Nonreacting Systems

If we express the composition of a phase in terms of the mole fractions of all the components, then (C 1) intensive variables are needed to describe the composition, if every component appears in the phase, because the mole fractions must sum to 1. In a system of p phases, p(C — 1) intensive variables are used to describe the composition of the system. As was pointed out in Section 3.1, a one-phase, one-component system can be described by a large number of intensive variables yet the specification of the values of any two such variables is sufficient to fix the state of such a system. Thus, for example, two variables are needed to describe the temperature and pressure of each phase of constant composition or any alternative convenient choice of two intensive variables. Therefore, the total number of variables needed to describe the state of the system is 

To calculate the number of degrees of freedom, we need to know the number of constraints placed on the relationships among the variables by the conditions of equilibrium. 

Mechanical Equilibrium. For a system of fixed total volume and of uniform temperature throughout, the condition of equilibrium is given by Equation (7.9) as 

If phase I of the system changes its volume, with a concurrent compensating change in the volume of phase II, then at constant temperature, it follows from Equation (7.39) that 

The equilibrium constraint of Equation (13.3) can be met only if = P, which is the condition for mechanical equilibrium. (We will discuss several special cases to which this requirement does not apply.) Or, to put the argument differently, if the pressures of two phases are different, the phase with the higher pressure will spontaneously expand and the phase with the lower pressure will spontaneously contract, with a decrease in A, until the pressures are equal. Thus, for p phases, p — independent relationships among the pressures of the phases can be written as follows  

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Model Reactions. Independent measurements of interfacial areas are difficult to obtain in Hquid—gas, Hquid—Hquid, and Hquid—soHd—gas systems. Correlations developed from studies of nonreacting systems maybe satisfactory. Comparisons of reaction rates in reactors of known small interfacial areas, such as falling-film reactors, with the reaction rates in reactors of large but undefined areas can provide an effective measure of such surface areas. Another method is substitution of a model reaction whose kinetics are well estabUshed and where the physical and chemical properties of reactants are similar and limiting mechanisms are comparable. The main advantage of employing a model reaction is the use of easily processed reactants, less severe operating conditions, and simpler equipment. 

Although derived for a reversible process, equation 46 relates properties only, irrespective of the process, and therefore apphes to any change in the equiUbtium state of a homogeneous, closed, nonreacting system. 

When an equilibrium reaction occurs in a vapor-hquid system, the phase compositions depend not only on the relative volatility of the components in the mixture, but also on the consumption (and production) of species. Thus, the condition for azeotropy in a nonreactive system = x, for all i) no longer holds true in a reactive system and must be modified to include reaction stoichiometry ... 

The transformed variables describe the system composition with or without reaction and sum to unity as do Xi and yi. The condition for azeotropy becomes X, = Y,. Barbosa and Doherty have shown that phase and distillation diagrams constructed using the transformed composition coordinates have the same properties as phase and distillation region diagrams for nonreactive systems and similarly can be used to assist in design feasibility and operability studies [Chem Eng Sci, 43, 529, 1523, and 2377 (1988a,b,c)]. A residue curve map in transformed coordinates for the reactive system methanol-acetic acid-methyl acetate-water is shown in Fig. 13-76. Note that the nonreactive azeotrope between water and methyl acetate has disappeared, while the methyl acetate-methanol azeotrope remains intact. Only... 

The volume fractions and mole fractions become identical in ideal gas mixtures at fixed conditions of pressure and temperature. In an isolated, nonreactive system, the molar composition does not vary with temperature. 

For a nonreactive system, the material balance may be done either on a mass or on a molar basis. 

This chapter reviews the reported effects of different types of energy on chemical processing. Many of them are already known for a long time, but were, until recently, mostly used in nonreactive systems such as separation or drying. The focus here is on the (assumed) mechanism, reported effects, and known industrial applications of reactive chemical systems. 

We should also point out that the adsorption equilibrium constants appearing in the Hougen-Watson models cannot be determined from adsorption equilibrium constants obtained from nonreacting systems if one expects the mathematical expression to yield accurate predictions of the reaction rate. One explanation of this fact is that probably only a small fraction of the catalyst sites are effective in promoting the reaction. 

Parametric sensitivity analysis showed that for nonreactive systems, the adsorption equilibrium assumption can be safely invoked for transient CO adsorption and desorption, and that intrapellet diffusion resistances have a strong influence on the time scale of the transients (they tend to slow down the responses). The latter observation has important implications in the analysis of transient adsorption and desorption over supported catalysts that is, the results of transient chemisorption studies should be viewed with caution, if the effects of intrapellet diffusion resistances are not properly accounted for. 

