Averaging theory

In order to overcome these problems, one approach, originally developed by Whitaker [420], Slattery [359], and Anderson and Jackson [17], involves the method of volume averaging. Using volume averaging theory, Whitaker and coworkers [193,264,268,337,436] found the effective diffusion tensor for a two-phase system to be given by... 

FIG. 22 Monte Carlo simulations (solid lines) in isotropic systems and comparison to results of volume averaging theory (horizontal bars). (Reproduced with kind permission of Kluwer Academic Publishers from Ref 398, Fig. 6 and 8, Copyright Kluwer Academic Publishers.)... 

Perrin model and the Johansson and Elvingston model fall above the experimental data. Also shown in this figure is the prediction from the Stokes-Einstein-Smoluchowski expression, whereby the Stokes-Einstein expression is modified with the inclusion of the Ein-stein-Smoluchowski expression for the effect of solute on viscosity. Penke et al. [290] found that the Mackie-Meares equation fit the water diffusion data however, upon consideration of water interactions with the polymer gel, through measurements of longitudinal relaxation, adsorption interactions incorporated within the volume averaging theory also well described the experimental results. The volume averaging theory had the advantage that it could describe the effect of Bis on the relaxation within the same framework as the description of the diffusion coefficient. 

According to the coherent averaging theory,3,4,53 the zero-order average Hamiltonian can be obtained straightforwardly... 

If the pulse strength of the simultaneous PIPs is further increased to 2/i = 44.48 kHz and 2nAfx=x (m= 1), (fi/Af) in Eq. (89) is no longer a small value and the Magnus-expansion in the coherent averaging theory may not converge. Consequently, the compensation in the null region vanishes almost completely. 

No reliable mixture-averaged theory is available for computing the thermal diffusion coefficient D[. When thermal diffusion is important, the rigorous multicomponent theory described next should be used to obtain D[. 

The mathematical model consists of four balance equations. These equations have been obtained in [3] by use of Volume Averaging Theory also called hybrid mixture theory, [11]-[13], The mass balance of the dry air includes both diffusive and advective components of the mass fluxes,... 

This spatial averaging theory could also be extended to the non-isothermal reacting flows to perform spatial averaging of coupled species and energy balances. This has been discussed in detail by Chakraborty and Balakotaiah (2004). It should also be pointed out that the above averaging theory is applicable to discrete interacting units, which is called L-S averaging in finite dimensions. In this case, Eq. (8) is modified to... 

At this point of the discussion the zero- and higher-order terms will not be derived explicitly, but we will return to the van Vleck approach at a later stage, where we will treat the BMFT case. Following Goldman s derivation [98,99] and Mehring s secular averaging theory [14] the result of the van Vleck transformation yields... 

Two comments (1) This exercise shows that the Duffing oscillator has a frequency that depends on amplitude w = 1-i- a -I-O( ), in agreement with (7.6.57). (2) The Poincare-Lindstedt method is good for approximating periodic solutions, but that s all it can do if you want to explore transients or nonperiodic solutions, you can t use this method. Use two-timing or averaging theory instead. 

The area averaging theory described in this section is based on the papers by [229, 243, 44, 46, 47, 22, 10, 24, 115, 135, 16[. The main object in this section is to provide the necessary theorems to derive the cross-sectional averaged equations that coincide with the conventional ID multiphase chemical reactor model from first principles. To formulate these theorems we chose to adopt the same concepts as were used deriving the corresponding single phase equations... 

The averaging theory of Suman and Ruth (58) for a nonconducting solid matrix porous medium leads to... 

Abstract We investigate the dependence of multiconfigurational perturbation theory framework on the choice of the Fermi-vacuum. A new formulation, based on a posteriori averaging is suggested. The averaged theory is invariant with respect to Fermi-vacuum choice but enhances the intruder effect. The performance of the averaged formulation is illustrated on the ethylene rotational potential curve. 

RSDI RSDI (Simple (Based on Average) Theories)... 

Stephan EA, Chase GG. Development of volume-average theory for deep-bed filtration. AIChE J. 2000 46 1918. 

Whitaker, S. 1998. The Method of Volume Averaging. Theory and Applications of Transport in Porous Media. Dordrecht, the Netherlands Kluwer Academic Publ. 

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