Thermodynamic homotopy

Another implementation of homotopy-continuation methods is the use of problem-dependent homotopies that exploit some physical aspect of the problem. Vickeiy and Taylor [AIChE J., 32, 547 (1986)] utilized thermodynamic homotopies for K values and enthalpies to gradually move these properties from ideal to ac tual values so as to solve the MESH equations when veiy nonideal hquid solutions were involved. Taylor, Wayburn, and Vickeiy [I. Chem. E. Symp. Sen No. 104, B305 (1987)] used a pseudo-Murphree efficiency homotopy to move the solution of the MESH equations from a low efficiency, where httle separation occurs, to a higher and more reasonable efficiency. 

The examples tested by Taylor et al. (80) for the efficiency homotopy were for moderate- or narrow-boiling mixtures. No wide-boiling mixtures were tested. Since the temperature profiles at the intermediate values of E yy will be flat and not broad, the homotopy may be best for the moderate- and narrow-boiling systems. Most of the mixtures were nonideal and the efficiency homotopy should lessen the effect of nonideal If-values where E yy acts as a damper on the if-values. The efficiency homotopy does not work for purity specifications because the purity will not be satisfied in solutions of early values of E yy-Vickery and Taylor (81) presented a thermodynamic homotopy where ideal If-values and enthalpies were used for the initial solution of the global Newton method and then slowly converted to the actual If-values and enthalpies using the homotopy parameter t, The homotopy functions were embedded in the If-value and enthalpy routines, freeing from having to modify the MESH equations. The If-values and enthalpies used are the homotopy functions ... 

The global Newton methods, such as the Naphtali-Sandholm method (Sec. 4.2.9), are often used to solve highly nonideal systems. These are frequently prone to failure. Good explanations of the theory of homotopy methods are provided by Seader (86) and Wayburn (83). A homotopy method can greatly expand the global Newton method ability to solve difficult nonideal systems. Homotopy methods have been associated with the Naphtali-Sandholm method, where the derivatives of the if-values and enthalpies with respect to all compositions directly appear within the Jacobian. Using a thermodynamic homotopy for another method such as a Tomich has not been presented in the literature. 

In the category of physical continuation methods are the thermodynamic homotopies of Vickery and Taylor [AIChE J., 32, 547 (1986)] and a related method due to Frantz and Van Brunt (AIChE National Meeting, Miami Beach, 1986). Thermodynamic continuation has also been used to find azeotropes in multicomponent systems by Fid-kowski et al. [Comput. Chem. Engng., 17, 1141 (1993)]. Parametric continuation methods may be considered to be physical continuation methods. The reflux ratio or bottoms flow rate has been used in parametric solutions of the MESH equations [Jelinek et al., Chem. Eng. Sd.,28, 1555(1973)]. 

Systems with highly nonideal VLE suffer from requiring very good initial profiles The sneaking-up technique can be used by first solving the column with a simple approximation of the VLE and then slowly introducing the nonideal VLE. This is described by Brierley and Smith (106) and is also the thermodynamic homotopy of Vickery and Taylor (81). As stated in Secs. 4.2.9 and 4.2.12, this can occur in the global Newton methods. The inside-out methods avoid these problems in their use of simple VLE models. 

The number of equations, M5C + 1), for a large number of trays and components, can be excessive. The global Newton method will suffer from the same problem of requiring initial values near the answer. This problem is aggravated with nonequilibrium models because of difficulties due to nonideal if-values and enthalpies then compounded by the addition of mass transfer coefficients to the thermodynamic properties and by the large number of equations. Taylor et al. (80) found that the number of sections of packing does not have to be great to properly model the column, and so the number of equations can be reduced. Also, since a system is seldom mass-transfer-limited in the vapor phase, the rate equations for the vapor can be eliminated. To force a solution, a combination of this technique with a homotopy method may be required. 

Reliable and fast equilibrium calculations (or so-called flash calculations) are the mechanism by which thermodynamic properties are used in industry. This area has received much attention in the past. Algorithms include successive substitution with acceleration and stability analysis,Inside-Out and Interval methods, Homotopy continuation methods with application to three-phase systems, and systems with simultaneous physical and chemical equilibrium. An area of recent focus is the flash algorithm for mixtures containing polydisperse polymers. However, many challenging problems remain. 

The NEQ model requires thermodynamic properties, not only for calculation of phase equilibrium but also for calculation of driving forces for mass transfer. In addition, physical properties such as surface tension, diffusion coefficients, and viscosities, for calculation of mass (and heat) transfer coefficients and interfacial areas are required. The steady-state model equations most often are solved using Newton s method or by homotopy-continuation. A review of early applications of NEQ models is available [5]. 

---