Dimensional length scales

As described earlier in this book, the dendritic architecture is perhaps one of the most pervasive topologies observed at the macro and micro-dimensional length scales (i.e. jum-m). At the nanoscale (molecular) level there are relatively few natural examples of this architecture. Most notable are probably the glycogen and amylopectin hyperbranched structures that Nature uses for energy storage. 

Figure 1.22 A comparison of dimensional length scales (A) for poly(amido-amine) (PAMAM) dendrimers Nc = 3, Nb = 2 (NH3 core) and various biological entities (e.g. proteins, DNA and lipid bilayers)...

Biomineralizatoin involves the exploitation of liquid crystal or self ordering properties of amphiphilic molecules or polymers to design inorganic materials with porous structures ordered over several dimensional length scales. 

Using the inviscid assumption for calculation of the dissipation rate, the non-dimensional length scale and time scale respectively reduce to... 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

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... 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

It is very useful to minimize significant variables by expressing the equations in dimensionless terms. To do this we need a characteristic length to normalize z. If we had a finite source diameter, D, it would be a natural selection however, it does not exist in the point source problem. We can determine this natural length scale, zc, by exploring the equations dimensionally. We equate dimensions. From Equation (10.22),... 

As discussed in Section 4.3, the linear-eddy model solves a one-dimensional reaction-diffusion equation for all length scales. Inertial-range fluid-particle interactions are accounted for by a random rearrangement process. This leads to significant computational inefficiency since step (3) is not the rate-controlling step. Simplifications have thus been introduced to avoid this problem (Baldyga and Bourne 1989). 

Figure 5.18. The diffusion flamelet can be approximated by a one-dimensional transport equation that describes the change in the direction normal to the stoichiometric surface. The rate of change in the tangent direction is assumed to be negligible since the flamelet thickness is small compared with the Kolmogorov length scale. The flamelet approximation is valid when the reaction separates regions of unmixed fluid. Thus, the boundary conditions on each side are known, and can be uniquely expressed in terms of .

However, in the case of large Kn, the no-slip approximation cannot be applied. This implies that the mean free path of the liquid is on the same length scale as the dimension of the system itself. In such a case, stress and displacement are discontinuous at the interface, so an additional parameter is required to characterize the boundary condition. A simple technique to model this is the one-dimensional slip length, which is the extrapolation length into the wall required to recover the no-slip condition, as shown in Fig. 1. If we consider... 

Three-dimensional electrode nanoarchitectures exhibit unique structural features, in the guise of amplified surface area and the extensive intermingling of electrode and electrolyte phases over small length scales. The physical consequences of this type of electrode architecture have already been discussed, and the key components include (i) minimized solid-state transport distances (ii) effective mass transport of necessary electroreactants to the large surface-to-volume electrode and (iii) magnified surface—and surface defect—character of the electrochemical behavior. This new terrain demands a more deliberate evaluation of the electrochemical properties inherent therein. 

The other type of model is the macrohomogeneous model. These models are macroscopic in nature and, as described above, have every phase defined in each volume element. Almost all of the models used for fuel-cell electrodes are macrohomogeneous. In the literature, the classification of macrohomogeneous models is confusing and sometimes contradictory. To sort this out, we propose that the macrohomogeneous models be subdivided on the basis of the length scale of the model. This is analogous to dimensionality for the overall fuel-cell models. 

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