Infinite tubes

For illustration, consider the simplest type of diffusion, described by the partial differential equation (2.5.3), also called linear diffusion. The system will be represented by an infinite tube closed at one end (for x = 0) and initially filled with a solution with concentration c°. Diffusion is produced by very fast removal (e.g. by precipitation or an electrode reaction) of the dissolved substance at the x = 0 plane (the reference plane). The initial concentration c° is retained at large distances from this reference plane (x— < >). The initial condition is thus... 

These expressions reduce even further to particularly simple forms for the case of an infinite tube or where 6 = 1. Thus letting subscript 00 refer to the infinite tube, or open vessel, case we find as shown in the third sketch of Table II... 

For temporal moments it is best to consider a semi-infinite tube and let c(0, t) = S(t) be the ideal tracer. Initially, c(z, 0) = 0 and c must remain finite as z — 00. These are not the most sophisticated conditions, as we know from Example 3, but they give the swiftest answer. If we take the Laplace transform of Eq. (261), we have... 

Consider an infinite tube whose axis is parallel to the axis Ox of a rectangular co-ordinate system Oxyz- Let S denote the domain occupied by the interior of the tube in the plane Oyz, and let s be its area and the curve T its perimeter. In steady uniform flow the velocity u is everywhere in the direction Ox and is a function of y and z given by... 

In addition to the infinite tubes, polycatenanes, and polyrotaxanes, all of which are polymeric complexes, a great number of silver coordination polymers with ID, 2D and 3D framework structures have been reported more recently. Selected examples and their specific topologies of some of these are listed in Table 11.1. Schematic structures of the corresponding organic ligands are shown in Figure 11.23.9,58-75... 

Volume surrounding infinite tube Tube diameter D, ... 

The cyclodextrins crystallize in patterns which form cages and channels illustrated in Fig. 18.2 [558, 5651. These patterns can be classified into the two cages of types A and B, where the cavity of each cyclodextrin is closed on both ends by adjacent molecules, and the channel type C, where cavities of adjacent cyclodextrins merge into infinite tube-like cavities. 

Brian Pickrell assumed the polar regions to be stretched out into a rectangular map projection prior to fitting the Earths on a cylinder. If the Earth were an infinite tube, like a giant paper-towel roll, gravity would decrease with distance as /r, rather than l/i. Normal satellite orbits would be impossible. Pickrell suggests that... 

The radiation dose from gas-turbine depends on its detail design configuration, arrangement, shieldings, etc. In this calculation, it is assumed that (a) FPs plateout on the inner surface of infinite tube, (b) an evaluation point is 1 m from the tube and (c) shielding material is 3 cm of iron. 

Taking as before the approximation of an infinite tube, and integrating the corresponding Navier-Stockes equation gives the velocity profiles... 

Imagine the following schematic apparatus. Take a semi-infinite tube and fill it with one stable stationary state, call it 1, and take another semi-infinite tube and fill it with the other stable stationary state, call it 3. Both tubes are at the same external constraints, of temperature, pressure and concentration of species. Place the tubes lengthwise together, see Fig. 5.2a, at first with a partition between them. 

In addition to the basis-set problem, it should be noted that band structure results (as well as results derived from these, e.g., transport data) ate usually obtained by considering tubes of infinite length. The infinite-tube results, however, are quite different from those obtained for tubes of finite length. [5, 6, 7] Taking into account that realistic calculations have to be performed for tubes of finite length and that ab initio methods are not easily applicable to systems consisting of a moderate number of atoms ( 1000 atoms are necessary for simulations of finite tubes), it can be seen that a practical calculation would require use of semi-empirical methods such as the TB. 

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