One-phase mixture

Fig. 5. Phase behavior of blends of a styrene—acrylonitrile copolymer containing 19 wt % of acrylonitrile with other SAN copolymers of varying AN content and as a function of the molecular weight of the two copolymers (° ) one-phase mixture ( ) two-phase mixtures as judged by optical clarity. Curve...

The volume of fluid (VOF) approach simulates the motion of all the phases rather than tracking the motion of the interface itself. The motion of the interface is inferred indirectly through the motion of different phases separated by an interface. Motion of the different phases is tracked by solving an advection equation of a marker function or of a phase volume fraction. Thus, when a control volume is not entirely occupied by one phase, mixture properties are used while solving governing Eqs (4.1) and (4.2). This avoids abrupt changes in properties across a very thin interface. The properties appearing in Eqs (4.1) and (4.2) are related to the volume fraction of the th phase as follows ... 

The fugacity coefficient of species i in a one-phase mixture is obtained from the PR-EOS using IPVDW as follows ... 

Separation operations are interphase mass transfer processes because they involve the creation, by the addition of heat as in distillation or of a mass separation agent as in absorption or extraction, of a second phase, and the subsequent selective separation of chemical components in what was originally a one-phase mixture by mass transfer to the newly created phase. The thermodynamic basis for the design of equilibrium staged equipment such as distillation and extraction columns are introduced in this chapter. Various flow arrangements for multiphase, staged contactors are considered. 

The extensive Gibbs free energy of a one-phase mixture of N components with - -1 degrees of freedom requires N + 2 independent variables for complete description of this thermodynamic state function. Hence,... 

Figure 13 illustrates the phase behavior of CO2 and water. As shown, the mutual solubilities (water in the CO2 phase and CO2 in the water phase) are small, i.e., < 5 mol%, and subsequent downstream contamination with H2O in an actual process is usually neither detrimental nor hazardous. Depending on the temperature and pressure, the water-C02 system may exhibit liquid-liquid-vapor equilibrium (LLV, e.g., at 25 °C and 6.4 MPa), or LLE, etc. At temperatures above the critical temperature of CO2, the mixture critical point reaches hyperbaric conditions (> 100 MPa) and therefore a one-phase mixture is impractical. 

As the temperature is increased, the hmits of this two-phase coexistence contract, until eventually they coalesce to produce a homogeneous, one-phase mixture at T, the critical solution temperature. This is sometimes referred to as the critical con-solute point. 

As an example of an open system, consider a fixed (control) volume that is open to steady-state mass and energy transfers with its surroundings. Crossing the system boundaries are ports through which one-phase mixtures of C components enter and leave the system. For steady flow situations, we must have at least one inlet and one outlet, so > 2. The system is in thermal contact with its surroundings and an interaction exists by which shaft work is done, either on or by the system. Note that we do not consider a work mode that could change the size or shape of the control volume. 

Figure 3.9 Schematic of a one-phase mixture immersed in a TP reservoir. The mixture is open to a single inlet (stream 1) through which a small amount of pure component 1 is added.

Consider two systems, 1 and 2. System 1 is a one-phase mixture of C components, with mole numbers N. This mixture fills a rigid vessel of volume Vj, and the vessel is immersed in a heat bath maintained at temperature Tj. System 2 is another sample of the same mixture, having the same C components and the same mole numbers N. System 2 fills the cylinder of a piston-cylinder apparatus. The cylinder is immersed in a heat bath at T2. A constant external pressure is imposed on the mixture at equilibrium the system pressure P2 balances that external pressure. Therefore, system 2 is at constant pressure, while system 1 is at constant volume. 

Figure 6.1 To obtain changes in prop>erties of one-phase mixtures, our basic strategy is to compute deviations relative to some ideality. In route lA (left) the ideality is the ideal gas and the deviations are the residual properties. In route IB (right) the ideality is an ideal solution and the deviations are the excess properties. In addition, we could use the relations in 5.3 to compute residual properties from excess properties and vice versa.

The middle envelope is the spinodal the set of states that separate metastable states from unstable states. Recall from 8.3 that one-phase mixtures become diffusionally unstable before becoming mechanically unstable. Therefore, the mixture spinodal is the locus of points at which the diffusional stability criterion (8.3.14) is first violated that is, it is the locus of points having... 

Figure 8.12 shows that if a mixture is mechanically unstable, then it is also diffu-sionally unstable, because the line of incipient mechanical instability lies under the spinodal, or equivalently because Kj appears in both stability criteria (8.1.30) and (8.3.13). Moreover, a one-phase mixture may be diffusionally unstable but remain mechanically stable, because the spinodal lies above the line of incipient mechanical instability, or equivalently because the mechanical criterion (8.1.30) can be satisfied while the diffusional criterion (8.3.13) is violated. Further, Figure 8.12 contains states at which no differential stability criteria are violated, but at which one-phase mixtures are metastable rather than stable. This means that a violation of any differential stability criteria (thermal, mechanical, or diffusional) is only sufficient, but not necessary, for a phase separation to occur. 

