Steady-state conditions, pseudo

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. 

Under pseudo steady-state condition, the total or overall flux (J) is equal to the flux in each phase, i.e., J = Jr = Jf = Jm ... 

Assuming perfect mixing in each compartment of the membrane pack and reservoir of the ED unit shown in Figure 9, the solute concentration in any of them is uniform and equal to that of the outlet stream. Therefore, by assuming pseudo-steady state conditions in any compartment, the differential solute and water mass balances in the diluted (D) and concentrated (C) reservoirs can be written as follows ... 

The concept of atmospheric lifetime is useful in discussions of the atmospheric degradation of anthropogenic molecules [5]. It can be defined in several ways. Most simply put it can be expressed as the turnover time, which is the atmospheric burden of a given species divided by its rate of emission, assuming a constant emission rate and steady state condition. Alternatively, it can be stated as the reciprocal of the pseudo first order rate constant (k ) for its removal ... 

A detailed consideration of all these steps in the Overhauser triplet mechanism operating in the photoreduction of qulnone systems has been given by Adrian et al. (10). With some reasonable approximations (a pseudo-first-order reaction of the polarized radicals and steady-state conditions), it is possible to show that the rate of production of nuclear spin polarized products is... 

Granting pseudo-steady-state conditions, the three rates—diffusion of A through the boundary layer, diffusion through the layer of product, and reaction at the interface—are identical. By equatmg.the expressions for each of these processes, the concentratio.ij...(,Q4-)c can. be expressed in terms of the known (C ) and the radius of the unreacted core The three... 

The solution to the preceding coupled differential equations above can be simplified by assuming a steady-state condition. Furthermore under the pseudo-first-order conditions of [O2]0 > [SO3-], we can express the rate of S(IV) depletion as... 

By applying the pseudo steady state conditions to the free radicals, the concentration of hydroxyl radicals can be estimated by (Benitez F.J., 1994)... 

Figure 2 shows the behavior of the catalysts as a function of time on stream at 310 °C. Some of the samples were tested for more than 60 h because 40-50 hours were necessary for the CuZnAl s ple to obtain pseudo steady-state conditions. The sudden decreases in the conversion curves are caused by instability of the feed. The conversion obtained with the CuZnAl sample, dropped from an initial value of about 10 % to 5.5 % after 60 h on stream. However, even after 60 h on stream a slight deactivation could be observed. 

The diffusion cloud (flame) technique developed by Hartel and Polanyi in the 1930s is one of the early methods of studying rapid bimolecular chemical reactions imder pseudo-first-order, steady-state conditions. This method is the source of most measured rates for reactions of alkali metals with halogenated compounds and still serves as a basis for experimental and theoretical studies. In most applications of the technique, sodium metal is heated in an oven, mixed with an inert carrier gas, and allowed to diffuse into a backgroimd of a reactant gas. In very exothermic reactions the sodium flame is chemiluminescent otherwise the cloud is illuminated with a sodium resonance lamp. The reaction rate can be measured either by determining the distance the sodium diffuses until it all reacts or by spectroscopically measuring the total amount of sodium in the cloud. ... 

This chapter has been devoted to continuous reactors and their analyses. We have examined the powerful idealizations of the CSTR and PER. Pseudo-steady states and steady states have been covered as well as chemical equilibrium. We must remember that the steady-state condition can be far from the equilibrium condition, and the two time-independent situations must not be confused. The time-dependent CSTR and PER are interesting problems, but they are less often used than the steady-state solutions. 

For Eq. (3.10) and Eq. (3.12) the bimolecular rate constants can be replaced with pseudo unimolecular rate constants within the limits of either fractional electron capture or constant positive ion concentration. All the above reactions take place on a time scale that is fast relative to the time required for transport through the detector. Under steady state conditions the electron capture coefficient K (see Eq. 3.7) is given by... 

If experimental conditions are carefully selected so that all individual steps are either first-order or pseudo-first-order processes, then, under transient or pre-steady-state conditions, the reaction time courses will take the form of a sum of exponentials (i.e., a linear combination of the individual rate equations for each relaxation), such that the observable, time-dependent changes in absorbance, AZ, are given by the relationship... 

Such an enzyme under normal steady-state conditions tends to have the concentrations of its substrates and products at or near to their equilibrium values. Any enzyme enhances equally both the forward and the reverse rates of its catalyzed reaction. Those enzymes that catalyze seemingly irreversible reactions usually involve hydrolysis and since the water concentration in cells and tissues is thousands of times greater than metabolites, the equilibrium position of these reactions is poised in favor of the forward or hydrolytic process. In reactions not involving water, the net direction of the reactions depends on the relative concentration of the substrates and products. If the activity of the enzyme is sufficiently high, relative to others in a metabolic pathway, altering the activity of the enzyme simply makes both the forward and reverse reactions go faster, thus maintaining a pseudo-equilibrium. 

It may happen that some cells (usually the ones on the flame frontier) that simulate a CSTR are in an unstable condition such that the reactor continuously starts and stops. In this instance, the steady-state condition does not exist, although the overall condition seems to be steady. Dynamic simulation allows us to obtain this pseudo-steady-state condition, which is otherwise impossible to get directly just by solving the nonlinear system. 

Approximate Solution. Approximate solutions of the model equations (1) - (3) were obtained under pseudo steady state conditions and it can be shown that the... 

For most practical cases, the pseudo-steady-state assumption can be applied to the radicals in the polymer particles and in the aqueous phase. Therefore, Equations 6.8 and 6.9 are converted in algebraic equations by making the left-hand side equal to zero. Under pseudo-steady-state conditions the exact solution for is available in terms of Bessel functions [33 ], but it is not easy to use. A simpler and accurate equation for n is as follows [34] ... 

A set of hypothetical data shown in Table I was assumed true for an undercompacted, stress-sensitive reservoir whose pressures were to be matched. The match period in all runs was 200 days, at which time a pseudo-steady state condition in the reservoir would have been attained. A single producing well was located at the center of the reservoir and was allowed to produce for 200 days. The drawdown data for the 200 days were then matched. Two sets of simulated drawdown data were used. One set was obtained assuming there was no measurement error in the data and another set was obtained with 0.20 percent measurement error. Reservoir permeability and porosity were separately and jointly estimated, and pressure prediction for an additional 60 days was obtained. The confidence limits of the point estimates and the predicted performance (pressure) data at 95 percent confidence level were then calculated. The results are presented as follows ... 

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. 

Solution We assume a pseudo-steady-state condition since the level drops very slowly. As time progresses, the path length z increases. At any time t, Eq. (6.2-22) holds but the path length is z and Eq. (6.2-22) becomes as follows where N and z are now variables. 

---