Pseudosteady state

The system is at a pseudosteady state, i.e., the overall concentration in the tank is constant, and the effect of the rate of change of this composition is negligible. 

The average reaction rate for a pseudosteady state is calculated according to... 

During each run, the products are analyzed to determine "pseudosteady state conditions" and the temperature is varied to obtain an approximate measure of activity changes. These measurements are made during a span of several hours. 

The concentration of constituent B becomes negligible at the surface of the mineral grain. Gradually, the rate of mass diffusion of B (Eq. 5.21) through an increasing depleted layer (y) becomes slower and is equal to the rate of surface-controlled dissolution of A (Eq. 5.22). Thus, a pseudosteady state is attained and the depleted layer thickness stabilizes. The rates of reaction of solid layer diffusion (Eq. 5.21) and of surface controlled dissolution become equal ... 

Consequently, while I jump into continuous reactors in Chapter 3, I have tried to cover essentially aU of conventional chemical kinetics in this book. I have tried to include aU the kinetics material in any of the chemical kinetics texts designed for undergraduates, but these are placed within and at the end of chapters throughout the book. The descriptions of reactions and kinetics in Chapter 2 do not assume any previous exposure to chemical kinetics. The simplification of complex reactions (pseudosteady-state and equilibrium step approximations) are covered in Chapter 4, as are theories of unimolecular and bimolecular reactions. I mention the need for statistical mechanics and quantum mechanics in interpreting reaction rates but do not go into state-to-state dynamics of reactions. The kinetics with catalysts (Chapter 7), solids (Chapter 9), combustion (Chapter 10), polymerization (Chapter 11), and reactions between phases (Chapter 12) are all given sufficient treatment that their rate expressions can be justified and used in the appropriate reactor mass balances. 

Until recently, MR flow imaging in fixed beds was limited to imaging pseudosteady-state phenomena. Data acquisition times for 2-D flow images were typically several minutes, or even tens of minutes, and successful imaging of the velocity field requires the flow field to be stable over the data acquisition time. Therefore, flow imaging studies had been restricted to relatively low flow rates (Re <200 based on the diameter of the packing elements) (82). Figure 27 shows data for a narrow packed bed characterized by a column-to-particle diameter ratio of 2 spheres of diameter 19 mm were packed within a 38-mm-diameter column. Flows at two different values of Reynolds number are shown (Re — 200 and 300). 

Fig. 1.9. Consecutive first-order reactions and cubic autocatalysis, showing pseudosteady-state predictions for the intermediate concentrations. Initial concentrations and rate constants are given in Table 1.1.

This approach is based on the premise that Al can be used as a tracer for bottom sediment material and that the concentration of Al in resus-pendable surface sediment is fairly uniform basinwide. Detailed profiles of size-fractionated particulate aluminum concentrations spaced closely in time over the unstratified period show vertical concentration profiles at nearly uniform levels, indicating that a pseudosteady state had been achieved. The mean areal pool of Al during this period was designated as the net resuspended pool (80-90% settles from the water column by September), and the quantity of surface sediment required to supply this pool was calculated. 

On the mathematical status of the pseudosteady state hypothesis of biochemical kinetics (with F.G. Heineken and H.M. Tsuchiya). Math. BioscL 1,95-113 (1967). 

Assuming that the bubble phase is in a pseudosteady state, the bubble phase equations are... 

The pseudosteady-state assumption for the hydroxyl ions gives us... 

In this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

The assumption of pseudosteady state in the bubble phase and the general assumptions allow us to describe the bubble-phase temperature profile using the algebraic equation (7.35). 

Equations (7.29) and (7.33) represent the pseudosteady-state gas oil balances for the dense and bubble phases, respectively, while equations (7.30) and (7.34) do the same for gasoline. Equations (7.32) and (7.35) are used for the steady-state dense- and bubble-phase temperatures of the reactor, and equations (7.37) and (7.38) for the dense and bubble-phase temperatures in the regenerator. 

Several useful approximate analytical solutions to Eq. (4.8) were developed. A well-known example is Higuchi s equation, based on a pseudosteady-state approach7 ... 

Chemicals that are not highly lipid-soluble and that have relatively short half-lives (hours to days) do not build up a stable long-term reservoir. However, a pseudosteady state can be attained if exposure is nearly uniform and constant (Figure 5-8). Pseudosteady state refers to the case in which blood or tissue concentrations are changing and therefore are not completely stable. These blood or tissue concentrations fluctuate slightly and in a regular pattern around the average concentration. 

