Reformulation of equilibrium equations

The reformulated bulk equilibrium equations given in the previous Section should also be supplemented by the appropriate boundary conditions discussed in Section 2.6 which can also be reformulated. We treat separately each of the cases mentioned earlier. 

Equilibrium Constants For practical application, Eq. (4-336) must be reformulated. The initial step is elimination of the in favor of fugacities. Equation (4-74) for species i in its standard state is subtracted from Eq. (4-77) for species i in the equilibrium mixture, giving... 

Considering the shift reaction alone results in a second-order equation in the conversion 4, and this equation can be solved analytically after reformulation of the equilibrium equation as ... 

The principle of virtual displacements, given by Eq. (3.45), may be utilized to determine the equations of equilibrium. We will refrain from considering external loads. For the two-dimensional shell structure still with the transverse shear strains 7° and 7 and associated internal transverse forces Qx and Qs, the principle of virtual displacements may then be reformulated as follows ... 

In this Section we derive a particularly convenient and useful reformulation of the dynamic equations for nematic liquid crystals. The Ericksen-LesUe equations summarised in Section 4.2.5 can be reformulated in a manner similar to the reformulation of the equilibrium equations in Section 2.7 when it is supposed that... 

The expressions for 11 and 11 in (6.73) and (6.74), which also appear in the equilibrium equations (6.79) and (6.80), can be reformulated in general vector form following the methodology of Nakagawa [210]. It is a straightforward exercise to find the general form when... 

It is possible to obtain a reformulation of the equilibrium equations which, to some extent, parallels the reformulation for nematics carried out in Section 2.7. We do not pursue this aspect here, but refer the reader to the article by Leslie [181] which contains full details of the results, including the situation for generalised curvilinear coordinates. When a is a fixed constant vector in Cartesian coordinates we can set... 

In this concluding section of Part II, it is appropriate to summarize the major conceptual and mathematical features of the Gibbs formulation of thermodynamic equilibrium theory, as a preface to its geometrical reformulation in the ensuing Part III. The following equations (8.70)-(8.95) summarize the essential mathematical structure of the Gibbsian formalism, as erected on the historical foundation of Chapters 1-4 and exploited in the applications of Chapters 5-8. 

Clearly, not all the nontrivial algebraic equations that correspond to the equilibrium of the fastest dynamics in Equation (5.39) are linearly independent. Specifically, the last three equations can be expressed as functions of the first three, reformulating G1 as in Equation (5.13), with... 

Flash Calculations. The ability to carry out vapor-liquid equilibrium calculations under various specifications (constant temperature, pressure constant enthalpy, pressure etc.) has long been recognized as one of the most important capabilities of a simulation system. Boston and Britt ( 6) reformulated the independent variables in the basic flash equations to make them weakly coupled. The authors claim their method works well for both wide and narrow boiling mixtures, and this has a distinct advantage over traditional algorithms ( 7). 

To be able to measure the osmotic pressure n, a semipermeable membrane that permits passage of the solvent molecules but not the solute molecules is needed. This can, in practice, be realized only when there is a large disparity between the sizes of the solute and solvent molecules, as in a solution of a polymer in a small-molecule solvent. However, the existence of osmotic pressure can be envisioned, at least mentally, with any kind of solution, such as a solution of two small-molecule liquids or a miscible blend of two polymers. Equation (6.6) is thus valid for any two-component (amorphous) system, as long as it is in equilibrium and classical thermodynamics is applicable to it. For applications to these general cases, it is more convenient if Equation (6.6) is reformulated in terms of the free energy of mixing and no explicit reference to osmotic pressure is made in it. 

At the present stage, where we are just entering a non-equilibrium thermodynamical description of multiphase polymer systems, an ab initio theoretic derivation of equations (11.30) and (I l.3l)ff is still lacking. The a.m. thoughts and reformulations may at least lead to some important conclusions ... 

The early time diffusion equation, equation 2, can be reformulated to describe this rate test. The extent of swelling at vortex disappearance Is taken as M = 22.8 g/Wp. The new term, Wp, is the mass of polymer used in the vortex test per 50 mL of test fluid. This is usually held constant for the vortex test as described above, but generalizes the test for ratios of polymer to fluid other than the 2 50 described in the experimental section. The value of Mg is taken to be the equilibrium swelling capacity of the particular polymer sample. The radius of the particle, rQ is determined from the screen analysis of the polymer sample. 

Note that since the map is defined for i > 0, we need only to look for non-negative roots R = 0 corresponds to an equilibrium state, and the positive roots correspond to periodic orbits of the system (11.5.13). Since we have already examined an equation of this type in the preceding section (Eq. (11.4.12)), when analyzed the period-two orbits emerging from a perioddoubling bifurcation in the case of zero Lyapunov value, we can simply reformulate the main results. 

Although Equation 18.14 qualitatively took this dynamic equilibrium into account, Shoji and coworkers [106-108] reformulated the crack tip strain rate relationship to take into account these complex strain rate factors in front of an advancing crack fip and to include the expected contributions due to work-hardening coefficient, yield strength, degree of plastic consfrainf, and dynamic applied loads ... 

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