For steady-state design scenarios, the required vent rate, once determined, provides the capacity information needed to properly size the relief device and associated piping. For situations that are transient (e.g., two-phase venting of a runaway reactor), the required vent rate would require the simultaneous solution of the applicable material and energy balances on the equipment together with the in-vessel hydrodynamic model. Special cases yielding simplified solutions are given below. For clarity, nonreactive systems and reactive systems are presented separately. 

For a process that is already in operation, there is an alternative approach that is based on experimental dynamic data obtained from plant tests. The experimental approach is sometimes used when the process is thought to be too complex to model from first principles. More often, it is used to find the values of some parameters in the model that are unknown. Many of the parameters can be calculated from stcadystate plant data, but some parameters must be found from dynamic tests (e.g., holdups in nonreactive systems). 

Chemical reactors may look similar to other units of chemical processing and sometimes they behave similarly, but the nonisothermal chemical reactor has nonhnearities that never occur in nonreacting systems. 

In a dynamic and cross-linkable system, such as the curing of a thermoset that contains a thermoplastic, the phase separation is more complicated than nonreaction system. The phase separation is controlled by the competing effects of thermodynamics and kinetics of phase separation and cure rate of thermoset resin (i.e. time dependent viscosity of the system). 

G. Coherence Control of Wavepackets Reactive and Nonreactive Systems... 

Deans, H. A. and Lapidus, L. A.I.Ch.E.JI. 6 (1960) 656, 663. A computational model for predicting and correlating the behavior of fixed-bed reactors I. Derivation of model for nonreactive systems, II. Extension to chemically reactive systems. 

One essential question is, How many variables must be specified to obtain a solution unique to the phase equilibrium calculation It is possible to have an infinite number of solutions to a problem if too few variables are specified—or no solution if too many variables are specified. One answer to this question is provided by Gibbs Phase Rule (Gibbs, 1928, p. 96), simply stated for nonreacting systems by the equation ... 

Section 6.2 deals with high-dimensional lumped nonreacting systems, with special emphasis on multitray absorption. 

Section 6.3 treats distributed nonreacting systems and specifically packed bed absorption, while Section 6.4 studies a battery of nonisothermal CSTRs and its dynamic behavior. 

In the previous chapters of this book we have dealt only with one phase systems. There the emphasis was mainly on reacting systems. Single-phase nonreacting systems such as mixers and splitters are almost trivial and shall not be dealt with at all. However, in the present section we deal with heterogeneous systems and here nonreacting systems are generally as nontrivial as reacting systems. 

For no reactions, i.e., a nonreacting system and single in- and output, the equations reduce to the trivial form n = for each component i. 

Equation (14-18) is the more useful one in practice. It requires either actual experimental HOG data or values estimated by combining individual measurements of HG and HL by Eq. (14-19). Correlations for Hg, Hl, and HOG in nonreacting systems are presented in Sec. 5. 

This shows that the natural variables of G for a one-phase nonreaction system are T, P, and n . The number of natural variables is not changed by a Legendre transform because conjugate variables are interchanged as natural variables. In contrast with the natural variables for U, the natural variables for G are two intensive properties and Ns extensive properties. These are generally much more convenient natural variables than S, V, and k j. Thus thermodynamic potentials can be defined to have the desired set of natural variables. 

Thus we have seen that all the thermodynamic properties of a one-phase nonreaction system can be calculated from G(T, P, ). 

This form of the fundamental equation for G applies to a system at chemical equilibrium. Note that the number D of natural variables of G is now C + 2, rather than Ns + 2 as it was for a nonreaction system (see Section 2.5). There are... 

The criterion of spontaneous change and equilibrium for a nonreaction system is dG < 0 at constant T, P, and n , but the criterion for a system involving chemical reactions is dG 0 at constant T, P, and nci. Therefore, to calculate the composition of a reaction system at equilibrium, it is necessary to specify the amounts of components. This can be done by specifying the initial composition because the initial reactants obviously contain all the components, but this is more information than necessary, as we will see in the chapter on matrices. 

For a reaction with positive gas mole change, Eq. (47) indicates that Kx decreases with pressure. Because ce is a monotonically increasing function of Kx, the equilibrium extent of a reaction with positive Avgas always decreases as pressure is increased. This is an example of Le Chatelier s principle, which states that a reaction at equilibrium shifts in response to a change in external conditions in a way that moderates the change. In this case, because the reaction increases the number of moles of gas and thus the pressure, the reaction shifts back to reactants. The isothermal compressibility of a reactive system can, therefore, be much greater than that of a nonreactive system. This effect can be dramatic in systems with condensed phases. For example, in the calcium carbonate dissociation discussed in Example 12, if the external pressure is raised above the dissociation pressure of C02, the system will compress down to the volume of the solid. Of course, a similar effect is observed in simple vaporization or sublimation equilibrium. As the pressure on water at 100°C is increased above 1.0 atm, all vapor is removed from the system. 

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