Flux dependence

Diffusion in the bulk crystals may sometimes be short circuited by diffusion down grain boundaries or dislocation cores. The boundary acts as a planar channel, about two atoms wide, with a local diffusion rate which can be as much as 10 times greater than in the bulk (Figs. 18.8 and 10.4). The dislocation core, too, can act as a high conductivity wire of cross-section about (2b), where b is the atom size (Fig. 18.9). Of course, their contribution to the total diffusive flux depends also on how many grain boundaries or dislocations there are when grains are small or dislocations numerous, their contribution becomes important. 

Flux depends on the product of membrane permeability of the solute times the concentration of the solute (summed over all charge state forms) at the water side of the donor surface of the membrane. This concentration ideally may be equal to the dose of the drug, unless the dose exceeds the solubility limit at the pH considered, in... 

The solids flux depends on the local concentration of solids, the settling velocity of the solids at this concentration relative to the liquid, and the net velocity of the liquid. Thus the local solids flux will vary within the thickener because the concentration of solids increases with depth and the amount of liquid that is displaced (upward) by the solids decreases as the solids concentration increases, thus affecting the upward drag on the particles. As these two effects act in opposite directions, there will be some point in the thickener at which the actual solids flux is a minimum. This point determines the conditions for stable steady-state operation, as explained below. 

The solids mixing study by injection of tracer particles indicated that the axial mixing of solids in the bubble street is apparently very fast. Radial mixing flux depends primarily on the bubble size, bubble velocity, and bubble frequency, which in turn depend on the size of the jet nozzle employed and the operating jet velocity. 

Some of the thermal and fluid factors affecting the burning rate are illustrated in Figure 9.1. Convective heat flux depends on the flow conditions and we see that both... 

As seen in equations (32)-(34), the forward adsorptive flux depends upon the concentration of free cell surface carriers. Unfortunately, there is only limited information in the literature on determinations of carrier concentrations for the uptake of trace metals. In principle, graphical and numerical methods can be used to determine carrier numbers and the equilibrium constant, As, corresponding to the formation of M — Rcen following measurement of [M] and (M —Rceii. For example, a (Scatchard) plot of (M — RCeii /[M] versus (M — RCeii should yield a straight line with a slope equal to the reciprocal of the dissociation constant and abscissa-intercept equal to the total carrier numbers (e.g. [186]). 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

However, this simple picture only applies to gases that do not undergo reactions in the boundary layers. For gases that do react, for example through hydration and acid-base reactions, the net flux depends on the simultaneous movement of all the solutes involved, and the flux will not be the simple function of concentration expressed in Equation (3.25). An example is CO2, which reacts with water to form carbonic acid and carbonate species-H2C03, HCOs and COs . The situation is complicated because the exchange of H+ ions in the carbonate equilibria results in a pH gradient across the still layer, and it is therefore necessary to account for the movement of H+ ions across the still layer as well as the movement of carbonate species. The situation is further complicated in the case of CO2 by the kinetics of hydration and dehydration, which may be slow in comparison with transport. 

Figure 13. Flux dependence on wall shear rate in laminar flow...

As pointed out above, the bond flux depends on the connectivity of the compound, that is, on the bond graph. This means that the length of a bond depends not only on its immediate environment, but also on the structure of the whole crystal or molecule of which the bond is part. Thus anions such as PO, which ideally are perfect tetrahedra, will often be distorted when they appear in crystals. However, this distortion can normally be predicted via the network equations provided the graph of the bond network is known. 

The effects of osmolarity on cells are well known from physiology. Semipermeable membranes allow a water flux depending on the number of solved ions. Therefore, a cell with its own tissue osmolarity of about... 

Clearly, the heat flux depends on z alone. Thus the heat flux is everywhere the same on the stagnation surface and is proportional to the nondimensional temperature gradient. Newton s law of cooling often provides a convenient way to represent wall heat flux,... 

The equation for the value of the velocity at each node is based on a momentum balance for each control volume. In the interior of the domain, the control volume has a momentum flux crossing each of the four sides. The flux depends on the sign of the velocity gradient and the outward-normal unit vector that defines the face orientation. In discrete, integral form, the two-dimensional difference equation emerges as... 

In such a material under these conditions, Fourier s law again pertains, but the thermal conductivity K depends on the direct coefficient Lqq, as in Eq. 2.25, as well as on the direct and coupling coefficients associated with electrical charge flow. In general, the empirical conductivity associated with a particular flux depends on the constraints applied to other possible fluxes. 

Each flux depends linearly on all the driving forces. 

That notorious pair, the Danckwerts boundary conditions for the tubular reactor, provides a good illustration of boundary conditions arising from nature. Much ink has been spilt over these, particularly the exit condition that Danckwerts based on his (perfectly correct, but intuitive) engineering insight. If we take the steady-state case of the simplest distributed example given previously but make the flux depend on dispersion as well as on convection, then, because there is only one-space dimension,/= vAc — DA dddz), where D is a dispersion coefficient. Then, as the assumption of steady state eliminates... 

Finally, it is worth reiterating that transport numbers are relatively complex functions of the concentration and mobility of all the ions present in the system. Thus, while the relationship between lidocaine molar fraction in the binary system (lidocaine-sodium) and the drug flux has been well defined [32,78], the results cannot be directly extrapolated to a different anodal composition. That is, the drug flux depends not only on its molar fraction [59], but also on the mobilities of the competing ions [115]. 

Diffusion of the solute is necessary for precipitate growth. The diffusional flux depends on both the diffusivity and the concentration gradient, which is proportional to cQ — ca. At low amounts of supercooling, the term ca — ca is nearly... 

The nitrogen supplies on land consist of the assimilable nitrogen in the soil VS2 0.19-104tkm-2, in plants (12 1091), and living organisms (0.2 1091). A diversity of nitrogen fluxes is formed here of the processes of nitrification, denitrification, ammonification, fixation, and river run-off. The intensities of these fluxes depend on climatic conditions, temperature regime, moisture, as well as the chemical and physical properties of soil. Many qualitative and quantitative characteristics of these dependences have been described in the literature (Hellebrandt et al., 2003). Let us consider some of them. 

Radiant Flux Dependence. Radiant flux can be measured by utilizing calibrated metallic grids and neutral density filters, or by filtering solutions of various absorbances without changing the geometry of the irradiating beam. For low radiant fluxes , a linear relation between the reaction rate and

values, the rate becomes proportional to... 

The upshot on the oscillation is a direct measure for the extent of perturbation on the metabolic network upon the uptake of a PAC. Glycolytic oscillations that are systematically perturbed by altered environmental conditions, i.e. exposure to the xenobiotic, constitute a direct and easily accessible measure of the intracellular behavior since the frequency and amplitude of oscillating metabolite concentrations and fluxes depend on both the perturbation and on most intracellular processes due to the coupled energy (ATP) and redox (NADH) balances (Fig. 3.4). 

The influence of any metabolite is described by a single bme-dependent variable, the concentration. The temporal evolution of the variables is then computed from ordinary differenbal equations (ODEs) that predict the immediate change of a variable according to the size of the fluxes that produce or degrade the particular metabolite. In turn, the size of each flux depends on a well defined set of variables, i.e. the instantaneous concentrations of substrates, products, cofactors and effectors, and a set of parameters. The state of the model, i.e. the values of all metabolite concentrations, evolves by updating the variables over subsequent time intervals which are repeated for different external conditions to yield theoretical time courses of concentrations and fluxes. 

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