Homotopy global

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. 

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. 

The global Newton method (Sec. 4.2.9) can be used for highly nonideal systems or reactive distillation systems with a homotopy forcing (Sec, 4.2.12) or relaxation technique (Sec, 4,2.11). 

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. 

Different solution approaches can be used to converge problems when a good initial point is not available. These include Homotopy or Continuation techniques (e.g., Allgower et al., 1990), Interval solvers (e.g., Neumaier, 1990) and Global Optimisation (e.g., Maranas and Floudas, 1995). Instead of using a different solution method, we propose to find a better initial point by using the same equations that have to be solved as guidance. 

The embedding chosen in (3.8.1) is called a global homotopy. It is a special case of a more general class of embeddings, the so-called convex homotopy... 

The existence of a solution path x s) at least in a neighborhood of (x(0),0) can be ensured by standard existence theorems for ordinary differential equations as long as Hx has a bounded inverse in that neighborhood. For the global homotopy (3.8.1) this requirement is met if F satisfies the conditions of the Inverse Function Theorem 3.2.1. 

For the global homotopy the Davidenko differential equation reads... 

Let us now consider the case where the global unstable set of the saddle-node periodic orbit L is not a manifold, but has the structure like shown in Fig. 12.4.1. This means that the integer m which determines the homotopy class of the curve fl jSq in the cross-section 5q x = —d is... 

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