## Dt

Vickery and Taylor (81) used a Naphtali-Sandholm method containing all of the MESH equations and variables [M2C + 3) equations] with the variables represented by x. H^. is the Jacobian from the Naphtali-Sandholm method solution of the known problem, G(x) = 0. This is numerically integrated from t = 0 to t - 1, finding a H, at each step and updating Hx when the solution is reached at each step. With Hx and H, known, dxjdt is solved, and with step size At, a new set of values for the independent variables x is found by Euler's rule x„ +! = x„ + At (dx/dt)n (4.116)

At t = 1, the values of x should yield the difficult solution F(x) = 0. This technique resembles the relaxation method, but only requires modifying the independent MESH functions to get the derivatives with respect to one term t. This is a purely mathematical approach and Ellis et al. (78) state that it can give negative flow rates at intermediate values of t, something that Ar-value and enthalpy routines may not tolerate. An alternative is a homotopy function that is rooted in the MESH equations themselves. The homotopy procedure is as follows:

1. Set initial temperatures and total vapor and liquid rates for each stage. Solve the column for the known solution.

2. The results from step 1 form the initial values x of the global Newton's method. Set t = 0.

3. Solve H(x,f) = 0 using the global Newton's method (Sec. 4.2.9). The /f-values and enthalpies used are determined by the current value of t.

5. Use Euler's rule to get a new set of variables x, and return to step 3.

The efficiency homotopy. Taylor, Wayburn, and Vickery (80) propose a physical homotopy based on a pseudo-Murphree efficiency. As stated earlier, the efficiency can be entered into the MESH by modifying the equilibrium equation [see Eq. (4.23)]. The pseudo-efficiency homotopy takes the column from little separation, such as i?MVy = 0.1, with little difference in composition between the top and bottom of the column, to maximum pseudo-efficiency, where E^¡j = 1.0, at the final solution. The pseudo-efficiency, EMV{i, represents the homotopy parameter t applied in the Naphtali-Sandholm method. The calculation sequence is as follows:

1. Initialize the MESH variables of the column with E^^ = 0.0. Taylor et al. (80) present suggestions for the initialization. Since Emvy ~ 0.0, all stages have the same temperature and constant molal overflow. Taylor et al. (80) also use a four-diagonal matrix method to solve for the initial-stage liquid compositions. These are normalized and used to find a better set of stage temperatures by solving for the bubble points. With E^y = 0.1, the compositions and temperatures are recalculated until a satisfactory initial temperature profile is reached.

2. With Emvy = 0.1, solve the Naphtali-Sandholm method for the column. The solution criteria for intermediate values of Ewvij (0-1,

0.4, and 0.7) should be looser than for the final value of Ewvij = 10.

1.e., a norm of the functions less than 10"1 instead of 10~4.

3. When the solution is reached at an intermediate value of Emv^ (0.1, 0.4, and 0.7), increase i!^ to the next value and restart the problem. The solution results from one value of E^a are used to initialize the next. When Ey^y = 1A the criteria of solution should be much more strict than that of other values of i?MVy •

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 ^MVy 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 Jf-values where acts as a damper on the it-values. The efficiency homotopy does not work for purity specifications because the purity will not be satisfied in solutions of early values of E^y-

Vickery and Taylor (81) presented a thermodynamic homotopy where ideal K-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:

where composition-independent if-values can be used. The thermodynamic homotopy is easier to implement and faster than those based on mathematics alone. This should be true for other physically based homotopies such as the above efficiency-based homotopy.

You may have at times come up with homotopy-like techniques for reaching a solution; what some engineers call "sneaking up on the answer." For instance, water often causes nonideal behavior and can make the solution difficult. One can first solve the column with water absent from the feed, then slowly increase the amount of water, solving the column at each increase, with all of the profiles from one solution used to initialize the next. Methods like these are described by Brierley and Smith (106).

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.

ASPENPlus of AspenTech, Cambridge, Massachusetts; PROCHEM of OLI Systems, Florham Park, New Jersey; and other programs use homotopies in their solution methods. The HOMDIS program, available from Dr. Warren Seider of the University of Pennsylvania, uses a homotopy in the solution of azeotropic and three-phase distillation columns.

### 4.2.13 Nonequilibrium or rate-based methods

Chapters 7 and 9 present that stage efficiency prediction and scaleup can be difficult and unreliable. Section 4.1.2 points out that the computational form in which stage efficiencies are often applied, as multipliers to the equilibrium if-values, may inadequately reflect actual equilibrium or column operation. For highly nonideal, polar, and reactive systems, such as amine absorbers and strippers, prediction and use of efficiencies is particularly difficult. In such mixtures, it is the mass transfer and not the equilibrium that often limits the separation.

Nonequilibrium methods attempt to get around the difficulty of predicting efficiencies by doing away with the equilibrium-stage concept. Instead, they apply a transport phenomena approach for predicting mass transfer rates. The mass transfer rates are calculated continuously along the column length and not in discrete equilibrium stages. This process is similar to the transfer unit concept (Sec. 10.3.1).

While the bulk vapor and liquid phases are not at equilibrium with each other, there is equilibrium at the interface between phases with a movement from the bulk phase through the interface (Fig. 4.3). The net loss or gain for a component at the interface is expressed in a rate form (hence the alternate name, rate-based methods):

Nf' = Nfj daj for the net loss by the liquid (4.121)

where N-f and Njj are vapor and liquid molar fluxes of the component at some point through the interface and da} is the small interface area through which the flux passes. There is also an energy rate equation between the two phases:

Ej" = ej da} for the net energy gain by the vapor (4.122 >

Ef ~ ef daj for the net energy loss by the liquid (4.1231

where ej and ef are energy fluxes through interfacial area dar

In nonequilibrium models, as in the other models, the subscript j is for the stage. In a trayed column, it is the actual tray. In a packed column, j is a section of packing. By convention, transfer is to be from the liquid to the vapor with the mass transfer rate to the vapor, taken as positive.

The movement of mass and energy from one phase through the interface to the other phase is illustrated in Fig. 4.3. One mass transfer rate is shown to the vapor from the liquid and one rate from the liquid to the vapor, but there is strictly only one independent rate, N]j, where JV* = N)" = Ny'. The mass transfer rates are dependent on the mass transfer coefficients for each phase. The coefficients are dependent on the composition in the bulk phase and at the interface; the temperature in the bulk phase and at the interface; the area of the interface. ay, and the mass transfer rates and coefficients of the other components.

The total energy transfer rates for each phase are dependent on heat transfer coefficients and the mass transfer rates of each compo-

Bulk vapor TVi

Interface

Vapor lo liquid transfer

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