Multiphase system, equilibrium state

Example 4.2 used the method of false transients to solve a steady-state reactor design problem. The method can also be used to find the equilibrium concentrations resulting from a set of batch chemical reactions. To do this, formulate the ODEs for a batch reactor and integrate until the concentrations stop changing. This is illustrated in Problem 4.6(b). Section 11.1.1 shows how the method of false transients can be used to determine physical or chemical equilibria in multiphase systems. 

Equilibrium in multiphase and/or multireaction systems. If more than one phase is present in the system, a criterion of phase equilibria has to be satisfied together with the chemical equilibrium criterion. For instance, in a gas-liquid system components are in chemical equilibrium in the phase where the reaction occurs, but vapour-liquid equilibria between the gas and the liquid phases must also be taken into account. To determine the equilibrium composition of a reacting mixture in both phases, chemical equilibrium constants as well as data concerning vapour-liquid equilibria for all components of the reaction mixture should be known. In the equilibrium state ... 

The following criterion of phase equilibrium can be developed from the first and second laws of thermodynamics the equilibrium state for a closed multiphase system of constant, uniform temperature and pressure is the state for which the total Gibbs energy is a minimum, whence... 

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

We may, with appropriate attention to the reference state, develop the relations in terms of components between the intensive variables pertinent to multiphase systems that contain species other than the components. Such relations would be rather complex, because no account would be taken of the effect of the chemical reactions that occur in the system. All deviations from ideality would appear either in the activity coefficients for substances in condensed phases or in the coefficients used in some equations of state for the gas phase. Simpler relations are obtained when the conditions of phase equilibrium are based on species rather than components, once the species have been identified. 

Ordinarily, the system may consist of several phases, whose interior in the state of equilibrium is homogeneous throughout its extent. The system, if composed for instance of only liquid water, is a single phase and if made up for instance of liquid water and water vapor, it is a two phase system. The single phase system is frequently called a homogeneous system, and a multiphase system is called heterogeneous. 

Distinguish between intensive and extensive variables, giving examples of each. Use the Gibbs phase rule to determine the number of degrees of freedom for a multicomponent multiphase system at equilibrium, and state the meaning of the value you calculate in terms of the system s intensive variables. Specify a feasible set of intensive variables that will enable the remaining intensive variables to be calculated. 

To describe the state of a reaction in a phase, we need to know the stoichiometric coefficients, j, and the chemical potential, pi, for each species in the reaction. For reaction equilibrium, the quantity AG = E Vi pi = 0 (as is T diS). For a possible, or spontaneous, reaction, AG < 0. For multireaction systems, complete equilibrium corresponds to dG = 0 for the system, that is, the Gibbs energy of the phase is a minimum. The total internal entropy production must vanish for the entire system. Similar consideration apply to multiphase systems. An expression analogous to equation 39 for dE, but for fixed T and p conditions, is ... 

DYNAMICS OF DISTRIBUTION The natural aqueous system is a complex multiphase system which contains dissolved chemicals as well as suspended solids. The metals present in such a system are likely to distribute themselves between the various components of the solid phase and the liquid phase. Such a distribution may attain (a) a true equilibrium or (b) follow a steady state condition. If an element in a system has attained a true equilibrium, the ratio of element concentrations in two phases (solid/liquid), in principle, must remain unchanged at any given temperature. The mathematical relation of metal concentrations in these two phases is governed by the Nernst distribution law (41) commonly called the partition coefficient (1 ) and is defined as = s) /a(l) where a(s) is the activity of metal ions associated with the solid phase and a( ) is the activity of metal ions associated with the liquid phase (dissolved). This behavior of element is a direct consequence of the dynamics of ionic distribution in a multiphase system. For dilute solution, which generally obeys Raoult s law (41) activity (a) of a metal ion can be substituted by its concentration, (c) moles L l or moles Kg i. This ratio (Kd) serves as a comparison for relative affinity of metal ions for various components-exchangeable, carbonate, oxide, organic-of the solid phase. Chemical potential which is a function of several variables controls the numerical values of Kd (41). 

In this section we are interested in determining the amount of information, and its type, that must be specified to completely fix the thermodynamic state of an equilibrium single-component, multiphase system. That, is, we are interested in obtaining answers to the following questions ... 

To specify the thermodynamic state of any one phase of a single-component, multiphase system, two thermodynamic state variables of that phase must be specified that is, each phase has two degrees of freedom. Thus, it might appear that if V phases are present, the system would have TP degrees of freedom. The actual number of degrees of freedom is considerably fewer, however, since the requirement that the phases be in equilibrium puts constraints on the values of certain state variables in each phase. For example, from the analysis of Secs. 7.1 and 7.2 it is clear that at equilibrium the temperature in each phase must be the same. Thus, there are P — 1 reladons of the form... 

