Steady-state concentrations and fluxes

Table VI. Measured and Estimated OH Steady-State Concentrations and Fluxes in Sunlight-Irradiated Seawater and Fresh Water...

The tip-generated interfacial undersaturation is governed by the interplay between mass transport in the tip/substrate gap and the dissolution kinetics. This concept is illustrated in Figures 15 and 16. Figure 15a and b shows the radial dependence of the steady-state concentration and flux at the crystal/solution interface for a first-order dissolution process characterized by K, = 1, 10, and 100. For rapid kinetics (K, = 100), the dissolution process is able to maintain the interfacial concentration close to the saturated value and only a small depletion in the concentration adjacent to the crystal is observed over a radial distance of about one electrode dimension. Under these conditions, diffusion in the z-direction dominates over radial diffusion. As the rate constant decreases, diffusion is able to compete with the interfacial kinetics and consequently the undersaturation at the crystal surface... 

Figure 16a and b shows the effect of L on the radial dependence of the steady-state concentration and flux at the substrate/solution interface for a first-order dissolution process characterized by Ki = 10 and L = 0.1, 0.32, and 1.0. As the tip-substrate separation decreases, the effective rate of diffusion between the probe and the surface increases, forcing the crystal/so-lution interface to become more undersaturated. Conversely, as the UME is retracted from the substrate, the interfacial undersaturation approaches the saturated value, since the solution mass transfer coefficient decreases compared to the first-order dissolution rate constant. Movement of the tip electrode away from the substrate also has the effect of promoting radial diffusion, and consequently the area of the substrate probed by the UME increases. 

Once the model was complete, it was adjusted to a steady state condition and tested using historic carbon isotope data from the atmosphere, oceans and polar ice. Several important parameters were calculated and chosen at this stage. Sensitivity analysis indicated that results dispersal of the missing carbon - were significantly influenced by the size of the vegetation carbon pool, its assimilation rate, the concentration of preindustrial atmospheric carbon used, and the CO2 fertilization factor. The model was also sensitive to several factors related to fluxes between ocean reservoirs. 

In the steady state, the total flux is constant along the entire path. This condition (i.e., that of flux continuity) is a reflection of mass balance nowhere in a steady flux will the ions accumulate or vanish (i.e., their local concentrations are time invariant). The condition of continuity of the steady flux is disturbed in those places where ions are consumed (sinks) or produced (sources) by chemical reactions. It is necessary to preserve the balance that any excess of ions supplied correspond to the amount of ions reacting, and that any excess of ions eliminated correspond to the amount of ions formed in the reaction. 

In this section we want to discuss unsteady diffusion across a permeable membrane. In other words, we are interested in how concentration and flux change before reaching the steady state discussed in Section IV.B. The membrane is initially free of solute. At time zero, the concentrations on both sides of the membrane are increased, to C and c2. Equilibrium between the solution and the membrane interface is assumed therefore, the corresponding concentrations on the membrane surfaces are Kc, and Kc2. Fick s second law is still applicable ... 

The problem is to calculate the steady-state concentration of dissolved phosphate in the five oceanic reservoirs, assuming that 95 percent of all the phosphate carried into each surface reservoir is consumed by plankton and carried downward in particulate form into the underlying deep reservoir (Figure 3-2). The remaining 5 percent of the incoming phosphate is carried out of the surface reservoir still in solution. Nearly all of the phosphorus carried into the deep sea in particles is restored to dissolved form by consumer organisms. A small fraction—equal to 1 percent of the original flux of dissolved phosphate into the surface reservoir—escapes dissolution and is removed from the ocean into seafloor sediments. This permanent removal of phosphorus is balanced by a flux of dissolved phosphate in river water, with a concentration of 10 3 mole P/m3. 

The interpretation of these results has not been completely clarified. A plausible starting point might be to assume that in the steady state, all concentrations and fluxes in the crystal are functions of the single variable C — x — vetf, where x is position relative to the original (t = 0) crystal surface, t is the time since the start of etching, and vet is the etch velocity. At any f, the diffusion-drift flux J of hydrogen must obey... 

