Notes on Transport Phenomena

An early normalization of the Thiele modulus for an isothermal pellet and arbitrary kinetics was given by R. B. Bird, W. E. Stewart, and E. N. Lightfoot on pages 335-41 of their Notes on Transport Phenomena, the precursor to their well-known Transport Phenomena (New York John Wiley Sons, Inc., 1960). Slightly more general forms—all of them equivalent—have been given independently and almost simultaneously in ... 

R. B, Bird, W, E. Stewart, and E N. Lightfoot. Notes on Transport Phenomena. John Wiley Sons, New York, 1958,... 

It should be noted that the sign convention adopted here for components of the stress tensor is consistent with that found in many fluid mechanics and heat transfer books however, it is opposite to that found in some books on transport phenomena, e.g., Refs. 10,11, and 14. 

Though the discovery of electrokinetic phenomena is certainly as old as Reuss s experiments in 1808,2 when he noted the transport of water through a plug of quartz powder, theories of its relation to the double layer began seriously with Helmholtz in 1879.3 He presented a simplified conception of the double layer with a plane layer of positive charges on one side of the phase boundary, and a similar plane layer of negative charges on the other side. 

In this text, the conversion rate is used in relevant equations to avoid difficulties in applying the correct sign to the reaction rate in material balances. Note that the chemical conversion rate is not identical to the chemical reaction rate. The chemical reaction rate only reflects the chemical kinetics of the system, that is, the conversion rate measured under such conditions that it is not influenced by physical transport (diffusion and convective mass transfer) of reactants toward the reaction site or of product away from it. The reaction rate generally depends only on the composition of the reaction mixture, its temperature and pressure, and the properties of the catalyst. The conversion rate, in addition, can be influenced by the conditions of flow, mixing, and mass and heat transfer in the reaction system. For homogeneous reactions that proceed slowly with respect to potential physical transport, the conversion rate approximates the reaction rate. In contrast, for homogeneous reactions in poorly mixed fluids and for relatively rapid heterogeneous reactions, physical transport phenomena may reduce the conversion rate. In this case, the conversion rate is lower than the reaction rate. 

You are expected to set up a simple theory of the stmcture of your product, and of what happens with your product. You may need to set up balances and transport equations as you have learned in Transport Phenomena or Process Engineering. For some assignments the Notes on Colloids may help. Our experience is that many teams find this job difficult, so start early. 

Many of the cells listed in Table 7.1 and 7.2 are involved in active membrane flow and other mass-cooperative transport phenomena. Since cubic membranes offer a high surface to-volume ratio, they may also be actively involved in these processes, perhaps as membrane storage bodies, or as transport guides. It is of interest to note that aggregates of "s3maptic vesicles" often resemble cubic membranes (see Chapter 5 and [136]). This can be taken as an indication of a possible on-off mechanism of membrane continuity, which might accovmt for a regulative capacity of the release of transmitter substance. 

The first section (i.e., 1 in Table 2) serves as an introduction and defines the scope of the subject. As implied in the title, it is one of chemodynamics or the movement of chemicals. Chemical transport is the primary focus of the material. Critics have noted that production and degradation rates of chemical reactions are all but absent in the course syllabus. Environmental reaction is a very important but is also a very broad subject and its inclusion at even a basic technical level into EC would detract from the transport message. Two basic subjects are necessary for understanding transport. These are chemical equilibrium at interfaces and the fundamentals of transport phenomena. Highly condensed material on these two key subjects are presented in chapters 2 and 3. The last chapter, number 7, is on the fate and transport in water, air, and soil. These are the traditional subjects of environmental modeling which treat each of the three media separately and as isolated units from a multimedia perspective. Nevertheless, this approach is very appropriate for numerous EC applications. The section stresses the commonalities of fate and transport in the three media however, the brief coverage offered on each belies the importance of these respective intraphase transport topics. 

Note that the conservation equations can be distinguished from the transport equations since they do not contain any production or destruction terms. Nevertheless, the conservation equations may contain terms on the RHS expressing a divergence of fluxes related to transport phenomena. The way in which these flux terms are divided into divergence of transport fluxes or source terms is rather involved, but procedures exist based on a number of requirements on the two types of terms which determine this separation uniquely. 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

Recall that the models used in the developments just presented are based on the quasi-continuum approach. As already noted, this means that the gas-solid quasicontinuum is regarded as a single phase with properties of its own. These are not thermodynamic properties but effective properties that can be used to analyze transport phenomena within the continuum, as in any homogeneous system. 

Thus, from the point of view of modem phenomenological thermodynamics, the current outputs of classical equilibrium thermodynamics (e.g. the description of thermochemistry of mixtures) and the tasks of irreversible thermodynamics, like the description of linear transport phenomena and nonlinear chemical kinetics, are valid much more generally, e.g. even when all these processes mn simultaneously. As we noted above, these properties are not expected to be valid in any material models in some models the local equilibrium may not be valid, reaction rates may depend not only on concentrations and temperature, etc. 

Lar] Larsson, L.-E., Note on the Grain Boundary Penetration of Cu and Mn in y-Fe at 1100°C During Liquid-Solid Powder Alloying , Mater. Sci. Eng., 19, 241-244 (1975) (Experimental, Kinetics, Transport Phenomena, 10)... 

For certain electrochemical systems it is possible to find experimental conditions which minimise the interactions between the anode and the cathode. Both electrodes remain related to each other since they are crossed by the same current, yet the difference is that the mass transport phenomena occurring at both interfaces do not interact with each other. This type of scenario, which is typically sought after in analytical experiments, is explored in this paragraph, focused exclusively on describing one single electrochemical interface . Moreover, it is worth noting that the same approach can be applied to any interface, such as for instance an ionic junction. 

As already noted above, when there is a current flow then mass transport phenomena are automatically brought into play in addition to the electron transfer itself. The link between current and voltage generally involves all of the system s kinetic parameters. It is often interesting to consider the two following limiting cases, which are illustrated here on the E mechanism (remember we assume that vox = vRed = 1) ... 

This completes our description of the thermodynamic basis functions in terms of the configurational and momenta density functions obtained directly from the equilibrium solution to the Liouville equation. As will be shown in the next chapter, the nonequilibrium counterparts (local in space and time) of the thermodynamic basis functions can also be obtained directly from the Liouville equation, thus, providing a unified molecular view of equilibrium thermodynamics and chemical transport phenomena. Before moving on, however, we conclude this chapter by noting some important aspects of the equilibrium solution to the Liouville equation. 

The reactor configuration is so important because it determines the rates of the physical transport phenomena that accompany the chemical reaction, which in their turn determine to a great extent the outcome of the chemical operation. It is also very important to note that in most cases the essential features of the reactor configuration can be studied in volumes on the order of 1 cm, since such a volume will as a rule contain several particles, drops or bubbles. Also reactions in parallel gas/liquid-flow can be studied on a very small scale. This means that the effects of the configuration on the selectivity can be studied in small scale laboratory equipment. 

It is important to note at this stage, that the chemical interaction between molecules can be studied experimentally at any scale that is sufficiently larger than the molecular dimensions. A consequence is, for example, that unexpected side reactions, that are found when the reaction is carried out on a large scale (in a plant, or in the environment), may be studied in detail in the laboratory under well defined conditions. The phenomena of which the rates are essentially scale dependent are all physical in nature, and in this context they can be summarized as physical transport phenomena. These phenomena can be studied separately or in combination with chemical reactions. 

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