Flux bottleneck

Figure 5. Determination of carbon flux bottlenecks in the toluene bioconversion pathway using resting cells of P. putida EM2878, (a) Identification of T4MO rate limiting step. Arrow denotes time at which p-cresol was added to culture. Curves indicate amount of p-cresol added (A), 0 ppm (B), 10 ppm (C), 50 ppm (D), 150 ppm. (b) Transient accumulation of p-hydroxybenzyl alcohol in conversion of p-cresol to HBA. (k) p-cresol ( ) HBA HBZ IWi) p-hydroxybenzyl alcohol. Lines drawn to indicate trends. (Reproduced with permission from reference 26. Copyright 1999 The Royal Society of Chemistry.)...

A bottleneck in all membrane processes, applied in practice, is fouling and scaling of the membranes. These processes cause a decrease in water flux through the membrane and a decrease in retention. Much attention is paid, especially in case of nanofiltration and hyperfiltration, to prevent fouling of the membrane by an intensive pretreatment and the regular removal of fouling and scaling layers by means of mechanical, physical or chemical treatment. 

We have also shown that these bottleneck membranes can show the same selectivity but higher flux than the more conventional shape (Fig. 

A). To demonstrate this point, the rate and selectivity of transport across a conventional (Fig. 15A) and a bottleneck membrane were compared. Both membranes were able to cleanly separate from Ru(bpy)3 in the two-molecule permeation experiment (see below). Hence, these membranes showed comparable, excellent selectivity. However, as expected, the flux of across the bottleneck nanotubule membrane was dramatically higher than for the conventional nanotubule membrane (14 vs. 0.07 nmol hr cm ). 

Because of the nonuniform shape of the bottleneck tubules (Fig. 15B), it is difficult to extract an i.d. using the gas-flux method [71]. All bottleneck membranes were plated a pH 12 bath for a duration of 8 hours, t Quinine was excited at = 308 nm and detected at X = 403 nm. 

The approach pursued in this and the next chapter is focused on the common mathematical characteristics of boundary processes. Most of the necessary mathematics has been developed in Chapter 18. Yet, from a physical point of view, many different driving forces are responsible for the transfer of mass. For instance, air-water exchange (Chapter 20), described as either bottleneck or diffusive boundary, is controlled by the turbulent energy flux produced by wind and water currents. The nature of these and other phenomena will be discussed once the mathematical structure of the models has been developed. 

Before dealing with this and other examples, let us derive the mathematical tools which we need to describe the flux of a chemical across a simple bottleneck boundary. First, we recognize that for a conservative substance at steady-state, the flux, F(x), along the boundary coordinate x orthogonal to the boundary must be constant. According to Fick s first law (Eq. 18-6) the flux is given by ... 

Figure 19.5 In multibox models, the exchange between two fairly homogeneous regions is expressed as the exchange flux of fluid (water, air etc.), Q . Normalization by the contact area, A, yields the exchange velocity vex = QJA. This quotient can be interpreted as a bottleneck exchange velocity vex =...

Thus, we ought to know how the total concentration change from zone A to B, (CA - CB), is partitioned between the two bottlenecks. Remember that at steady-state the flux of a conservative substance along x must be constant (Eq. 19-1), thus ... 

In these equations we recognize expressions which by now should have become familiar to us. During the initial phase of the exchange process (t Zcm), boundary concentration and flux at the interface remind us of a (B-side controlled) bottleneck boundary with transfer velocity vbl = Db,/S (see Eq. 19-19). The concentrations on either side are C and CBq=CA/FA/B, where the latter is the B-side concentration in equilibrium with the initial A-side concentration CA. 

Due to the spherical geometry of the surface, the concentration profile across the boundary layer is no longer a straight line as was the case for the flat bottleneck boundary (Fig. 19.4). We can calculate the steady-state profile by assuming that CF and CFq = Cs/Fs/F are constant. Then, the integrated flux, ZF, across all concentric shells with radius r inside the boundary layer (r0 < r < r0 + 8) must be equal ... 

At first sight Eq. 19-63 does not resemble the type of equation that we found earlier for bottleneck boundaries (e.g., Eq. 19-19). That is not surprising since XF is an integrated flux (mass per unit time), while in the case of flat boundaries we have always dealt with specific fluxes (mass per unit area and time). If we divide XF by the surface of the sphere, 4n r, after some algebraic rearrangements we get ... 

The flux across a bottleneck boundary can be expressed either in terms of Fick s first law or by a transfer velocity. Explain how the two views are related. 

Assume that the concentrations on either side of a boundary, CA and CB, are constant. You calculate the flux across this boundary by treating it (a) as a bottleneck boundary and (b) as a wall boundary, respectively. How does the flux evolve as a function of time in these two models ... 

All the necessary tools to develop kinetic models for air-water exchange have been derived already in Chapters 18 and 19. However, we don t yet understand in detail the physical processes which act at the water surface and which are relevant for the exchange of chemicals between air and water. In fact, we are not even able to clearly classify the air-water interface either as a bottleneck boundary, a diffusive boundary, or even something else. Therefore, for a quantitative description of mass fluxes at this interface, we have to make use of a mixture of theoretical concepts and empirical knowledge. Fortunately, the former provide us with information which is independent of the exact nature of the exchange process. As it turned out, the different flux equations which we have derived so far (see Eqs. 19-3, 19-12, 19-57) are all of the form ... 

Note that the inverse of -Kha /ha is identical with aa which was introduced in Eq. 8-21. Here we choose the Annotation to indicate that the ratio is like a partition coefficient which appears in the flux (Eq. 20-1) if different phases or different chemical species are involved (see section 19.2 and Eq. 19-20). In order to show how the combination of both partitioning relationships, one between air and water (Eq. 20-42), the other between neutral and total concentration (Eq. 20-43), affect the air-water exchange of [HA], we choose the simplest air-water transfer model, the film or bottleneck model. Figure 20.11 helps to understand the following derivation. 

Thus, again, as in the pseudo-JT effect considered in Section 3 and, also, in the E <8> e case [7], the tunneling rate E is proportional to the probability flux through the bottleneck point of the potential barrier. Similar to equation (21), the right-side (9 > 0) ground-state WKB wave function under the barrier is... 

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