Limiting flux

For liquid boiling the designer is limited by a maximum flux q/A. This handbook cannot treat this subject in detail. For most applications assuming a limiting flux of 10,000 will give a ballpark estimate. 

The derivation of a steady-state solution requires boundary conditions, but no initial condition. Steady-state can be seen as the asymptotic solution (so never mathematically reached at any finite time [43]) of the transient, which -for practical purposes - can be approached in a reasonably short time. For instance, limiting-flux diffusion of a species with diffusion coefficient Di = 10-9 m2 s 1 towards a spherical organism of radius rQ = 1 jxm is practically attained at t r jDi = 1 ms. 

The bioaffinity parameter a basically reflects the free metal ion concentration, whereas the limiting flux ratio b reflects the total labile metal species concentration. Due to the complexation, the ratio a/b thus changes by a factor (1 +sKcl) in spherical geometry, while the factor (1 I sKc ) (l I Kc ) is required for planar geometry [26]. 

Despite its widely recognized limitations, flux balance analysis has resulted in a large number of successful applications [35, 67, 72 74], including several extensions and refinements. See Ref. [247] for a recent review. Of particular interest are recent efforts to augment the stoichiometric balance equations with thermodynamic constraints providing a link between concentration and flux in the constraint-based analysis of metabolic networks [74, 149, 150]. For a more comprehensive review, we refer to the very readable monograph of Palsson [50]. 

Availability of reactive iron in sediments also has been postulated to control S retention. Reactive iron may limit fluxes of S recycled from sediments by rendering sulfide immobile and less amenable to oxidation by bacteria or chemical agents. Availability of iron strongly influences total S content and isotopic signature of marine sediments (198). Canfield (94) ob-... 

The physical meaning of the quantities appearing in Eq. (5-12) is obvious JL = cx DT (4/3)/S is the limiting diffusion flux at Q on the metallic surface, free from porous layer and Jn-,x = D /SF is the limiting flux when the entire concentration gradient is located within the porous layer, i.e. when Q- . 

The effect of the gel layer on the flux through an ultrafiltration membrane at different feed pressures is illustrated in Figure 6.7. At a very low pressure p, the flux Jv is low, so the effect of concentration polarization is small, and a gel layer does not form on the membrane surface. The flux is close to the pure water flux of the membrane at the same pressure. As the applied pressure is increased to pressure p2, the higher flux causes increased concentration polarization, and the concentration of retained material at the membrane surface increases. If the pressure is increased further to p3, concentration polarization becomes enough for the retained solutes at the membrane surface to reach the gel concentration cgel and form the secondary barrier layer. This is the limiting flux for the membrane. Further increases in pressure only increase the thickness of the gel layer, not the flux. 

Plots of the limiting flux. /mux as a function of solution concentration for latex solution data are shown in Figure 6.9 for a series of latex solutions at various feed solution flow rates. A series of straight line plots is obtained, and these extrapolate to the gel concentration cgel at zero flux. The slopes of the plots in Figure 6.9 are proportional to the term D/S in Equation (6.3). The increase in flux resulting from an increase in the fluid recirculation rate is caused by the decrease in the boundary layer thickness S. 

Figure 6.9 Ultrafiltration flux with a latex solution at an applied pressure of 60 psi (in the limiting flux region) as a function of feed solution latex concentration. These results were obtained in a high-turbulence, thin-channel cell. The solution recirculation rate is shown in the figure [13]...

Experiments with different ultrafiltration membranes and the same feed solution often yield very different ultrafiltration limiting fluxes. But according to the model shown in Figure 6.6 and represented by Equation (6.2), the ultrafiltration limiting flux is independent of the membrane type. 

Tang, C. Y., and Leckie, J. O. (2007). Membrane independent limiting flux for RO and NF membranes fouled by humic acid. Environ. Sci. Technol. 41,4767 773. 

Bacchin, P. (2004) A possible link between critical and limiting flux for colloidal systems consideration of critical deposit... 

Although O2 reacts with proton sources to form HOO- (which dispropor-tionates via a second 02), with limiting fluxes of protons to control the rate of HOO- formation from 02, the rate of decay of HOO- is enhanced by reaction with the allylic hydrogens of excess 1,4-cyclohexadiene (1,4-CHD).25 Because HOO- disproportionation is a second-order process, low concentrations favor hydrogen-atom abstraction from 1,4-CHD. This is especially so for MezSO, in which the rate of disproportionation for HOO- is the slowest (PhCl > MeCN > H20 > DMF > Me O).16... 

Collman JP et al (2007) A cytochrome c oxidase model catalyzes oxygen to water reduction under rate-limiting flux. Science 315 1565-1568... 

In the longer term, the solution to limited flux is more—and more powerful— neutron sources. There is a considerable ongoing building program of both reactors and spallation sources. The FRM-II reactor (Munich, Germany) went critical for the first time in 2004. A replacement for the HIFAR reactor (Lucas Heights, Australia) is under construction and is scheduled for operation in late 2007. 

If there were no kinetic control (i.e., if all species could dissociate sufficiently rapidly to contribute fully to the metal flux) then, under the same conditions, the diffusion limited flux of the metal would be... 

For a particular diffusion layer thickness S the thermodynamic availability, as measured with an ISE or calculated from a speciation model, and the electrochemical availability, as measured by ASV, represent limiting cases of a continuum of trace metal availability. The nature of this continuum is most simply defined by considering the flux of the free metal ion across the diffusion layer to a surface which senses the metal availability. The ratio of the observed flux (J) to the limiting flux (J] ) is unity for ASV measurements under current limiting conditions and zero for ISE measurements. 

In simple metabolic pathways, with only one independent flux, the value of Cj is usually between zero and 1. Partial inhibition of an enzyme with zero flux control does not affect the flux, while an enzyme with a flux control coefficient of 1 is completely rate-limiting. Flux control coefficients between 0 and 1 have often been found (Groen et al. 1982 Ruijter et al. 1991 Snoep et al. 1996). In ideal metabolic pathways, for example, in the absence of metabolite channelling and coenzyme sequestration (Kholodenko et al. 1995), the sum of the flux control coefficients of all enzymes in the pathway must equal 1 ... 

Critical section—Also called terminal section, this is the location in a thickener where the limiting flux is transpiring. 

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