N

flow, calculations are readily made of remaining values of Vj and Lj, from which initial values of vi,j and (¡j are obtained from Eqs. (13-89) and (13-90) after obtaining approximations of xij and «¡,j from steps 4, 5, 8, 9, and 10 of the SR method. Alternatively, a much cruder but often sufficient estimate of Xy and yij is obtained by flashing the combined column feeds at average column pressure and a vapor-to-liquid ratio that approximates the ratio of overhead plus vapor-sidestream flows to bottoms plus liquid-sidestream flows. Resulting compositions are used as the initial estimate for every stage.

At the conclusion of each iteration, convergence is checked by employing an approximate criterion such as

where E, is a scale factor that is of the order of the average molal heat of vaporization. If we take e = N(2C +

converged values of the unknowns will generally be accurate, on the average, to from four or more significant digits.

During early iterations, particularly when initial estimates of the unknowns are poor, T and corrections to the unknowns will be very large. It is then preferred to utilize a small value of t in Eq. (13-101) so as to dampen changes to unknowns and prevent wild oscillations. However, the use of values of t much less than 0.25 may slow or prevent convergence.

It is also best to reset to zero or small values any negative values of component flow rates before initiating the next iteration. When the neighborhood of the solution is reached, T will often decrease by one or more orders of magnitude at each iteration, and it is best to set t = 1. Because the Newton method is quadratically convergent in the neighborhood of the solution, usually only three or four additional iterations will be required to reach the convergence criterion. Prior to that, it is not uncommon for T to increase somewhat from one iteration to the next. If the Jacobian tends toward a singular condition, it may be necessary to restart the procedure with different initial guesses or adjust the Jacobian in some manner.

Standard specifications for the Naphtali-Sandholm method are Qj (including zero values) at each stage at which heat transfer occurs and sidestream flow ratio s, or Sj (including zero values) at each stage at which a sidestream is withdrawn. However, the desirable block tridi-agonal 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. 13-7f it is desiredto specify a boil-up ratio rather than reboiler duty. Equation (13-95) for function HN is removed from the N(2C + 1) set of equations and is replaced by the equation

where the value of (VN/LN) is specified. Following convergence of the calculations, QN is computed from the removed equation.

All of the major computer-aided design and simulation programs have a simultaneous-correction algorithm. A Naphtali-Sandholm type of program, particularly suited for applications to distillation, extractive distillation, and azeotropic distillation, has been published by Fre-denslund, Gmehling, and Rasmussen (Vapor-Liquid Equilibria Using UNIFAC, a Group Contribution Method, Elsevier, Amsterdam, 1977). Christiansen, Michelsen, and Fredenslund [Comput. Chem. Eng., 3, 535 (1979)] apply a modified Naphtali-Sandholm type of method to the distillation of natural-gas liquids, even near the critical region, using thermodynamic properties computed from the Soave-Redlich-Kwong equation of state. Block and Hegner [Am. Inst. Chem. Eng. J., 22, 582 (1976)] extended the Naphtali-Sandholm method to staged separators involving two liquid phases (liquid-liquid extraction) and three coexisting phases (three-phase distillation).

Example 6: Calculation of Naphtali-Sandholm SC Method

Use the Naphtali-Sandholm SC method to compute stage temperatures and interstage vapor and liquid flow rates and compositions for the reboiled-stripper specifications shown in Fig. 13-53. The specified bottoms rate is equivalent to removing most of the nC5 and nC6 and some of the nC7 in the bottoms.

Stripper Reboiled Absorber
FIG. 13-53 Specifications for the calculation of a reboiled stripper by the Naphtali-Sandholm method.
Calculations were made with the Grayson-Streed modification of the Chao-Seader method for K values and the Lee-Kesler method for enthalpy departures. Initial estimates for stage temperatures and flow rates were as follows, where numbers in parentheses are consistent with specifications:

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