

As sketched in Figure 8.8b, the SECM tip electrogenerates a redox species that is transported to the substrate through the polymeric film. The measurement of a redox cycling or feedback allows for the quantification of the permeation of the redox species within the membrane. Steady-state measurements such as those obtained from these approach curves, ij - d, for example, presented in Figure 8.11, have been proposed. A complete theoretical analysis of the approach curve shape depending on the membrane characteristics (permeation and thickness) has been described. For thin membrane layers of thickness e, the approach curves follow the first-order approximation, and when fitted by Equations 8.3 to 8.5, an effective heterogeneous rate constant k ff = PDp/e is obtained from which the permeability of the electroactive species PDp ensues. 

In the first order approximation considering that the gas is not in the uniform steady-state, a first approximation to / is given by / = The solution... 

Figure 4.3 allows us to verify that the steady state is valid only in cases where the droplet does not disappear. The exponential included in expression (4.6) for the solubility at the interface of the smaller droplet is now predominant and induces a rapid increase in Cm- However, equation (4.14) cannot be used to compute the disappearance of the droplet because, in this case, the exponential of the Kelvin equation cannot be given a first-order approximation. Such other physical events as the desorption of the surfactant from the interface must be taken into account. 

When Nk is small this mean value is not very different from the geometric mean. The boundary values are readily determined from the initial value P(t=0) and the steady state voltage using the Nernst equation (see page 446) . The above equation is also useful if conductivity measurements are carried out in the stationary state of a chemical polarization and can be used to correct the first order approximation in the case of chains including a reversible and a blocking electrode [555]. 

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

With the concentration of I from the steady-state approximation, the pseudo-first-order rate constant is... 

Equations (4.1) or (4.2) are a set of N simultaneous equations in iV+1 unknowns, the unknowns being the N outlet concentrations aout,bout, , and the one volumetric flow rate Qout- Note that Qom is evaluated at the conditions within the reactor. If the mass density of the fluid is constant, as is approximately true for liquid systems, then Qout=Qm- This allows Equations (4.1) to be solved for the outlet compositions. If Qout is unknown, then the component balances must be supplemented by an equation of state for the system. Perhaps surprisingly, the algebraic equations governing the steady-state performance of a CSTR are usually more difficult to solve than the sets of simultaneous, first-order ODEs encountered in Chapters 2 and 3. We start with an example that is easy but important. 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

It is practical to make the approximation that CM(oo) m Cm (t). This is justified if the membrane is saturated with the sample in a short period of time. This lag (steady-state) time may be approximated from Fick s second law as tlag = h2 / (n2Dm), where h is the membrane thickness in centimeters and Dm is the sample diffusivity inside the membrane, in cm2/s [40,41]. Mathematically, xLAG is the time at which Fick s second law has transformed into the limiting situation of Fick s first law. In the PAMPA approximation, the lag time is taken as the time when solute molecules first appear in the acceptor compartment. This is a tradeoff approximation to achieve high-throughput speed in PAMPA. With h = 125 pm and Dm 10 7 cm2/s, it should take 3 min to saturate the lipid membrane with sample. The observed times are of the order of 20 min (see below), short enough for our purposes. Cools... 

The H20 exchange mechanism was studied by Hunt et al. (32) who reported that exchange between aqueous solvent and Fein(TPPS)(H20)2 occurs with a first-order rate constant (kex = 1.4xl07s-1 in water at 298 K) far exceeding the k0 s values determined at any [NO]. If the steady state approximation was applied with regard to the intermediate Fem(Por)(H20), the kohs for the exponential relaxation of the non-equilibrium mixture generated by the flash photolysis experiment would be,... 

The quantity and quality of experimental information determined by the new techniques call for the use of comprehensive data treatment and evaluation methods. In earlier literature, quite often kinetic studies were simplified by using pseudo-first-order conditions, the steady-state approach or initial rate methods. In some cases, these simplifications were fully justified but sometimes the approximations led to distorted results. Autoxidation reactions are particularly vulnerable to this problem because of strong kinetic coupling between the individual steps and feed-back reactions. It was demonstrated in many cases, that these reactions are very sensitive to the conditions applied and their kinetic profiles and stoichiometries may be significantly altered by changing the pH, the absolute concentrations and concentration ratios of the reactants, and also by the presence of trace amounts of impurities which may act either as catalysts and/or inhibitors. 

A large number of analytical solutions of these equations appear in the literature. Mostly, however, they deal only with first order reactions. All others require solution by numerical or other approximate means. In this book, solutions of two examples are carried along analytically part way in P7.02.06 and P7.02.07. Section 7.4 considers flow through an external film, while Section 7.5 deals with diffusion and reaction in catalyst pores under steady state conditions. 

In general, the effective overall rate constant associated with loss of reactants can be expressed in terms of the individual rate constants a-e in eq 3 by use of the steady state approximation. Simpler expressions can be obtained if the species related by diffusion (a,b) and activation (c,d) processes are assumed to be in thermal equilibrium. In such a case one finds straightforwardly that the effective first order rate constant, k (r), for electron transfer at separation r can be written as... 

Equations (5.16) of Table 5.1 refer to series first-order reactions. Of interest for the solvent extraction kinetics is a special case arising when the concentration of the intermediate, [Y], may be considered essentially constant (i.e., d[Y]/dt = 0). This approximation, called the stationary state or steady-state approximation, is particularly good when the intermediate is very reactive and present at very small concentrations. This situation is often met when the intermediate [Y] is an interfacially adsorbed species. One then obtains... 

Let us find the first-order (in rate limiting constants) approximation to the steady states. If 21 12 en fc 2 the rate-limiting constant for the... 

If we assume that the activity coefficients of X- and H20 are independent of the X- concentration at any given ionic strength, then the usual steady state treatment leads, without further approximation, to Equation 3, a relationship between the pseudo first-order rate constant and the other kinetic parameters. 

The steady-state approximation is often used for the atomic and free radical intermediates occurring in combustion processes. The validity of this approximation has been examined in connection with the theoretical calculation of laminar flame velocities (3, 20, 21) in premixed gaseous systems. The steady-state approximation is occasionally useful for obtaining first-order estimates for flame-propagation velocities but should probably not be used in estimating concentration profiles for reaction intermediates. Some additional observations on the steady-state approximation are contained in Appendix I. 

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