Pressure thermodynamic

The thermodynamic pressure is equal to the mechanical pressure due to force at equilibrium. While most problems of interest possess gradients in the intensive properties... 

In the case of a flowing fluid the mechanical pressure is not necessarily the same as the thermodynamic pressure as is the case in a static fluid. The pressure in a flowing fluid is defined as the average of the normal stress components. In the case of inelastic fluids, the normal stress components are equal and therefore, with the negative sign convention, equal to the pressure. It is for this reason that the pressure can be used in place of the normal stress when writing force balances for inelastic liquids, as was done in Examples 1.7-1.9. 

Another way around the problem of pressure-driven flow in the single-phase membrane was presented by Meyers.He worked around the problem by allowing for a discontinuity in pressure at the membrane/solution interface, even though the electrochemical potential of all soluble species is continuous. He argued that additional mechanical stresses compressing the membrane should be indistinguishable from the thermodynamic pressure, and thus, the thermodynamic pressure might be discontinuous at the interface. 

The phases Pi..Pj in the system may or may not be entirely uniform in temperature due to the need for heating/cooling, the presence of reaction, the degree of insulation etc. Also, the total thermodynamic pressure P in the system may not be entirely constant throughout the system even if low viscosity solvents are used in the syntheses. ... 

Figure 9.1 Effect of temperature on equilibrium conversion as predicted by thermodynamics (pressure fixed.)...

The effect of pressure on the measured bimolecular rate constant of the Diels-Alder reaction between maleic anhydride and isoprene was investigated in supercritical CO2 and subcritical propane. The reaction was carried out at 35°C in CO2 and 80°C in propane. The rate constants in supercritical CO2 agreed closely with the thermodynamic pressure effect predictions over the entire pressure range. The rate constants in the subcritical propane solvent significantly diverged from the thermodynamic pressure effect predictions and were found to deviate from this linear density dependence at the lower pressures studied. The results show solvent-solute and cosolvent-solute interaction (Reaves and Roberts, 1999). 

It has been suggested that the experimental isothermal kinetic rate constants of some reactions at near and supercritical conditions could not be explained solely by the thermodynamic pressure effect, but from the combination of local composition enhancement and density augmentation around reactants. 

The force acting on any differential segment of a surface can be represented as a vector. The orientation of the surface itself can be defined by an outward-normal unit vector, called n. This force vector, indeed any vector, has direction and magnitude, which can be resolved into components in various ways. Normally the components are taken to align with coordinate directions. The force vector itself, of course, is independent of the particular representation. In fluid flow the force on a surface is caused by the compressive (or expansive) and shearing actions of the fluid as it flows. Thermodynamic pressure also acts to exert force on a surface. By definition, stress is a force per unit area. On any surface where a force acts, a stress vector can also be defined. Like the force the stress vector can be represented by components in various ways. 

For a fluid at rest (i.e., when all strain rates vanish), the stress tensor must reduce to that caused by hydrostatic pressure. The thermodynamic pressure, as defined... 

It is convenient for subsequent derivations to introduce the notion of deviatoric normal stresses, r/ = m + p, meaning the fluid mechanical normal stress plus the thermodynamic pressure ... 

When rotated into the principal coordinates, all the shear stresses vanish, and it is reasonable to think of the average normal compressive stress as a pressure. It is apparent from Eq. 2.183, however, that the average compressive stress is not equal to the thermodynamic pressure p as evaluated from an equation of state. Stokes made this interesting observation and recognized its concomitant dilemma in his famous 1845 paper. He hypothesized that the dilemma could be resolved by assuming that... 

The CFD solution determines the pressure variation relative to a uniform thermodynamic pressure, p = pa + p. In this problem, pa = 10,000 Pa, and the sheet called relative p reports the local values of p. How does the pressure compare in magnitude to the other stress components What observations can be made about the effect of the pressure on the structure of the stress tensor ... 

By using the deviatoric stress tensor, Eq. 2.143, we can separate the thermodynamic pressure from the T W term as... 

The dynamics of the incompressible fluid flow depend on small changes in the pressure through the flowfield. These changes are negligible compared to the absolute value of the thermodynamic pressure. The reference value can then be taken as some pressure at a fixed point and time in the flow. Changes in pressure result from fluid dynamic effects and an appropriate pressure scale is where Vmax is a measure of the maximum velocity in... 

The temperature, composition, and density are presumed to have only radial variations. The pressure, however, is allowed to vary throughout the flow, but in a very special way as will be derived shortly. Also the magnitude of the pressure variations is assumed to be small compared to the mean thermodynamic pressure. Using these assumptions, and invoking the Stokes hypothesis to give X = —2p./3, we can reduce the mass-continuity and Navier-Stokes equations to the following ... 

In heterogeneous solid state reactions, the phase boundaries move under the action of chemical (electrochemical) potential gradients. If the Gibbs energy of reaction is dissipated mainly at the interface, the reaction is named an interface controlled chemical reaction. Sometimes a thermodynamic pressure (AG/AK) is invoked to formalize the movement of the phase boundaries during heterogeneous reactions. This force, however, is a virtual thermodynamic force and must not be confused with mechanical (electrical) forces. 

At this point we believe that changing local compositions is the dominant effect in the apparent rate increase for the reaction of benzophenone triplet and isopropanol in supercritical CO2 and this conclusion is supported by subsequent studies of the reaction of the triplet with other quenchers (Roberts et al., 1991). However, one must also consider the thermodynamic pressure effect on the rate constant in terms of transition state theory and possible cage effects. 

The thermodynamic pressure effect on the reaction rate constant can be explained in terms of transition state theory (Evans and Polanyi, 1935), when the reactants are in thermodynamic equilibrium with a transition state. Once the transition state complex is formed it proceeds directly to products. With this analysis the pressure effect on the reaction rate constant can be given as follows ... 

In the case of hydrostatic pressure arising from gravitational forces, this may be removed by simply neglecting the gravitational body force in the N-S Equation (5). For the thermodynamic pressure, it is less obvious what can be done. However it may be noticed that at equilibrium, the N-P Equation (1) is satisfied and the ion fluxes are zero. It can now be seen that the thermodynamic... 

For nonideal systems, the fugacities of component i in the vapor and in the liquid play the same role as the component partial pressure in the vapor and the component vapor pressure in the liquid. The fugacity can be regarded as a thermodynamic pressure, For equilibrium, vapor fugacity is equal to liquid ftigacity, i.e.,... 

A particular globule, which has the pressure P2 inside, feels outside, in its vicinity, the micropressure pp. Only a macroscopic part of the microemulsion, which of course should be large compared to the size of a globule (in order to contain a large number of globules), feels the thermodynamic pressure (which is equal to the external pressure p). The local condition of mechanical equilibrium of the interface of the globules demands P2 pi. However, in contrast to the case of a liquid droplet surrounded by... 

In three directions, there will be nine stress components Ty. The viscous forces appear only when there are velocity gradients within the fluid. The forces per unit area (molecular stresses) acting on the body, it, both by the thermodynamic pressure and by the viscous stresses are given by... 

The orientation of the two classes of complexes in each membrane is such that, in the absence of any thermodynamic back-pressure from a proton gradient, they would pump protons in the same direction. The means by which this assembly can be used for the continuous transduction of energy from substrate oxidation or photon capture to ATP synthesis can best be understood if the thermodynamic pressures operative on the ATP synthase are examined in detail. 

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