Equilibrium Processes in Linear Fluid

A linear solvation energy relationship (LSER) has been developed to predict the water-supercritical CO2 partition coefficients for a published collection of data. The independent variables in the model are empirically determined descriptors of the solute and solvent molecules. The LSER approach provides an average absolute relative deviation of 22% in the prediction of the water-supercritical CO2 partition coefficients for the six solutes considered. Results suggest that other types of equilibrium processes in supercritical fluids may be modeled using a LSER approach (Lagalante and Bruno, 1998). 

Consider first an equilibrium process in the linear fluid model where (3.220), (3.221) are valid through all the body and persistently (at least for considered time interval) practically this is achieved by the discussed below [14, 18, 92-95]. 

Figure 9.4 (a) Solution to exp( m n) (M,t )+/ [2(Mi ) ] fraction of the adsorbate remaining in the fluid phase-linear equilibrium, (b) Fraction of adsorbate remaining in fluid phase-low time range. [From O.A. Hougen, C.C. Watson and R.A. Ragatz, Chemical Process Principles, Vol. Ill, with permission of John Wiley and Sons, New York, (NY), (1957).]... 

The validity of (3.220) throughout the body is expressed by Killing s theorem (3.18), which is that the motion of a linear fluid body in an equilibrium process is rigid. This means that a frame fixed with such a body exists in which... 

Now, we restrict ourselves to some equilibrium process persisting in one equilibrium state of the linear fluid model in the sense of the property S4 from Sect. 1.2 (one equilibrium from those more possible which is compatible with the given boundary and external conditions). Such an equilibrium state may be achieved if no radiation heat transfer is considered... 

Even this reversible process is rather a special one. We note it here to demonstrate that in the model of the linear fluid equality (in entropic inequality) is possible, see (1.35), and to show that entropy may be calculated with the precision of a constant, see (1.40), cf. application of reversible processes in Sect. 1.4. An equilibrium state is also an equilibrium process formed by a unique state with (3.231), cf. definition below (2.11). 

We now deduce basic thermodynamic properties of the mixture of fluids with linear transport properties discussed in Sect. 4.5. Among others, we show that Gibbs equations and (equilibrium) thermodynamic relationships in such mixtures are valid also in any non-equilibrium process including chemical reactions (i.e. local equilibrium is proved in this model) [56, 59, 64, 65, 79, 138]. 

Therefore, the classical relations of thermochemistry were obtained. Especially, the Gibbs equations (4.201)-(4.206) are valid in arbitrary process in this chemically reacting mixture of fluids with linear transport properties, i.e. the principle of local equilibrium is valid in this mixture. But we show in the following relations that this accord with classical thermochemistry (e.g. [138]) is not quite identical indeed, if we differentiate (4.211) and use (4.22), (4.23) we obtain... 

These and all previous results of thermodynamic mixture which also fulfil Gibbs-Duhem equations (4.263) show the complete agreement with the classical thermodynamic of mixtures but moreover all these relations are valid much more generally. Namely, they are valid in this material model—linear fluid mixture—in all processes whether equilibrium or not. Linear irreversible thermodynamics [1-4], which studies the same model, postulates this agreement as the principle of local equilibrium. Here in rational thermodynamics, this property is proved in this special model and it cannot be expected to be valid in a more general model. We stress the difference in the cases when (4.184) is not valid—e.g. in a chemically reacting mixture out of equilibrium—the thermodynamic pressures P, Pa need not be the same as the measured pressure (as e.g. X =i Pa) therefore applications of these thermodynamic... 

Definition of equilibrium is motivated similarly as in Sects. 1.2, 2.1, 2.2 and 3.8 [39, 52, 53, 56, 79, 98, 142, 143] (for non-linear models, see, e.g. [60, 71, 72]). For the regular linear fluid mixture model summarized at the end of previous Sect. 4.6, we define equilibrium by zero entropy production (4.301) as an equilibrium process going persistently through a unique equilibrium state, which is possible, as we shall see, if the body heat source is zero (4.303) and at zero rates of chemical reactions (4.302). By regularity conditions (see 1,2,3 at the end of Sect. 4.6), we exclude some unusual processes compatible with zero entropy production. We apply the regularity conditions on equilibrium states (moreover, regularity condition 3 follows for stable equilibrium states which will be discussed later in this Sect. 4.7). 

The initial concentration distribution is therefore simply translated at the velocity of the liquid steady flow and full equilibrium between the liquid and its matrix require that the amount of element transported by the concentration wave is constant. In more realistic cases, either the flow is non-steady due to abrupt changes in fluid advection rate or porosity, or solid-liquid equilibrium is not achieved. These cases may lead to non-linear terms in the chromatographic equation (9.4.35) and unstable behavior. The rather complicated theory of these processes is beyond the scope of the present book. 

Adsorption of organics using activated carbon follows a linear isotherm (x/m)KC(. To process a volume V of fluid containing an initial concentration Co of organics, it is suggested to divide the total mass M of sorbent into three equal parts and contact the fluid with M/3 mass of sorbent in three consecutive batch contacts, allowing equilibrium to be reached with each contact. 

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