Homogeneous equilibrium method

In a series of papers by Leung and coworkers (AlChE J., 32, 1743-1746 [1986] 33, 524-527 [1987] 34, 688-691 [1988] J. Loss Prevention Proc. Ind., 2[2], 78-86 [April 1989] 3(1), 27-32 [Januaiy 1990] Trans. ASME J. Heat Transfer, 112, 524-528, 528-530 [1990] 113, 269-272 [1991]) approximate techni ques have been developed for homogeneous equilibrium calculations based on pseudo-equation of state methods for flashing mixtures. 

Sutherland (1975). Orifice flow rates are underpredicted by about the same factor with the energy balance method and with the NEM. Discharge predictions for short (0.2-m) pipes are overpredicted by the energy balance method. In this region, the assumption of homogeneous equilibrium is not justified. A model that takes slip velocity into account may improve predictions for short pipes. 

The purpose of this chapter is to outline the simplest methods of arriving at a description of the distribution of species in mixtures of liquids, gases and solids. Homogeneous equilibrium deals with single phase systems, such as electrolyte solutions (e.g., seawater) or gas mixtures (e.g., a volcanic gas). Heterogeneous equilibrium involves coexisting gaseous, liquid and solid phases. 

Assumption (f) of isothermal flow means that the method is different to the homogeneous equilibrium model (which assumes adiabatic flow). The difference between the two assumptions is usually small. The isothermal flow assumption gives a slightly simpler method and yields a conservative low value of G for relief sizing purposes. The DIERS Project Manual1111 gives the alternative version of Tangren et al/s method, which assumes adiabatic flow and is therefore equivalent to the HEM. 

The Omega method calculates the two-phase flow capacity per unit area, G, of a nozzle or pipe of constant diameter. It evaluates the homogeneous equilibrium model (see 9.4.1) for two-phase flow. The Omega method is particularly convenient, when applicable, because it does not require the use of a computer. All properties can often be evaluated at the conditions in the upstream vessel, (which are known). Most other methods to evaluate G for two-phase flow require the use of appropriate computer codes (see Annex 4). Exceptions are given in 9.4. 

J L Woodward, "An Amended Method for Calculating Omega for a Homogenous Equilibrium Model of Predicting Discharge Rates", J Loss Prev Process Ind, 8 (5), 253-259, 1995... 

The emphasis of this book is entirely on analytical, mechanistic (homogeneous), kinetic (homogeneous), and synthetic (laboratory-scale) applications. Physical electrochemistry is not a direct concern, and equilibrium methods (potentiometry) are intentionally omitted. There is no attempt to include specific chemical examples except where they are particularly illustrative and have pedagogical value. No extensive review of the original literature is included, but references to key reviews and papers of historical interest are emphasized. Authors have selected experimental approaches that work best and have commented freely on outmoded or underdeveloped methods. The authors and editors have made value judgments that undoubtedly will disappoint some readers. 

Hopefully, we have hinted that there are interesting nonequilibrium systems that can be explored using homogeneous, field-driven simulations. Because these approaches are often extensions of equilibrium methods, it is natural to first present the equilibrium foundation, as is done in the next section. From there, we will be able to develop the theoretical basis for NEMD simulations and practical guidelines for implementing them. Once the tools are in place, we will discuss applications and the kinds of question that can be tackled by NEMD methods. 

The computer code for the electrode equilibration model (EHDRIFT) was written in PASCAL for use on a microcomputer. The program calculates the rest potential which is the EMF value where the currents sum to zero. If the system is in homogeneous equilibrium the rest potential will represent the system Eh. The numerical algorithm uses Eulers method (il) to integrate Equation 2, which involves recalculation of the aqueous concentrations (Equations 4 and 5) at each time step. A full listing of the source code can be found in Kempton (12). 

Submarine slopes are commonly fairly gentle, uniform, and homogeneous over considerable horizontal distances (see Figure 11.8). Thus in many cases infinite slope limiting equilibrium methods are applicable. Assume an element of an infinite slope as shown in Figure 11.10. 

It is also necessary that the condensate be a homogeneous mixture.. Thus, if the condensate separates into two layers, the operation is not satisfactory. The other vapor-liquid equilibrium methods are suitable for multilayer systems either in the stiU or in the vapor sample. 

The competitive adsorption isotherms were determined experimentally for the separation of chiral epoxide enantiomers at 25 °C by the adsorption-desorption method [37]. A mass balance allows the knowledge of the concentration of each component retained in the particle, q, in equilibrium with the feed concentration, < In fact includes both the adsorbed phase concentration and the concentration in the fluid inside pores. This overall retained concentration is used to be consistent with the models presented for the SMB simulations based on homogeneous particles. The bed porosity was taken as = 0.4 since the total porosity was measured as Ej = 0.67 and the particle porosity of microcrystalline cellulose triacetate is p = 0.45 [38]. This procedure provides one point of the adsorption isotherm for each component (Cp q. The determination of the complete isotherm will require a set of experiments using different feed concentrations. To support the measured isotherms, a dynamic method of frontal chromatography is implemented based on the analysis of the response curves to a step change in feed concentration (adsorption) followed by the desorption of the column with pure eluent. It is well known that often the selectivity factor decreases with the increase of the concentration of chiral species and therefore the linear -i- Langmuir competitive isotherm was used ... 

Equilibrium Compositions for Single Reactions. We turn now to the problem of calculating the equilibrium composition for a single, homogeneous reaction. The most direct way of estimating equilibrium compositions is by simulating the reaction. Set the desired initial conditions and simulate an isothermal, constant-pressure, batch reaction. If the simulation is accurate, a real reaction could follow the same trajectory of composition versus time to approach equilibrium, but an accurate simulation is unnecessary. The solution can use the method of false transients. The rate equation must have a functional form consistent with the functional form of K,i,ermo> e.g., Equation (7.38). The time scale is unimportant and even the functional forms for the forward and reverse reactions have some latitude, as will be illustrated in the following example. 

We have considered thermodynamic equilibrium in homogeneous systems. When two or more phases exist, it is necessary that the requirements for reaction equilibria (i.e., Equations (7.46)) be satisfied simultaneously with the requirements for phase equilibria (i.e., that the component fugacities be equal in each phase). We leave the treatment of chemical equilibria in multiphase systems to the specialized literature, but note that the method of false transients normally works quite well for multiphase systems. The simulation includes reaction—typically confined to one phase—and mass transfer between the phases. The governing equations are given in Chapter 11. 

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