Global Newton methods

The global Newton methods are the most sensitive of the rigorous methods to the quality of the initial values and often require initial values near the answer. While they are the most powerful in solving nonideal mixtures, it may be necessary to use another rigorous method, such as a BP or SR method, to develop initial values approaching the solution before the global Newton method takes over the calculation. 

The method of Gallun and Holland is the broadest application of the MESH equations in a global Newton method and may solve the widest range of columns. Formulations by Gallun and Holland (40) for distillation columns included adding the total material balance to give freedom in specifications or to substitute these for the equilibrium equations for more ideal mixtures. 

Gallun and Holland (40) also presented solution techniques for the sparse matrices of the global Newton method. There are many other... 

The matrix solution techniques of the block-banded formulations of Naphtali and Sandholm 42) and of Holland (6) are generally simpler than that of the other global Newton methods. Also, the Naphtali-Sandhoha and almost hend methods are better suited for nonideal mixtures than other global Newton methods. 

Any of the global Newton methods can be converted to a relaxation form in Ketchum s method by making both the temperatures and the liquid compositions time dependent and by having the time step increase as the solution is approached. The relaxation technique should be applied to difflcult-to-solve systems and the method of Naphtali and Sandholm (42) is best-suited for nonideal mixtures since both the liquid and vapor compositions are included in the independent variables. Drew and Franks (65) presented a Naphtali-Sandholm method for the dynamic simulation of a reactive distillation column but also stated that this method could be used for finding a steady-state solution. 

The time step, At, is used to switch the method from being a relaxation method to a global Newton method. When the time step is small, e.g., if = 0.1, then the changes in the independent variables are small. The method performs like a damped Newton-Raphson method, where the steps are small but in the direction of the solution and without any oscillation. When the value of At is large, i.e., At = 1000, the method performs like a Newton-Raphson method. The value of At at each column trial determines the speed and stability of the method, The units of the time step are the same as the flows to and from the column. The calculation sequence of the Ketchum method is as follows ... 

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 Taylor method. Krishnamurthy and Taylor (88, 89) present and test a nonequilibrium model which includes rate equations for mass transfer, and sometimes reaction, among the traditional MESH equations. These include individual mass and energy balances in the vapor and the liquid and across the interface. An equilibrium equation exists for the interface only. The solution method for these equations is the same as that of the block-banded matrices of the global Newton methods and the style of the method is similar to the Naphtali-Sandholm (Sec. 4.2.9). 

There are (5C + 1) equations per section of packing or per tray. The N 5C +1) equations of a complete column are arranged to have the block-banded form of a global Newton method and can be solved by the same numerical methods. The independent equations for a tray or a section of packing are... 

The independent variables for this global Newton method will be the bulk component vapor and liquid flow rates compositions at the interface for each component less one a mass transfer rate for each component and the temperatures of the bulk vapor, bulk liquid, and the interface ... 

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. 

For some methods, many specifications may not be used simultaneously and there is little freedom in setting specifications. In the 21V Newton or global Newton methods, there can usually be only one specification equation, such as Eqs. (4.133) or (4.134). The new equation will be sensitive to only a few of the independent variables, and if a second equation is added, any attempt to manipulate the Jacobian ma- rix will fail. For this reason, specifying both top and bottom purities often fails when using such methods,... 

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). 

Nonequilibrium methods (Sec. 4,2.13) tend to be global Newton methods extended to solve mass-transfer-inhibited systems. Nonequilibrium methods are not yet completely extended to more common systems, but these methods should see the greatest amount of development in distillation modeling. 

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