Some transient problems tend to a trivial (and useless) steady-state solution without flux and concentration profiles. For instance, concentration profiles due to limiting diffusion towards a plane in an infinite stagnant medium always keep diminishing. Spherical and disc geometries sustain steady-state under semi-infinite diffusion, and this can be practically exploited for small-scale active surfaces. 

Let us first calculate steady-state concentrations when all Na weathered from evaporites has been transported to the sea and all the parameters become time-invariant. At steady-state, fluxes between reservoirs must be equal... 

The elements of the matrix A are fully specified by the stoichiometry matrix N and the metabolic state of the system. Usually, though not necessarily, the metabolic state corresponds to an experimentally observed state of the system and is characterized by steady-state concentrations S° and flux values v(S°). 

To evaluate model, we focus on the experimentally observed metabolic state of the pathway. However, in the case of sustained oscillations, the (unstable) steady state cannot be observed directly. We thus approximate the metabolic state by the average observed concentration and flux values, as reported in Refs. [101, 126], See Table VII for numeric values. 

At this point, our notion and implications of the term stability must be clarified. At the most basic level, and as utilized in Section VILA, dynamic stability implies that the system returns to its steady state after a small perturbation. More quantitatively, increased stability can be associated with a decreased amount of time required to return to the steady state as for example, quantified by the largest real part within the spectrum of eigenvalues. However, obviously, stability does not imply the absence of variability in metabolite concentrations. In the face of constantperturbations, the concentration and flux values will fluctuate around their... 

The last issue that remains to be addressed is whether the MBL results are sensitive to the characteristic diffusion distance L one assumes to fix the outer boundary of the domain of analysis. In the calculations so far, we took the size L of the MBL domain to be equal to the size h - a of the uncracked ligament in the pipeline. To investigate the effect of the size L on the steady state concentration profiles, in particular within the fracture process zone, we performed additional transient hydrogen transport calculations using the MBL approach with L = 8(/i — a) = 60.96 mm under the same stress intensity factor Kf =34.12 MPa /m and normalized T-stress T /steady state distributions of the NILS concentration ahead of the crack tip are plotted in Fig. 8 for the two boundary conditions, i.e. / = 0 and C, =0 on the outer boundary. The concentration profiles for the zero flux boundary condition are identical for both domain sizes. For the zero concentration boundary condition CL = 0 on the outer boundary, although the concentration profiles for the two domain sizes L = h - a and L = 8(/i - a) differ substantially away from the crack tip. they are very close in the region near the crack tip, and notably their maxima differ by less than... 

An alternative mechanism has been proposed by Schneider (1988) who considers that ferritin could be also filled via a transient, mononuclear Fe " species. This species is similar to Fe in size, but is more versatile in its interaction with the protein shell. Experiments have shown that as the pH of a system containing diferric-trans-ferrin and ferritin was lowered very slowly from 7.5 to 5.0, monomeric Fe was released from the transferrin and redeposited in the ferritin (Glaus, 1989). Calculations of the iron flux across the cell membrane and estimates of the rates of interaction of the mononuclear species with ferritin and with the cell mitochondria indicated that the steady state concentration of the mononuclear Fe species would be sufficiently low for this species alone to enter the protein shell and be deposited as the iron core. Uptake of this species by the protein shell is about fiftyfold faster than the rate of hydrolytic polymerization or even of the dimerization of Fe (tiy2 1 vs. 50 ms). This hypothesis suggests an interesting direction for further research. 

