Internal gradient

Fig. 9. Schematic diagram and internal gradient for an 8.75 million-SWU/yr plant.

It should be noted that the decomposition shown in Eq. 3.7.2 is not necessarily a subdivision of separate sets of spins, as all spins in general are subject to both relaxation and diffusion. Rather, it is a classification of different components of the overall decay according to their time constant. In particular cases, the spectrum of amplitudes an represents the populations of a set of pore types, each encoded with a modulation determined by its internal gradient. However, in the case of stronger encoding, the initial magnetization distribution within a single pore type may contain multiple modes (j)n. In this case the interpretation could become more complex [49]. 

Porous catalyst particles are complex devices with appreciable internal gradients of temperature and composition, but these factors can be taken account of by the concept of catalyst effectiveness which is sometimes calculable. 

An external gas pressure gradient applied between anode and cathode sides of the fuel cell may be superimposed on the internal gradient in liquid pressure. This provides a means to control the water distribution in PEMs under fuel cell operation. This picture forms the basis for the hydraulic permeation model of membrane operation that has been proposed by Eikerling et al. This basic structural approach can be rationalized on the basis of the cluster network model. It can also be adapted to include the pertinent structural pictures of Gebel et and Schmidt-Rohr et al. ... 

Similarly, when internal gradients correspond to differences in concentration or temperature between the external surface of the catalyst particle and its centre, the rate in the particle is substantially different from that which would prevail if the concentration or temperature were the same throughout the particle. The catalytic reaction is then said to be influenced by internal mass or heat transfer, and, when this influence is the dominant one, the rate corresponds to a regime of internal mass or heat transfer. 

Paul Weisz suggested in a lucid note published in 1973 that cells, and indeed even entire organisms, have evolved in a way that maintains unity effectiveness factor [24]. That is, the size of the catalytic assembly is increased in nature as the overall rate at which that assembly operates decreases, and the relationship between characteristic dimension and activity can be well approximated by the observable modulus criterion for reaction limitation. It is possible that Weisz s arguments may fail under process conditions, and internal gradients within a compartment or cell may be important. However, at present it appears that the most important transport limitations and activities in cells are those that operate across cellular membranes. Therefore, to understand and to manipulate key transport activities in cells, it is essential that biochemical engineers understand these membrane transport processes and the factors influencing their operation. A brief outline of some of the important systems and their implications in cell function and biotechnology follows. 

The results showed that the activity is improved by increasing the linear velocities and liquid-to-gas ratios. The absence of internal diffusion limitations was checked by means of the Weisz-Prater criterion. It appeared that internal gradients were negligible. 

This example also shows the effects of mass- and enei y-transfer resistances within the catalyst pellet. The temperature increases toward the center of the pellet and increases the rate, but the oxygen concentration goes down, tending to reduce the rate. The global value of 49.8 x 10" is the resultant balance of both factors. Hence the net error in using the bulk conditions to evaluate the rate would be [(49.8 — 43.6)/49.8] (100), 12.5%. In this case the rate increase due to external and internal thermal effects more than balances the adverse effect of internal mass-transfer resistance. The procedure for calculating the effects of internal gradients on the rate is presented in Chap. 11. 

These calculations show sources and values of the possible properties mismatch in properties of the graded joints. For instance, such large differences and anisotropy in thermal conductivity confirm that heat flow in non-steady conditions would affect the temperature distribution in the joint quite significantly. In this case additional thermal stresses could be generated by internal gradients of temperature. 

II. Thus an internal gradient of Ca " would contribute to the turnover of actln filaments in migrating cells. 

External signals (e.g., growth factors and chemoattractants) Induce the assembly and organization of the cytoskeleton and the establishment of an Internal gradient of trimeric G proteins and calcium (see Figure 19-29). The resulting polarization of the cell leads to locomotion. 

Robinson SA and Osmond CB (1994) Internal gradients of chlorophyll and carotenoid pigments in relation to photo-... 

To this point we have dealt only with transport effects within the porous catalyst matrix (intraphase), and the mathematics have been worked out for boundary conditions that specify concentration and temperature at the catalyst surface. In actual fact, external boundaries often exist that offer resistance to heat and mass transport, as shown in Figure 7.1, and the surface conditions of temperature and concentration may differ substantially from those measured in the bulk fluid. Indeed, if internal gradients of temperature exist, interphase gradients in the boundary layer must also exist because of the relative values of the pertinent thermal conductivities [J.J. Carberry, Ind. Eng. Chem., 55(10), 40 (1966)]. 

For interphase limitations (boundary layer effects) the situation seems, at first glance, as simple as that for internal gradients, since most correlations for heat-and mass-transfer eoeffieients show a proportionality to the flow velocity of the surrounding fluid, u", where normally 0.6 < n < 1. At the lower velocities associated in particular with laboratory reactor operation, however, n tends to be closer to 0.6 than to 1, and the transport coefficients become insensitive to flow velocity and changing flow velocity is not an effective diagnostic. 

The temperature gradient in the gas film just outside a catalyst pellet is steeper than the internal gradient at the surface, because the solid conductivity is generally several times greater than the thermal conductivity of the... 

Combining external and internal gradients also an effect on the possibte unstable behavior of the catalyst pellet This could be studied by solving the complete transient Eq. 3.7.a-8, 9 together with the boundary conditions Eq. 3.7.b-3, 4. However, b use of the mathematical complexity, most information concerns the steady-state situation. 

In discussing the preliminary design of fixed bed reactors in Sec. 11.3 we mentioned that adiabatic operation is frequently considered in industrial operation because of the simplicity of construction of the reactor. It was also mentioned why straight adiabatic operation may not always be feasible and examples of multibed adiabatic reactors were given. With such reactors the question is how the beds should be sized. Should they be designed to have equal ATs or is there some optimum in the AT s, therefore in the number of beds and catalyst distribution In Section 11.3. this problem was already discussed in a qualitative way. It is taken up in detail on the basis of an example drawn from SOj oxidation, an exothermic reversible reaction. To simplify somewhat it will be assumed, however, that no internal gradients occur inside the catalyst so that the effectiveness factor is one. 

The mathematical model used by Capelli et al. to simulate the reactor may be classified as a one-dimensional heterogeneous model considering external and internal gradients, but not axial difiusion and conduction. The steady-state model equations are, in terms of the partial pressures... 

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