New method for designing distillation columns of multicomponent mixtures

This paper presents a series of calculation procedures for computer design of ternary distillation columns overcoming the iterative equilibrium calculations necessary in these kind of problems and, thus, reducing the calculation time. The proposed procedures include interpolation and intersection methods to solve the equilibrium equations and the mass and energy balances. The calculation programs proposed also include the possibility of rigorous solution of mass and energy balances and equilibrium relations. 1.Introduction The initial research covering the theoretical aspects of distillation date from 1893 (Sorel, 1893). The remarkable feature in the development of these calculations was the publication of the first graphical methods (Ponchon, 1921 and McCabe and Thiele, 1925). Later, in the 50's, the development of computers allowed the appearance of several convergence procedures based on the Newton-Raphson method (i.e., Nielsen, 1957; Greenstad et al., 1958). The constant development of computers has allowed more exact solutions for both, steady and non-steady state problems (i.e., Gallun and Holland, 1980 and 1982). The time required for applying such methods is usually long, and various procedures have been developed in order to accelerate the calculations (i.e., Broyden, 1965; Vickery and Taylor, 1986). The methods developed to simulate the operation of distillation columns are aimed at finding the separation which can be obtained from a given feed, operation conditions and column configuration (i.e., for a given number of trays and specified location of the feeds and side streams). On the other hand, the design methods focus on the determination of the best column configuration for a specified separation. Both problems require different approaches to find the solution. Simulation problems can be solved by any kind of approximated or rigorous method whereas the design problems involve more dificulties, such as finding approximate solutions from the minimum number of trays and minimum reflux (Underwood, 1932) or empirical correlations between tray number and reflux (Erbar and Maddox, 1961). Rigorous tray to tray methods often show convergence problems whereas the sucessive approximation methods are not directly applicable (Ricker and Greens, 1974). NEW METHOD FOR DESIGNING DISTILLATION COLUMNS OF MULTICOMPONENT MIXTURES A. Marcilla, A. Gomez & J.A. Reyes-Labarta Latin American Applied Research an International Journal of Chemical Engineering. 1997, vol. 27, 1-2, 51-60. ii In modern distillation practice, rigorous methods are the primary design and optimization tools and the role of shortcut calculations is restricted to eliminating the least desirable design options. In very common situations, the product composition requirements are fixed and the optimum design is that which minimizes the cost of achieving this requirements. Kister (1992) presented a design procedure that optimizes the product purity versus the distillation cost, where the ideal feed point can be determined by graphical, shortcut or rigorous techniques. Small variations are introduced to the feed point, and their effect on the top and bottom compositions is estimated by a rigorous or shortcut method. The results are substitued into the objective function, and the next test begins with the feed entering a stage for which the objective function is lower. The rigorous methods generally involve the following steps: 1. Specification of variables 2. Initialization of MESH variables 3. Actual calculation 4. Test of solution 5. Exit and engineering evaluation The existing calculation methods differ in their selection of independent and dependent variables and in the procedure used to solve the MESH equations (usually tridiagonal matrix or Newton-Raphson methods). In this work, different tray to tray calculation methods are suggested for distillation columns, that combine original approximate and rigorous procedures to solve the mass and energy balances and the equilibrium relations, avoiding the iterations necessary for equilibrium calculations and optimising the feed tray location. The calculation procedure suggested is based on a extension of the graphical method of Ponchon-Savarit for complex columns and for multicomponent mixtures proposed by Marcilla et al. (1995). The methods proposed in this work are design methods in the sense that knowing the product compositions and reflux calculate the number of stages and the optimum feed stage location. The proposed procedures have been applied to two ternary mixtures, both ideal and non-ideal respectively, and the results obtained have been compared with those provided by a conventional rigorous method, showing a very good agreement. The effect of the different calculation and interpolation methods suggested has been analysed. 2. Calculation procedure An extension of The PonchonSavarit method to solve the problem of the separation of a binary mixture in a complex column was proposed by Marcilla et al. (1995). The extension of this procedure to multicomponent mixtures can be easily made if explicit functions for the saturated vapor enthalpy, saturated liquid enthalpy and saturated liquid composition of the type H=H( yi,T,P), h=h(xi ,T,P) and xi=x( yi, T,P) are available. In the other case, approximate methods are necessary. In this work, we have generated these enthalpy functions from fittings of the NEW METHOD FOR DESIGNING DISTILLATION COLUMNS OF MULTICOMPONENT MIXTURES A. Marcilla, A. Gomez & J.A. Reyes-Labarta Latin American Applied Research an International Journal of Chemical Engineering. 1997, vol. 27, 1-2, 51-60. iii equilibrium data in all the composition range to polynomial functions. These functions have been combined with different suggested interpolation methods in order to obtain the equilibrium compositions at each stage. Figure 1 shows a sketch of the location of the net flow points (i) and the graphical solution of a simple column for a ternary system, as in the clasical Ponchon-Savarit method for binary mixtures. Obviously, systems with more than three components (and ternary systems with great difficulties) cannot be graphically represented and the problem must be solved by the use of computational methods. Nevertheless, these methods are still based on the same geometrical concepts as those used for the binaries in combination with the algorithms that involve the corresponding equations for the equilibrium conditions and mass and enthalpy balances. Equations (1) and (2) show the composition (k) and enthalpy (Mk), respectively, of the net flow (k) in the section k of the column of Figure 2, in the Ponchon Savarit diagram: k j k p k j D j k A k j