Method of Naphtali and Sandholm

This method was aimed at overcoming some of the weaknesses of other methods. 

While certain methods may be better suited for wide-boiling or narrow-boiling mix- 

The simultaneous solution uses the Newton-Raphson method, which is based on linearizing the model equations. Two characteristics are inherent in this method. Since the equations are highly nonlinear, the success of linearization usually requires good starting values. On the other hand, as the solution is approached, the linearized equations become progressively more accurate and convergence is accelerated. 

The Newton-Raphson simultaneous solution procedure is formulated in a generalized terminology by representing the model equations by a function vector g, and all the variables by a variable vector w. The system of equations is written as 

Naphtali and Sandholm (1971) grouped the equations by stages, writing Equation 13.46 in its expanded form as follows  

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The linearisation method of Naphtali and Sandholm has been used by Fredenslund et al. (1977) for the multicomponent distillation program given in their book. Included in then-book, and coupled to the distillation program, are methods for estimation of the liquid-vapour relationships (activity coefficients) using the UNIFAC method (see Chapter 8, Section 16.3). This makes the program particularly useful for the design of columns for... 

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. 

Solution. A digital computer program for the method of Naphtali and Sandholm as used. The X-values and enthalpies were assumed independent of composition and ere computed by linear interpolation between tabular values given at 100°F increments rom 0°F to 400°F (-17.8 to 204.4°C). The tabular values were computed from the quations given in Example 12.8, except for the following values at 400°F. 

With analytical derivatives, the TAYLOR method should handle all problems. If not, composition derivatives will be added to the matrix solution in the manner of Naphtali and Sandholm (16). 

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. 

The SC procedure of Naphtali and Sandholm is developed in detail because it utilizes many of the mathematical techniques presented in Section 15.3 on tearing methods. A computer program for their method is given by Fredenslund, Gmehling, and Rasmussen. ... 

King, 1971 Naphtali and Sandholm, 1971 Newman, 1963 and Tomich, 1970). Moreover the choice of appropriate computation procedures for distillation, absorption, and extraction is highly dependent on the system being separated, the conditions of separation, and the specifications to be satisfied (Friday and Smith, 1964 Seppala and Luus, 1972). The thermodynamic methods presented in Chapters 3, 4, and 5, particularly when combined to... 

The class of simultaneous solution methods in which all of the model equations are solved simultaneously using Newton s method (or a modification thereof) is one class of methods for solving the MESH equations that allow the user to incorporate efficiencies that differ from unity. Simultaneous solution methods have long been used for solving equilibrium stage simulation problems (see, e.g., Whitehouse, 1964 Stainthorp and Whitehouse, 1967 Naphtali, 1965 Goldstein and Stanfield, 1970 Naphtali and Sandholm, 1971). Simultaneous solution methods are discussed at length in the textbook by Henley and Seader (1981) and by Seader (1986). 

One of the more robust methods for solution of multiconponent distillation and absorption problems was developed in a classic paper by Naphtali and Sandholm 119711. This method is available in many commercial simulators. Naphtali and Sandholm developed a linearized Newtonian method to solve all the equations for multicomponent distillation simultaneously. 

Christiansen et al. (54) applied the Naphtali-Sandholm method to natural gas mixtures. They replaced the equilibrium relationships and component vapor rates with the bubble-point equation and total liquid rate to get practically half the number of functions and variables [to iV(C + 2)]. By exclusively using the Soave-Redlich-Kwong equation of state, they were able to use analytical derivatives of revalues and enthalpies with respect to composition and temperature. To improve stability in the calculation, they limited the changes in the independent variables between trials to where each change did not exceed a preset maximum. There is a Naphtali-Sandholm method in the FraChem program of OLI Systems, Florham Park, New Jersey CHEMCAD of Coade Inc, of Houston, Texas PRO/II of Simulation Sciences of Fullerton, California and Distil-R of TECS Software, Houston, Texas. Variations of the Naphtali-Sandholm method are used in other methods such as the homotopy methods (Sec. 4,2.12) and the nonequilibrium methods (Sec. 4.2.13). 

Standard specifications for the Naphtali-Sandholm method are Q-(including zero values) at each stage at which heat transfer occurs and sidestream flow ratio Sj or Sj (including zero values) at each stage at which a sidestream is withdrawn. However, the desirable block tridiagonal structure of the Jacobian matrix can still be preserved when substitute specifications are made if they are associated with the same stage or an adjacent stage. For example, suppose that for a reboiled absorber, as in Fig. 13- it is desired to specify a boil-up ratio rather than reboiler duty. Equation (13-95) for function is removed from the N(2C + 1) set of equations and is replaced by the equation... 

Use the Naphtali-Sandholm SC method to compute stage temperatures and interstage vapor and liquid flow rates and compositions for the rehoiled-stripper specifications shown in Fig. 13-53. The specified hottoms rate is equivalent to removing most of the nCs and nCe and some of the nC in the hottoms. 

The Naphtali-Sandholm (42) method. This method chooses the stage temperatures and component vapor and liquid rates from the MESH variables as the independent variables of the Newton-Raphson calcu-... 

The functions and variables are salved together using a large -Jacobian of size N(2C + 1) x N(2C + 1). When originally presented, The Naphtali-Sandholm method used derivatives of IT-values and enthalpies with respect to composition and temperature, but it was not stated whether these are analytical or numerical derivatives. 

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 H when the solution is reached at each step, With Hj. and H, known, dxjdt is solved, and with step size t, a new set of values for the independent variables x is found by Euler s rule... 

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). 

Naphtali-Sandholm SC Method This method employs the equilibrium-stage model of Figs. 13-48 and 13-49 but reduces the number of variables by 2N so that only N(2C + 1) equations in a like number of unknowns must be solved. In place of W Lj, Xij, and yt j, component flow rates are used according to their definitions ... 

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