Quasi-static equilibrium

If surface diffusion or vapor transport is rapid enough, the pores will maintain their quasi-static equilibrium shape, illustrated in Fig. 16.1 in the form of four cylindrical sections of radius R.6 The dihedral angle at the four intersections with grain boundaries, ip, will obey Young s equation, ip is related to 6 by sin( /j/2) = cos 0. 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

The stress expression in Equation (3) is used in the following quasi-static equilibrium equation assuming the acceleration term to be negligible. 

The ultimate equilibrium would be that of a drop or film of liquid that spreads until it thins to a monomolecular layer. Actually, a quasi-static equilibrium will be reached when the viscous forces exceed the gravitational and surface tension forces and the thinning of the film becomes imperceptibly slow. However, as discussed below, there are surface chemical and capillary forces acting at the three phase boundary that cause liquids to diffuse from the edge of a drop or a film. There is a school of thought and some experimental evidence that when a liquid film becomes very thin its chemical characteristics become different than that of the bulk liquid because of orientation effects induced by the solid substrate. Derjaguin strong advocate... 

The pressure underneath the screen was increased in fixed, quasi-static increments for precise pressure measurements and to allow a quasi-static equilibrium for the contact angle between the pore throat and solution and for the contact angle that is eventually pinned at the pore mouth. Bubble point was thus taken as the point when a visible gas bubble detached and broke away from the screen. 

If the free and trapped electrons are in local quasi-static equilibrium, it is possible to combine the continuity equations for free and trapped electrons into a single equation for the free electrons with values of the effective electron diffusion coefficient and lifetime and respectively, that depend strongly on the free electTOTi density due to trapping and detrapping as described above [18]. [2],... 

An interesting application of these methods to determine the quasi static equilibrium of a railway boogie can be found in [SFR91]. 

If the grains of sand are small, each step does not represent a very large departure from equilibrium between p and ptxt. This process is an example of a quasi-static process that is, one in which the process is never far from equilibrium during the expansion. 

Two theories exist for the lack of isotope fractionation in chondmles. Both rely on the concept of quasi-statical (i.e., almost equilibrium) exchange of isotopes between liquid and gas followed by permanent loss of rock-forming elements to the escaping gas phase. One model (Alexander 2003) implies that the chondmles formed in such close proximity to one... 

Any finite expansion that occurs in a finite time is irreversible. A reversible expansion can be approximated as closely as desired, and the values of the thermodynamic changes can be calculated for the limiting case of a reversible process. In the limiting case, the process must be carried out infinitely slowly so that the pressure P is always a well-defined quantity. A reversible process is a succession of states, each of which is an equilibrium state, in which the temperature and pressure have well-defined values such a process is also called a quasi-static process. 

In considering physicochemical equilibria, that is to say, if one is interested in the internal constitution of a system in equilibrium when changes of phase and chemical reactions are admitted, one introduces the constitutive coordinates this being the number of moles of the ith constituent Ct in the a th phase. The definitions of Equations (10) through (12) remain unaltered, for die nf do not enter into the description of the interaction of the system with its surroundings. Let an amount dnf of C be introduced quasi,statically into the a th phase of the system. The work done on K shall be fi dnt> The quantity fif so defined is the chemical potential of C, in die ct th phase. It is in general a function of all the coordinates of K. Then, identically. 

The equilibrium between IH- (or 2H-) and 4H-forms is expected to be different in excited and unexcited molecules, hence interference with a molecule to a disruptive extent may without paradox afford results at odds with those obtained by gentler methods on quasi-static molecules. 

At this point we may introduce a better definition for quasi-static or reversible processes. These are changes in a system that result from an imbalance of only those forces that maintain a system at equilibrium. 

When the plasma is not in local thermal equilibrium (LTE), the electron number densities cannot be determined on the basis of the Saha equation. Irrespective of the plasma being in local thermal equilibrium or not, the electron number density can be derived directly from the Stark broadening of the Hg line or of a suitable argon line. This contribution to broadening is a result of the electrical field of the quasi-static ions on one side and the mobile electrons on the other side. As described in Ref. [17] it can be written as ... 

The mere fact that a substantial change can be broken down into a very large number of small steps, with equilibrium (with respect to any applied constraints) at the end of each step, does not guarantee that the process is reversible. One can modify the gas expansion discussed above by restraining the piston, not by a pile of sand, but by the series of stops (pins that one can withdraw one-by-one) shown in figure A2.1.3. Each successive state is indeed an equilibrium one, but the pressures on opposite sides of the piston are not equal, and pushing the pins back in one-by-one will not drive the piston back down to its initial position. The two processes are, in fact, quite different even in the infinitesimal limit of their small steps in the first case work is done by the gas to raise the sand pile, while in the second case there is no such work. Both the processes may be called quasi-static but only the first is anywhere near reversible. (Some thermodynamics texts restrict the term quasi-static to a more restrictive meaning equivalent to reversible , but this then leaves no term for the slow irreversible process.)... 

We consider an arbitrary adiabatically isolated system consisting of two parts in thermal contact. The system is at a uniform temperature t, and we assume that the equilibrium states of each of the parts can be characterized by t and one other parameter. Thus, for a quasi-static process,... 

Spontaneous processes, in their turn, are subdivided into reversible processes (equilibrium, quasi-static) and irreversible processes or unequilibrium. Real processes, as a rule, are irreversible as they occm with loss of energy to the surrounding medium, and that is why the system cannot spontaneously return to its previous state. 

The nucleate boiling superheat AT has been described as a function of the heat flow-density complex equations. There exist essentially tw o kinds of boiling correlations the transport equations valid for a special configuration and equations closely related to the equilibrium thermodynamics of quasi-static systems. The transport equations determine the details of the heat transport mechanism with consideration of the force field and special conditions of each particular system, but they are so complex that it is difficult to arrive at satisfactory equations which will be valid in the entire range of pressures from the triple to the critical point. 

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