FIGURE 5-8 Blood concentrations of rapidly cleared chemical to which there is frequent and nearly uniform exposure. Highlighted line ( — ) is mean blood concentration. Under these exposure conditions, biomarker concentration will be within a factor of 2 of mean after first several hours. Simplifying assumption of pseudosteady state (mean concentration is approximated by concentration found at any sampling time) may suffice for estimating exposure dose from blood concentration under these circumstances. 

In summary, conversion of biomonitoring data to exposure dose requires knowledge of chemical elimination rate and Vd and requires that conditions be approximately pseudosteady state. It may be useful to estimate dose from body burden however, this cannot be used to interpret an individual s biomonitoring result, because the elimination rate and Vd would not be known. Reasonable bounds on elimination rate and Vd could be used to calculate an upper end of daily dose that is still compatible with the biomonitoring results (for example, when both Vd and elimination rate are high). 

The Higuchi model is an approximate solution in that it assumes a pseudosteady state , in which the concentration profile from the dispersed drug front to the outer surface is linear. Paul and McSpadden [24] have shown that the correct expression can be written as ... 

A + B can be limited by the rate of breaking apart the encountered pair AB. For the pathways of Figure 6, the encountered species is BPA. Rate constants kj, k i, and k3 are intrinsic, whereas k2 is the inverse of the duration of an encounter. Rate constant k2 is proportional to the diffusion coefficient of the encountered species, which will be high in the gas phase and several orders of magnitude lower in the liquid phase. Allowing BPA to exist in a pseudosteady-state permits derivation of a rate expression for the pathways of Figure 6 as ... 

However, because of the pseudosteady state assumptions involved, Higuchi s equation is only valid when the drug loading is in excess of the drug solubility (A>>C ). At the limit of A+C, Higuchi s equation gives a result 11.31 smaller than the exact solution. Lee (4)recently presented a simple analytical solution for this problem which is uniformly valid over all A/Cg values ... 

Equation (6.89) is identical with Equation (6.72) derived under the pseudosteady-state approximation. The error caused by Higuchi s equation increases with decreasing values of A/Cs. 

The turnover numbers and activation energies for methanation over all four metals freshly reduced and under sulfur-free conditions (Table XVI) were in good agreement with values reported for supported metal catalysts (220). At 673 K, Ni and Ru exhibited only very slow losses in activity apparently due to slow carbon deposition, whereas Co and Fe underwent rapid, severe carbon deactivation after maintaining their fresh catalyst activity for a few hours. After rapid deactivation the final steady-state activity, which was about 100-fold lower than the activity of the fresh catalyst, was approached slowly this activity region was referred to as the lower pseudosteady state. Likewise, the fresh catalyst behavior was referred to as the upper pseudo-... 

Activation energies for methanation over Ni and Ru were the same for both the fresh and the aged catalysts (Table XVII). In contrast, activation energies for methanation over aged Co and Fe were lower by 50 and 25 kJ/mol, respectively as compared to fresh Co and Fe (Table XVII). In the case of Co the CO partial pressure dependence changed from a negative order in the upper pseudosteady state to a positive-order dependence in the lower pseudosteady state (Table XVIII). The dependence on H2 partial pressure was positive one-half order in both upper and lower pseudosteady states. For Fe the kinetic behavior was not investigated. 

Several methods have been proposed to overcome this problem. In one, the styrene and part of the butadiene are charged initially with butadiene metered at a rate equivalent to its incorporation into the chain. A second approach involves adding both monomers at a relatively slow rate so that the equilibrium monomer concentration reaches a pseudosteady state that will produce polymer at the desired composition.32 This process can be done in either a batch or a continuous mode.33... 

A kinetic expression for the above reaction can be obtained by assuming a pseudosteady state for the adsorbed species A(ad]) and P(ad3). The measure of the concentrations of these species is the degree of occupation of the surface, 0A and 0P, respectively 0A and Qp are defined such that they range from zero (no active surface area covered at all) to unity (all active surface area covered). We further assume a pseudosteady state for the degrees of occupation of the catalytically active surface area, so that 0A and 0P do not change in time, so d6Jdt = d6,Jdt = 0. If the concentration of active sites in the catalyst pellet is C, then the free sites, on which A can be adsorbed, are given by C ( 1 -0A- 0B) and the rate of adsorption by... 

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