From these equations, it is evident that to determine the mass distribution between the phases, we need to specify a sufficient number of variables of the individual phases to fix the thermodynamic state of each phase (i.e., the degrees of freedom F) and V — thermodynamic properties of the multiphase system in the form of Eq. 7.6-3. For example, if we know that steam and water are in equilibrium at some temperature T (which fixes the single-degree freedom of this two-phase system), the equation of state or the steam tables can be used to obtain the equilibrium pressure, specific enthalpy, entropy, and volume of each of the phases, but not the mass distribution between the phases. If, in addition, the volume tor enthalpy or entropy, etc.) per unit mass of the two-phase mixture were known, this would be sufficient to determine the distribution of mass between the two phases, and then all the other overall thermodynamic properties. 

Specification of the Equilibrium Thermodynamic State of a Multicomponent, Multiphase System 387... 

SPECIFICATION OF THE EQUILIBRIUM THERMODYNAMIC STATE OF A MULTICOMPONENT, MULTIPHASE SYSTEM THE GIBBS PHASE RULE... 

It is also of interest to determine the amount and type of additional information needed to fix the relative amounts of each of the phases in equilibrium, once their thermodynamic states are known. We can obtain this from an analysis that equates the number of variables to the number of restrictions on these variables. It is convenient for this discussion to write the specific thermodynamic properties of the multiphase system in terms of the distribution of mass between the phases. The argument could be ba.sed on a distribution of numbers of moles however, it is somewhat more straightforv/ard on a mass basis because total mass, and not total moles, is a conserved quantity. Thus, we will use w to represent the mass fraction of the ith phase.- Clearly the w must satisfy the equation... 

The equilibrium state of a system at constant temperature and pressure is characterized by a minimum in the Gibbs free energy of the system. For a multicomponent, multiphase system, the minimum free energy corresponds to uniformity of the chemical potential (gi) of each component throughout the system. For a binary system, the molar free energy (G) and chemical potentials are related by Equation (2.1),... 

The processes discussed in this chapter demonstrate the great variety of phase equilibrium that can arise beyond the basic vapor-liquid problems discussed in most of the previous chapters. Many other systems could be included The adsorption of gases onto solids (used in the removal of pollutants from air), the distribution of detergents in water/oil systems, the wetting of solid surface by a liquid, the formation of an electrochemical cell when two metals make contact are all examples of multiphase/multicomponent equilibrium. They all share one important common element their equilibrium state is determined by the requirement that the chemical potential of any species must be the same in any phase where the species can be found. These problems are beyond the scope of this book. The important point is this The mathematical development of equilibrium (Chapter 10) is extremely powerful and encompasses any system whose behavior is dominated by equilibrium. 

In a multicomponent, multiphase system at equilibrium, ji, is the same in every phase, but in most cases /i° and therefore fi, — fi° is different for solids, liquids, gases, and solutes (we know this without knowing the numerical value of either term). Thermodynamic properties are determined and tabulated for substances in these various standard states, and how they relate to one another in chemical reactions can be seen when we consider the equilibrium constant (Chapter 9). 

Frequently encountered in nature and process industries, multiphase flows may comprise various states of matter, e.g., gas and solid in fluidization gas and liquid in bubble column and gas, liquid, and solid in airlift slurry bed (Mudde, 2005). In this article, the term phase in multiphase flow is related to the aggregative state of flow, which is normally far from equilibrium states. And it is different from the phase for a thermodynamic equilibrium system, where the phase is used to refer to a set of equilibrium states that can be demarcated in terms of state variables by a phase boundary on a phase diagram. As a result, it is possible to have a gas—soHd flow mixture with more than two phases, which can be classified by size, density of particles, or by the states of dispersion, e.g., poly disperse multiphase flow pCue and Fox, 2014) and dilute—dense, gas—soHd multiphase flow (Hong et al, 2012). 

Direct numerical simulation (DNS) is a powerful tool to investigate the velocity distribution and density fluctuation of multiphase flow systems, thus facilitates revealing the mechanisms of nonequilibrium behavior. Due to its high demand in computing resources, current DNS is largely hmited to simulations over static arrays of particles or smaU-sized, periodic flow domains, which are expected to be close to local equilibrium states. Thus, the nonequilibrium characteristics of multiphase flow are hard to be fuUy revealed. Recent release of hybrid computing hardware boosts the rapid development of DNS with respect to the scales of time and space... 

Pure substances may occur in a variety of phases, depending on the boundary conditions. Thus, increasing the temperature of a solid at constant pressure causes its fusion and finally its vaporization to a gas. In pure substances, phases correspond to the states of aggregation. In a heterogeneous system, on the other hand, a number of phases in the same state of a regation may coexist (see earlier examples). Here, we shall describe the conditions under which a multiphase system is in equilibrium in other words, when does a homogeneous system dissociate into different phases. ... 

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