Phase equilibrium condition

Physical constraints, imposed by mass, energy, and momentum balances, chemical and phase equilibrium conditions, and rate phenomena. 

This chapter focuses on the question, What are hydrates and Chapter 3 concerns the question, How do hydrates form Although a few thermodynamic properties are discussed in this chapter, the phase equilibrium conditions are considered in Chapters 4, 5, and 6. 

This equation can be used to show that fiAx = nAp = HA, HBx = HBp = /phase equilibrium conditions are inserted in equation 8.2-5, it becomes... 

The solution of Eq. (26) for the phase equilibrium condition gives P(T) (e.g., vapor pressure as a function of temperature) or T(P) (e.g., melting point as a function of pressure). A single point on one of these curves can be obtained by measurement or calculation.1 We will now show how thermodynamics can be used to obtain the slope of these curves. Equation (26), pa = pp, holds at phase equilibrium. If we change either P or T and the system remains at equilibrium, Eq. (26) must still hold ... 

It is not straightforward to calculate phase equilibria or chemical potentials in a simulation, and special techniques must be used. This has been an active research area over the past few years [15]. Several methods have been proposed, and these can be divided into direct methods, in which the coexistence properties of the phases are calculated directly, and indirect methods, where the chemical potential is first calculated and then used to determine the phase equilibrium conditions (Table 4). 

In systems where heterogeneous chemical equilibria prevail, both chemical and phase equilibrium conditions must be simultaneously satisfied. In practice, this means that the chemical equilibrium condition—Eq. (51) in the discrete description, and Eq. (60) in the continuous one—must be satisfied in one phase, and the phase equilibrium condition [/( or fi(x) to be the same in all phases] must be satisfied this clearly guarantees that the chemical equilibrium condition is automatically satisfied in all phases. 

Eq. (14) can be also used to derive an expression for the solubility of a protein in a mixed solvent as a function of the cosolvent concentration [29,30]. Indeed, by combining Eq. (14) with the phase equilibrium condition one obtains the following equation [30]... 

In the present study, CH4 was adopted as help gas from a viewpoint to develop the gas-hydrate technology mentioned above. Ten C6-C8 hydrocarbons, that have various sizes and shapes, were added to CH4+H2O system for phase equilibrium measurement. The objective is to study the relations between the equilibrium conditions and the molecular sizes of LGS. The phase equilibrium relations for CH4+dimethylcyclohexane (DMCH) stereo-isomers, CH4+methylcyclohexane (MCH), CH4+cyclooctane (c-Octane), and CH4+methylcyclopentane (MCP) s-H hydrate systems were measured under four-phase equilibrium condition up to 10 MPa. The four phases were gas, water, liquid LGS, and s-H hydrate phases. In addition, the single-crystals of s-H hydrates were analyzed by Raman spectroscopy in order to obtain Raman spectra about the s-H hydrates. The CH4+1,1-DMCH and CD4+1,1-DMCH s-H hydrate systems were analyzed under four-phase equilibrium conditions up to 30 MPa. 

CD4 was used to avoid the overlap of Raman peaks and to clarify the Raman spectra derived from only 1,1-DMCH, because the Raman peaks of CH4 and 1,1-DMCH overlap around 2900 cm . Figures 3 (a) and (b) are the Raman spectra of 1,1-DMCH observed in the CD4+1,1-DMCH hydrate system under four-phase equilibrium condition at 294.06 K and 14.2 MPa. The Raman spectrum of liquid LGS phase is shown in Figure 3 (a), that of s-H hydrate phase is Figure 3 (b). Figure 3 (a) also agrees well with the reference The Raman spectrum of 1,1 -DMCH changed apparently by enclathration. 

Figure 4 (a) shows the Raman spectra of 1,1-DMCH and CH4 in the CH4+1,1-DMCH hydrate system under four-phase equilibrium condition at 298.28 K and 25.5 MPa. All... 

Figure 5 Raman spectra of CD4 for the CD4+1,1-DMCH hydrate system under four-phase equilibrium condition at 294.06 K and 14.2 MPa. (a) hydrate phase, (b) gas phase.

The pressure-temperature relations for the CH4 s-H hydrates were measured under four-phase equilibrium condition. The s-H hydrate formation was observed in the CH4+CI-DMCH, CH4+MCH, CH4+c-Octane, CH4+c >l,2-DMCH, CH4+c/ -l,4-DMCH and CH4+MCP systems. The equilibrium pressure of each s-H system increases in that order at isothermal condition and it is lower than that of pure CH4 s-I hydrate. The 1,1-DMCH molecule has the best molecular-size to fit the E-cage cavity and constructs the CH4 s-H hydrate at the lowest pressure in the all LGS of present study. 

At the three-phase equilibrium condition, the chemical potential (or fugacity) of water in the hydrate phase... 

Three-phase equilibrium conditions Hydrate formation conditions Gas phase... 

This fundamental criterion of equilibrium may be applied to practical situations such as determining phase equilibrium conditions. The relationship 1.8 is combined with the first law to arrive at the following expression ... 

A bubble point or dew point calculation consists of determining the pressure at a given temperature, or the temperature at a given pressure, which would result in X S that satisfy Equation 2.16 or 2.17. In addition, the phase equilibrium condition. Equation 2.12, must be satisfied. 

With constant distillate rate and composition, the amount and composition of the residue at any point in time can be determined by material balance. The number of stages is also fixed, so that leaves the reflux ratio (or UV ratio) as the only variable to be adjusted to satisfy the phase equilibrium conditions. 

Assuming phase equilibrium conditions, a problem solution strategy may be used... 

The field of membrane separations is radically different from processes based on vapor-liquid or fluid-solid operations. This separation process is based on differences in mass transfer and permeation rates, rather than phase equilibrium conditions. Nevertheless, membrane separations share the same goal as the more traditional separation processes the separation and purification of products. The principles of multi-component membrane separation are discussed for membrane modules in various flow patterns. Several applications are considered, including purification, dialysis, and reverse osmosis. 

This reaction is simply a restatement of the phase-equilibrium condition of equality of chemical potentials of a given component in the phases in which it is found. 

For the liquidus curve in the pre-eutectic region of the phase diagram with a simple eutectic, the phase-equilibrium condition means AG — 0, which can be expressed by the following equation ... 

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