Characteristic length parameter

The effects of miniaturization on the performance of an analytical separation system are often discussed in terms of a reduction of a characteristic length parameter (e.g., column diameter dc or particle diameter dp) and the associated consequences for lateral zone dispersion processes and their interplay with longitudinal (axial) zone dispersion. A rigid discussion of the physical-chemical basis is beyond the scope of this chapter. For a general account in terms of scaling laws and proportionality considerations, the reader is referred to the references [12,14]. A comprehensive and detailed description with emphasis on the underlying physical-chemical processes can be found in the book by Giddings [20]. 

It is worthwhile to examine how reasonably well a single characteristic length parameter describes reaction/diffusion in a finite cylinder, a very common catalyst pellet configuration. The pellet shown has a cylinder length 2xp and radius Rp. 

Spherical Diffusion Model. This simple model assumes that diffusion occurs within a spherical particle. The model, however, cannot yield the diffusion coefficient directly since it contains a dimensional length parameter whose numerical value depends on the assumed diffusion mechanism. If intradiffusion predominates, the characteristic length parameter is assumed to be the size of a single crystal of the adsorbent. Consequently, the resulting diffusion coefficients are very small. If interdiffusion predominates, the characteristic length parameter is assumed to be the adsorbent particle diameter. The diffusion coefficient values in this case are much higher than the former ones. 

The values of some of the parameters in these equations, such as the diffusion coefficient D and the characteristic length parameter d, will depend on specific models and definitions (see below). Using the definitions of the Sherwood number, Sh = kd/D, the ratio of total and molecular mass transfer (with k the mass transfer coefficient), and the Schmidt number. Sc = r]/pD the ratio of momentum and molecular mass transfer, the equation can be written as ... 

Fully developed nonisothermal flow may also be similar at different Reynolds numbers, Prandtl numbers, and Schmidt numbers. The Archimedes number will, on the other hand, always be an important parameter. Figure 12.30 shows a number of model experiments performed in three geometrically identical models with the heights 0.53 m, 1.60 m, and 4.75 m." Sixteen experiments carried out in the rotxms at different Archimedes numbers and Reynolds numbers show that the general flow pattern (jet trajectory of a cold jet from a circular opening in the wall) is a function of the Archimedes number but independent of the Reynolds number. The characteristic length and velocity in Fig. 12.30 are defined as = 4WH/ 2W + IH) and u = where W is... 

Selecting the values of the parameters for the calculations we have in mind a 1 1 aqueous 1 m solution at a room temperature for which the Debye length is 0.3 nm. We assume that the non-local term has the same characteristic length, leading to b=. For the adsorption potential parameter h we select its value so that it has a similar value to the other contributions to the Hamiltonian. To illustrate, a wall potential with h = 1 corresponds to a square well 0.1 nm wide and 3.0 kT high or, conversely, a 3.0 nm wide square well of height 1.0 kT. 

Eqs. (74-78) contain two dimensional parameters, and D, and two dimensionless parameters, A and e. This means that any characteristic length scale i and growth velocity v of the possible structures can be presented in the form... 

The above relation is obtained if the constant a obtained from the work of Zel dovich [7] is assumed to be a = Nu.U (which was not suggested in the work of Zel dovich [7]), where k is the thermal dif-fusivity and d is the characteristic length. However, Nu/Pe is a parameter in the solution of energy equation for flames with heat losses above certain value for which no solution of the energy equation exist (see, e.g., Ref. [8], p. 108)... 

Flow of the liquid past the electrode is found in electrochemical cells where a liquid electrolyte is agitated with a stirrer or by pumping. The character of liquid flow near a solid wall depends on the flow velocity v, on the characteristic length L of the solid, and on the kinematic viscosity (which is the ratio of the usual rheological viscosity q and the liquid s density p). A convenient criterion is the dimensionless parameter Re = vLN, called the Reynolds number. The flow is laminar when this number is smaller than some critical value (which is about 10 for rough surfaces and about 10 for smooth surfaces) in this case the liquid moves in the form of layers parallel to the surface. At high Reynolds numbers (high flow velocities) the motion becomes turbulent and eddies develop at random in the flow. We shall only be concerned with laminar flow of the liquid. 

Curves showing the cnrrent densities as functions of x are presented for two val-nes of electrode thickness in Fig. 18.5. The parameter L has the dimensions of length it is called the characteristic length of the ohmic process. It corresponds approximately to the depth x at which the local current density has fallen by a factor of e (approximately 2.72). Therefore, this parameter can be nsed as a convenient characteristic of attenuation of the process inside the electrode. 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

Equation 9 shows that the chemical capacitance in this case is similar to that derived previously in eq 6 for a thin film (Ci) however, in the co-limited situation the important length parameter is not L but rather a characteristic utilization length given by... 

For CO 0, Eq. (11-7) reduces to the stream function for steady creeping flow past a rigid sphere, i.e., Eq. (3-7) with k = co. The parameter 3 may be regarded as a characteristic length scale for diffusion of vorticity generated at the particle surface into the surrounding fluid. When co is very large, 3 is small, and the flow can be considered irrotational except in the immediate vicinity of the particle. In the limit co go, Eq. (11-7) reduces to Eq. (1-29), the result for potential flow past a stationary sphere. 

Assuming a quasiuniform power distribution in the throughput or in the volume, a characteristic length of the dispersion space becomes irrelevant. In the relevance list, Equation (66), the parameter d must be cancelled. The target number = 32/ must be dropped and the dimensionless numbers La and Oh must be built by J32 instead of d. At given and constant material conditions pjpd, 9, Ci = const.), the process characteristics will be represented in the following pi space ... 

Flexibility in the choice of parameters and their reliable extrapolation within the range covered by the dimensionless numbers. These advantages become clear if one considers the well-known Reynolds number. Re = vL/v, which can be varied by altering the characteristic velocity V or a characteristic length L or the kinematic viscosity v. By choosing... 

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