A stable one-phase mixture satisfies (8.4.6), but the converse is not true a mixture obeying (8.4.6) might be stable or metastable. However, if a mixture violates (8.4.6), then the mixture is definitely unstable and not observable. 

Since mole fractions and fugadties are always positive, (8.4.7) suggests that stable one-phase mixtures have... 

Therefore, of the alternatives a, P, and yin Figure 8.18, the stable one-phase mixture is that which has the lowest value for f2- We would compute f2 from an appropriate equation of state. If two of those states had the same value of f2, then a two-phase equilibrium situation could occur. The condition (8.4.9) together with minimization of f2 give us sufficient tools for determining the stability of states proposed for binary mixtures. Note we can make such judgements without solving the phase-equilibrium problem. We illustrate with an example. 

Step 7. Identify the root having the lowest value of f2 as the stable one-phase mixture at the proposed T, P, and From Table 8.2 we see that the stable one-phase mixture is root p. Therefore root a, which is our proposed mixture, is not a stable one-phase mixture. Further, Figure 8.18 shows that root a satisfies the requirement on the derivative (8.4.8), so the proposed mixture is not unstable. Hence, it must be metastable it might be observed, but more likely it will split into two phases. To find the compositions of those phases, we would solve the phase-equilibrium problem. Other procedures for identifying stable one-phase mixtures include the tangent-plane method which originates with Gibbs [15] and has been fully developed by Michelsen, especially for multi-component mixtures [16]. 

At a given temperature, if A < 2 then the binary is a stable one-phase mixture at all compositions. However if at some other temperature, A > 2, then over some range of X the mixture is either metastable or unstable and a phase split can occur. 

Following the procedure outlined in 8.4.4, use the Redlich-Kwong equation (8.2.1) to compute the fugacity /i(xi) for the following mixtures. Prepare plots of your results and identify the regions over which one-phase mixtures are definitely stable and definitely not stable. Will phase splits occur from those situations that are not stable Let the first named component be 1. 

As in 3.6.2, we select the system to be a control volume that is open to steady-state energy and mass transfers. The system contains T homogeneous phases in which (R independent chemical reactions are occurring. Material crosses system boundaries via Np inlet and outlet streams each stream is a one-phase mixture of C components. For... 

In fact, the mixture at x has the smallest fugacity of any stable, one-phase mixture that is rich in component 1. 

We also assume pure 2 is a stable phase, so again by continuity, mixtures from Xj = 0 to x must be stable single phases, and by (F.0.2) their fugacities must increase mono-tonically from 0 to In fact, the mixture at x has the largest fugadty of any stable, one-phase mixture that is rich in component 2. Hence,... 

If only the two phases, a and p, form as a result of the phase split, then one-phase mixtures at compositions between x and can only be metastable or unstable. Therefore, all stable one-phase mixtures are boxmded by [0, x ] or by [x, 1], so they all are described by (F.0.6). Hence at the given T and P, aU stable one-phase mixtures must satisfy (F.0.1). QED... 

Therefore all stable one-phase mixtures still obey (F.0.1). 

Mixtures having f > /pure l are either metastable (they satisfy (F.0.2)) or unstable (they violate (F.0.2)). We caution that while stable one-phase mixtures must obey (F.0.1), the converse is not true mixtures satisfying (F.0.1) are not necessarily stable. They could be stable, unstable, or metastable. 

Consider the phase diagram for a binary polymer-solvent mixture in which the solvent is roughly the size of one monomer unit (segment). Starting from a one phase mixture, a decrease in temperature can lead to separation into polymer-rich and polymer-lean phases at the upper critical solution temperature (UCST) phase boundary. In general a polymer solution also phase separates as temperature is increased to the lower critical solution temperature (LCST) phase boundary [40]. In near critical and supercritical fluids, the driving force for phase separation at the LCST phase boundary is the difference in the compressibility of polymer and solvent, which becomes large... 

For the description of mixtures of substances, FLUENT provides the species model. This model calculates the convection, diffusion, and reaction equations for each component in a mixture. This allows the volumetric reactions, surface reactions, and reactions at phase boundaries to be modeled. For the analysis of one-phase mixtures, the conservation of mass equation for a component i can be formulated by accounting for the local mass fraction Yj ... 

To test this prediction of the crossover from mean-field to Ising-t5q)e behavior, very precise small-angle neutron scattering measurements were completed on blends of deuterated polystyrene (d-PS) and poly(vinyl methyl ether) (PVME) at the critical concentration for a series of temperatures as the one-phase mixture approaches the temperature of phase separation (ie, the critical point Tc in Fig. 7) (61). The data were analyzed by fitting the measured I (q) to the random phase approximation to estimate l(g=0) for each temperature, where the temperature is controlled to 0.01 K... 

Certain classes of polymer blends are known to be truly miscible and form homogeneous, one-phase mixtures. A specific interaction at the submolecular level, e.g.. 

They showed that between Tg and bimodal, the one-phase mixtures exist in a hindered thermodynamic state, because the crystallisation of the TMBPA-PC component is a very slow process which requires very long annealing. With increasing amounts of PS in the mixture the crystallisation half time of TMBPA-PC is reduced. 

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