Equilibrium design

As the pressure is increased in the system, the equilibrium designated as reaction (9.2) shifts to Ti305(l) because the pressure effect overrides the increase in temperature with pressure that would make the shift to the gaseous products. Indeed, at a given assigned enthalpy, the Ti305(l) mole fraction increases as the total pressure increases. For the equilibrium designated as... 

Here Z represents a catalyst surface site (active centre). The two final steps are in equilibrium, designated by the symbol —. "The natural classification of simple (elementary) reactions by the number of molecules involved simultaneously in the reaction belongs to Van t Hoff. If the reaction involves one molecule (reaction A - B), it is classified as first-order (monomolecular). In cases where two molecules take part in the reaction (e.g. 2 A - B or A + B - C), the reaction is said to be second-order (bimolecular). With the participation of three molecules (3 A -> B or 2 A + B -> C), the reaction is specified as third-order (termolecular). The simultaneous interaction of more than three reactants is believed to be highly improbable. 

Table 8.1 describes the steps of the methodology in more detail. The procedure starts with the Problem definition production rate, chemistry, product specifications, safety, health and environmental constraints, physical properties, available technologies. Then, a first evaluation of feasibility is performed by an equilibrium design. This is based on a thermodynamic analysis that includes simultaneous chemical and physical equilibrium (CPE). The investigation can be done directly by computer simulation, or in a more systematic way by building a residue curve map (RCM), as explained in the Appendix A. This step will identify additional thermodynamic experiments necessary to consolidate the design decisions, mainly phase-equilibrium measurements. Limitations set by chemical equilibrium or by thermodynamic boundaries should be analyzed here. 

To make Equation (8.21) tractable for design purposes, it is necessary to make the assumption that thermodynamic equilibrium exists at the interface, i.e., yl = Ki or y,, /x, int = Ki, where Kj is the conventional thermodynamic equilibrium ratio and the asterisks denote an equilibrium composition. Note that the equilibrium designation depends on the concentration units used. For example, if gas concentrations are in partial pressures and liquid compositions are in molar concentrations, a Henry s law coefficient. Hi = must be used. 

Furnace buyers may not be familiar with furnace technology, and they may be obligated to buy the least-expensive bid. For example, the energy need following a delay is much higher than this equilibrium design. 

Phases gas-liquid, liquid-liquid, gas-liquid biosolids. Very fast reactions, essentially plug flow for both G and L. High capacity, high conversion in both gas and liquid phases. Difficult to control temperature, adiabatic. Conversion is often limited by equilibrium. Design like an absorber. Section 4.8. Dimensionless Hatta number, 0.3 < Ha < 0.6 and = 10-100. 

If a reaction is reversible, both products and reactants will be present during that reaction. If the reaction of A -l- B to give C -i- D is reversible, all four components—A, B, C, and D—will be present. In principle, the reaction will reach an equilibrium condition and at that point the amounts of A-D will be relatively constant. If the reaction comes to equilibrium (designated by the stacked forward and reverse reaction arrows), the position of the equilibrium is defined by the equilibrium constant, K (K q). The value of K is determined by the concentrations of all species, K = products/reactants, where products are on the right side of the reaction and reactants are on the left side of the equation. In this particular reaction, K = [C][D]/[A][B]. 

The composition profiles are quite different between the two cases. Figure 9.17 presents the results for the equilibrium design. A comparison with Figure 9.14b shows that the reactant compositions are much smaller for the equilibrium model (peak methanol concentration = 5 mol%) compared with those for the kinetic model (peak methanol concentration = 25 mol%). 

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

In Equation (24), a is the estimated standard deviation for each of the measured variables, i.e. pressure, temperature, and liquid-phase and vapor-phase compositions. The values assigned to a determine the relative weighting between the tieline data and the vapor-liquid equilibrium data this weighting determines how well the ternary system is represented. This weighting depends first, on the estimated accuracy of the ternary data, relative to that of the binary vapor-liquid data and second, on how remote the temperature of the binary data is from that of the ternary data and finally, on how important in a design the liquid-liquid equilibria are relative to the vapor-liquid equilibria. Typical values which we use in data reduction are Op = 1 mm Hg, = 0.05°C, = 0.001, and = 0.003... 

Null, H. R., "Phase Equilibrium in Process Design," John Wiley, New York (19 70). ... 

In modern separation design, a significant part of many phase-equilibrium calculations is the mathematical representation of pure-component and mixture enthalpies. Enthalpy estimates are important not only for determination of heat loads, but also for adiabatic flash and distillation computations. Further, mixture enthalpy data, when available, are useful for extending vapor-liquid equilibria to higher (or lower) temperatures, through the Gibbs-Helmholtz equation. ... 

The most frequent application of phase-equilibrium calculations in chemical process design and analysis is probably in treatment of equilibrium separations. In these operations, often called flash processes, a feed stream (or several feed streams) enters a separation stage where it is split into two streams of different composition that are in equilibrium with each other. 

Smith, B. D., "Design of Equilibrium Stage Processes," McGraw-Hill, New York (1963). 

It is important to keep in mind that the phases are mutually in equilibrium. In particular, the designation is a reminder that the solid surface must be in equilibrium with the saturated vapor pressure and that there must therefore be an adsorbed film of film pressure (see Section X-3B). Thus... 

In this section we concentrate on the electronic and vibrational parts of the wavefimctions. It is convenient to treat the nuclear configuration in temis of nomial coordinates describing the displacements from the equilibrium position. We call these nuclear nomial coordinates Q- and use the symbol Q without a subscript to designate the whole set. Similarly, the symbol v. designates the coordinates of the th electron and v the whole set of electronic coordinates. We also use subscripts 1 and ii to designate the lower and upper electronic states of a transition, and subscripts a and b to number the vibrational states in the respective electronic states. The total wavefiinction f can be written... 

In order for this to work, the force field must be designed to describe inter-molecular forces and vibrations away from equilibrium. If the purpose of the simulation is to search conformation space, a force field designed for geometry optimization is often used. For simulating bulk systems, it is more common to use a force field that has been designed for this purpose, such as the GROMOS or OPLS force fields. 

Rapid Approximate Design Procedure for Curved Operating and Equilibrium Lines. If the operating or the equihbrium line is nonlinear, equation 56 is of Httie use because will assume a range of values over the tower. The substitution of effective average values for m and... 

The recommended design procedure uses the values of (E /and m from Figures 7 and 8 in equation 56 and yields a very good estimation of Alp despite the curvature of the operating and the equilibrium lines. This value differs from A/q obtained by equation 49 because of the /(I — y) term in the latter equation. A convenient approach for purposes of approximate design is to define a correction term AA/q which can be added to equation 55 ... 

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