A typical chemical system is the oxidative decarboxylation of malonic acid catalyzed by cerium ions and bromine, the so-called Zhabotinsky reaction this reaction in a given domain leads to the evolution of sustained oscillations and chemical waves. Furthermore, these states have been observed in a number of enzyme systems. The simplest case is the reaction catalyzed by the enzyme peroxidase. The reaction kinetics display either steady states, bistability, or oscillations. A more complex system is the ubiquitous process of glycolysis catalyzed by a sequence of coordinated enzyme reactions. In a given domain the process readily exhibits continuous oscillations of chemical concentrations and fluxes, which can be recorded by spectroscopic and electrometric techniques. The source of the periodicity is the enzyme phosphofructokinase, which catalyzes the phosphorylation of fructose-6-phosphate by ATP, resulting in the formation of fructose-1,6 biphosphate and ADP. The overall activity of the octameric enzyme is described by an allosteric model with fructose-6-phosphate, ATP, and AMP as controlling ligands. 

At quasi-steady state, the two fluxes, Eqs. 1 and 2, have to be equal. This allows us to calculate the boundary layer concentration Cf ... 

In the steady limit (times 10ns or more are usually sufficient to attain this in mobile solvents), the flux of B towards an A reactant a distance r away is [B]0DR/r2, from eqn. (6), and the current is 47rUJ [B]0, from eqn. (7), at any separation between B and A. This constancy reflects the steady concentration of B around A achieved at long times. At shorter times the steady-state concentration of B is not established and the particle current of B is both time- and space-dependent. 

Eqn. (8.6) describes the steady state concentration profile of an (A, B) alloy which has been exposed to the stationary vacancy flux j°. The result is particularly simple if the mobilities, b are independent of composition, that is, if P = constant. From Eqn. (8.6), we infer that, depending on the ratio of the mobilities P, demixing can occur in two directions (either A or B can concentrate at the surface acting as the vacancy source). The demixing strength is proportional toy°-(l-p)/RT, and thus directly proportional to the vacancy flux density j°, and to the reciprocal of the absolute temperature, 1/71 For p = 1, there is no demixing. 

In concluding, let us comment on the time needed to attain the steady state after establishing the surface activities. Two transient processes having different relaxation times occur I) the steady state vacancy concentration profile builds up and 2) the component demixing profile builds up until eventually the system becomes truly stationary. Even if the vacancies have attained a (quasi-) steady state, their drift flux is not stationary until the demixing profile has also reached its steady state. This time dependence of the vacancy drift is responsible for the difficulties that arise when the transient transport problem must be solved explicitly, see, for example, [G. Petot-Er-vas, et al. (1992)]. 

SimpleBox is a multimedia mass balance model of the so-called Mackay type. It represents the environment as a series of well-mixed boxes of air, water, sediment, soil, and vegetation (compartments). Calculations start with user-specified emission fluxes into the compartments. Intermedia mass transfer fluxes and degradation fluxes are calculated by the model on the basis of user-specified mass transfer coefficients and degradation rate constants. The model performs a simultaneous mass balance calculation for all the compartments, and produces steady-state concentrations in... 

Mass transfer in catalysis proceeds under non-equilibrium conditions with at least two molecular species (the reactant and product molecules) involved [4, 5], Under steady state conditions, the flux of the product molecules out of the catalyst particle is stoi-chiometrically equivalent (but in the opposite direction) to the flux of the entering reactant species. The process of diffusion of two different molecular species with concentration gradients opposed to each other is called counter diffusion, and if the stoichiometry is 1 1 we have equimolar counter diffusion. The situation is then similar to that considered in the case of self-or tracer diffusion, the only difference being that now two different molecular species are involved. Tracer diffusion may be considered, therefore, as equimolar counter diffusion of two identical species. 

This type of diffusion/reaction mechanism has been treated semi-analyti-cally by Albery et al. [42, 44, 45], under steady-state conditions and its applications to amperometric chemical sensors has been described by Lyons et al. [46]. In both models, only diffusion and reaction within a boundary layer is considered, while the effect of concentration polarisation in the solution is neglected. Thus, to apply the model to an experimental system it is necessary to be able to accurately determine the concentration of substrate at the polymer/solution interface. Assuming that the system is in the steady state, the use of the rotating disc electrode allows simple determination of the substrate concentration at the interface from the bulk concentration and the experimentally determined flux using [47]... 

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