First equivalent formulations

The first quantitative formulation of the decomposition process was made by Marcellin, who did not attempt a solution. An interesting analysis was later made by Polanyi and Wigner, who treated the molecule as an elastic medium in which decomposition occurred when certain elastic waves (equivalent to the atomic vibrations) reinforced each other sufficiently to break a bond. The form of the law they then deduced was the same as that derived from discrete models and has been of considerable use in discussing the decomposition of unstable atomic nuclei, which is a related problem. 

In the general iterative approach, one first determines the equilibrium state for the product composition at an initially assumed value of the temperature and pressure, and then one checks to see whether the energy equation is satisfied. Chemical equilibrium is usually described by either of two equivalent formulations— equilibrium constants or minimization of free energy. For such simple problems as determining the decomposition temperature of a monopropellant having few exhaust products or examining the variation of a specific species with temperature... 

We begin with an abstract of the physics that underlies the kinetics of bond dissociation and structural transitions in a liquid environment. Developed from Einstein s theory of Brownian motion, these well-known concepts take advantage of the huge gap in time scale that separates rapid thermal impulses in liquids (< 10 s) from slow processes in laboratory measurements (e.g. from 10 s to min in the case of force probe tests). Three equivalent formulations describe molecular kinetics in an overdamped liquid environment. The first is a microscopic perspective where molecules behave as particles with instantaneous positions or states x(t) governed by an overdamped Langevin equation of motion,... 

The purpose of this study is to review the recently developed available methods and propose an effective method for evaluating dynamic structural reliability. The general performance function is investigated, and equivalence of several first-order formulations is discussed. A point estimate... 

Equivalence of the first order formulations and the graphical interpretation of the method... 

In the case of a nonlinear limit state surface, the reliability index fails to be constant under different but mechanically equivalent formulations of the performance function. The issue of a lack of invariance was first recognized by Ditlevsen... 

Micellar/polymer (MP) chemical enhanced oil recovery systems have demonstrated the greatest potential of all of the recovery systems under study (170) and equivalent oil recovery for mahogany and first-intent petroleum sulfonates has been shown (171). Many somewhat different sulfonate, ie, slug, formulations, slug sizes (pore volumes), and recovery design systems were employed. Most of these field tests were deemed technically successful, but uneconomical based on prevailing oil market prices (172,173). 

The design of a AA-size alkaline manganese dioxide cell is shown in Fig. 1 (Sec. 3.1). Primary and secondary alkaline batteries are constructed in the same way and can be manufactured on essentially the same machinery. The separator material, electrode formulation, and the Mn02 Zn balance are different. Rechargeable cells are zinc-limited to prevent a discharge beyond the first electron-equivalent of the MnOz reduction. The electrolyte is 7-9 mol L KOH. The electrode reactions are ... 

You can now see why for line A a branch-and-bound technique is not required to solve the design problem. Because of the way the objective function is formulated, if the ratio (pd ps) = 1, the term involving compressor i vanishes from the first summation in the objective function. This outcome is equivalent to the deletion of compressor i in the execution of a branch-and-bound strategy. (Of course the pipeline segments joined at node i may be of different diameters.) But when... 

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