Analogy Theories

Results from philosophical considerations on language show that attempts to define life lead to a dilemma, similar to that which occurred when trying to define water before molecular theory existed. Since no analogous theory of the nature of living systems exists, an infinite controversy as to the definition of life is unavoidable (Cleland and Chyba, 2002). 

Here r is a characteristic time constant, usually called the dielectric relaxation time. To conform to analogous theories of visco-elastic behaviour we should really use the term dielectric retardation time, because it refers to a gradual change in a strain (the polarisation or resulting electric displacement) following an abrupt change in stress (the applied field). Dielectric relaxation time is, however, still most commonly used in spite of this inconsistency. By integration of Equation (3.12) ... 

More recently, Kauh et al. (K3) calculated the local heat- and mass-transfer coefficients for a flat plate by extending the turbulent analogy theory treatment of Kestin and Persen (K4) and Gardner and Kestin (Gl), and by employing the Spalding (S13) equation for the law of the wall. 

Reviews of several extensions of the analogy theory are given in the reports by Sherwood [133] and Sideman and Pinczewski [135]. 

The situation is analogous to momentum flux, where the relative Importance of turbulent shear to viscous shear follows the same general pattern. Under certain ideal conditions, the correspondence between heat flow and momentum flow is exact, and at any specific value of rjr the ratio of heat transfer by conduction to that by turbulence equals the ratio of momentum flux by viscous forces to that by Reynolds stresses. In the general case, however, the correspondence is only approximate and may be greatly in error. The study of the relationship between heat and momentum flux for the entire spectrum of fluids leads to the so-called analogy theory, and the equations so derived are called analogy equations. A detailed treatment of the theory is beyond the scope of this book, but some of the more elementary relationships are considered. 

There are, therefore, theories that relate the (pd and absolute values of which can not be experimentally measured, the -potential is a directly measurable quantity, which, along with the thickness of the ionic atmosphere, is an important characteristic of the diffuse part of the electrical double layer. 

The simplest theories of polymer adsorption are mean-field theories. In fact, it is possible simply to apply the square gradient theory of surface segregation developed in section 5.1 to polymer solutions by setting the degree of polymerisation of one of the components to unity. An analogous theory was developed by de Gennes (1981) and applied to adsorption from theta solvents (under which circumstances mean-field theory can be expected to be reasonably accurate) by Klein and Pincus (1982). 

The analogy theories derive their name from the similarity of the transport process mechanisms and the mathematical descriptions of the phenomena. For the purposes of this section, the mass and momentum flux relationships are needed. Each is quantified by a simple equation using gradients of concentration and velocity and the respective coefficients of diffiisivity as constants. 

Both the theoretical and empirical results of the analogy theories relate one transport coefficient to another. In this case, they are k, the MTC, and Cf, the coefficient of skin friction or the momentum transfer coefficient. This is a fortuitous outcome, because as will be seen in the next few paragraphs much information is available on the coefficients of friction, Cf, for fluid moving over solid surfaces in natural systems. 

The skin friction coefficient at the atmosphere soil surface. This section provides means for obtaining numerical estimating the coefficient of skin friction on the air side of large, flat terrestrial or water surfaces during neutral stability and windy conditions. The Cf is then used with the appropriate analogy theory algorithm to obtain a numerical estimate of the air-side MTC. 

Solution The analogy theory equations provide an approach to estimating MTC for the case of a fiilly developed and steady-state boundary layer. The traditional turbulent BL theory correlation (Equation 2.18) suggests that the MTC decreases with increasing fetch as The Chilton-Colbum analogy result, Equation 2.20 will be used, it is the simplest and based on experimental data. From Table 2.1, Cf = 0.0044 and jo = 2E - 3m is assumed appropriate for farmland, however a 30% variation about this Cf is arguable, based on values of similar terrain types in the table. Equation 2.22a is the alternative choice for estimation Cf. Using Equation 2.23 to obtain yields 21.1 and a Cf = 0.0022 results. The MTC computation is... 

It appears that the analogy theory estimate of 880 cm/h is approximately 3x higher than the turbulent boundary layer result. However, both values fall within the range of reported field observations see Thibodeaux (1996, p. 390). 

VA viscosity in good solvents. The correction of excluded volume effect is made by using the Flory-Krigbaum-Orofino theory of the second virial coefficient A 2 or other analogous theories (12). 

Rather than assuming that the cross-links are fixed in space in a network at rest, James and Guth [27] proposed an analogous theory to that of Flory, but one where the cross-links fluctuate in space around a mean position and that the mean positions deform affinely. These fluctuations reduce the force needed for a given deformation by the following relation ... 

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