Volume elements

The RTD of a chemical reactor can be determined by using tracer substances and, consequently, by tracing every volume element that passes through the reactor. We attempt to determine the length of the time the volume elements reside in the reactor. In other words, we attempt to determine the RTD of the species. 

FIGURE 4.5 An example of the implementation of a marker substance (tracer) technique, together with its response curve. 

For a tracer to be suitable for dynamic studies, it should satisfy certain criteria. The most important ones are listed below  

In an industrial context, we often use radioactive isotopes as tracers. These do break down, but the breakdown process is well known, defined, and easy to take into account. The isotopes can be introduced into the system in several ways, which makes it significantly easier to attain the correspondence mentioned above. Keeping in mind the radiation risks, we generally choose isotopes with a relatively short half-life. Species other than radioactive ones with other measurable properties can also be utilized as tracers. 

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The general analysis, while not difficult, is complicated however, the limiting case of the very elongated, essentially cylindrical drop is not hard to treat. Consider a section of the elongated cylinder of volume V (Fig. II-18h). The centrifugal force on a volume element is u rAp, where w is the speed of revolution and Ap the difference in density. The potential energy at distance r from the axis of revolution is then w r Apfl, and the total potential energy for the... 

The total change d.S can be detennined, as has been seen, by driving the subsystem a back to its initial state, but the separation into dj.S and dj S is sometimes ambiguous. Any statistical mechanical interpretation of the second law requires that, at least for any volume element of macroscopic size, dj.S > 0. However, the total... 

In an irreversible process the temperature and pressure of the system (and other properties such as the chemical potentials to be defined later) are not necessarily definable at some intemiediate time between the equilibrium initial state and the equilibrium final state they may vary greatly from one point to another. One can usually define T and p for each small volume element. (These volume elements must not be too small e.g. for gases, it is impossible to define T, p, S, etc for volume elements smaller than the cube of the mean free... 

If there are more than two subsystems in equilibrium in the large isolated system, the transfers of S, V and n. between any pair can be chosen arbitrarily so it follows that at equilibrium all the subsystems must have the same temperature, pressure and chemical potentials. The subsystems can be chosen as very small volume elements, so it is evident that the criterion of internal equilibrium within a system (asserted earlier, but without proof) is unifonnity of temperature, pressure and chemical potentials tlu-oughout. It has now been... 

The volume of a Y -space-volume-element does not change in the course of time if each of its points traces out a trajectory in Y space determined by the equations of motion. Equivalently, the Jacobian... 

Such an ensemble of systems can be geometrically represented by a distribution of representative points m the F space (classically a continuous distribution). It is described by an ensemble density fiinction p(p, q, t) such that pip, q, t)S Q is the number of representative points which at time t are within the infinitesimal phase volume element df p df q (denoted by d - D) around the point (p, q) in the F space. 

Finally, the assumed spherical synnnetry of the interactions implies that the volume element r 2 is dri2- For angularly-dependent potentials, the second virial coefficient... 

Because p(r) is spherically syimnetric, the number of electrons in this volume element is p(r),di. Letting v = Or cos a, da = -dxJ(Qr sina). Then, making all substitutions. 

Figure Bl.8.1. The atomic scattering factor from a spherically synnnetric atom. The volume element is a ring subtending angle a with width da at radius r and thickness dr.

For homogeneous particles, it represents the number of distances within the particle. For inhomogeneous particles, it has to take into account the different electron density of the volume elements. Thus it represents the number of pairs of difference in electrons separated by the distance r. A qualitative description of shape and internal structure of the... 

The second of Pick s laws expresses the change in concentration of a species at a point as a fimction of time due to difflision (figure B 1.28.2). Plence, the one-dimensional variation in concentration of material within a volume element bounded by two planes v and x + dx during a time interval dt is expressed by dc fx.,t)ldt) = D... 

Many-body problems wnth RT potentials are notoriously difficult. It is well known that the Coulomb potential falls off so slowly with distance that mathematical difficulties can arise. The 4-k dependence of the integration volume element, combined with the RT dependence of the potential, produce ill-defined interaction integrals unless attractive and repulsive mteractions are properly combined. The classical or quantum treatment of ionic melts [17], many-body gravitational dynamics [18] and Madelung sums [19] for ionic crystals are all plagued by such difficulties. 

In one of the earliest DFT models, the Thomas-Fermi theory, the kinetic energy of an atom or a molecule is approximated using the above type of treatment on a local level. That is, for each volume element in r space, one... 

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

By Max Bom s postulate, the produet of /(a ) and its complex conjugate r / (A ) times an infinitesimal volume element d x is proportional to the probability that a paitiele will be in the volume element d x... 

Note, these many "eoeffieients" are the elements whieh make up the Jaeobian matrix used whenever one wishes to transform a funetion from one eoordinate representation to another. One very familiar result should be in transforming the volume element dxdydz to... 

In addition, the volume element of interest is not the box dx dy dz shown in Fig. 1.6a but, rather, a spherical shell of radius r and thickness dr as shown in Fig. 1.6b. The result of expressing the volume element in spherical coordinates and integrating over all angles is the replacement... 

Figure 1.6 A flexible coil attached at the origin at one end and (a) in a volume element dx dy dz at the other end and (b) in a spherical shell of volume 47rr dr. (Reprinted from Ref. 4, p. 116.)...

We desire to use the probability function derived above, so we recognize that the mass contribution of the volume element located a distance r from an axis through the center of mass is the product of the mass of a chain unit mp times the probability of a chain unit at that location as given by Eq. (1.44). For this purpose, however, it is not the distance from the chain end that matters but, rather, the distance from the center of mass. Therefore we temporarily identify the jth repeat unit as the center of mass and use the index k to count outward toward the chain ends from j. On this basis, Eq. (1.49) may be written as... 

If we were required to pack beads in a beaker, we know from experience that by jostling the container we could achieve some compaction or decrease in free volume. In fact, we can picture the flow of a huge array of beads through a pipe by considering the beaker as a volume element in that pipe. By vibration, the beads are jostled downward that is, the holes work their way to the top. 

Equation (1.41) gives the probability of finding one end of a chain with degree of polymerization n in a volume element dx dy dz located at x, y, and z if the other end of the chain is located at the origin. We can use this relationship to describe the unstretched chain shown in Fig. 3.2a all that is required is to replace n by n, the degree of polymerization of the subchain. Therefore for the unstretched chain (subscript u) we write... 

In the volume elements describing individual subchains, the x, y, and z dimensions will be different, so Eq. (3.32) must be averaged over all possible values to obtain the average entropy change per subchain. This process is also easily accomplished by using a result from Chap. 1. Equation (1.62) gives the mean-square end-to-end distance of a subchain as n, 1q, and this quantity can also be written as x + y + z therefore... 

We defined the equation of motion as a general expression of Newton s second law applied to a volume element of fluid subject to forces arising from pressure, viscosity, and external mechanical sources. Although we shall not attempt to use this result in its most general sense, it is informative to consider the equation of motion as it applies to a specific problem the flow of liquid through a capillary. This consideration provides not only a better appreciation of the equation of... 

Figure 9.5a shows a portion of a cylindrical capillary of radius R and length 1. We measure the general distance from the center axis of the liquid in the capillary in terms of the variable r and consider specifically the cylindrical shell of thickness dr designated by the broken line in Fig. 9.5a. In general, gravitational, pressure, and viscous forces act on such a volume element, with the viscous forces depending on the velocity gradient in the liquid. Our first task, then, is to examine how the velocity of flow in a cylindrical shell such as this varies with the radius of the shell.

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