Tearing algorithms for separation process simulation

Computers them. Emgng, Vol. 14, No. 8, pp. 90-905, Printed in Great Britain. All rights TEARING 1990 reserved C o pyrig ht ALGORITHMS FOR SEPARATION SIMULATION L. N. Department (Received of Chemical 13 April Engineering, 1989; JbzaI revision a nd SRIDHAR Clarkson received PROCESS A. LUCIA? University, 12 March 0098-1354/90 53.00 + 0.00 1990 Pergamon Press plc 0 Potsdam, 1990; received NY 13699-5705, for publication U.S.A. 27 March 1990) A bstract-A modified sum-rates method based on insights provided by an analysis given by Sridhar and Lucia (Ind. Engng Chem. Res. 28, 793-803, 1989) is presented. Newton’s method is used to accelerate the inner loop of combined mass balance and phase equilibrium equations and to solve the outer loop of energy balance equations. A ll partial derivative information is obtained in analytical form. Several literature examples are used to show that the proposed algorithm is more reliable and more efficient than traditional sum-rates methods and provides the capability of solving problems involving intermediate and narrow boiling mixtures. 1. INTRODUCTION cess model equations easily. The model equations that must be solved are the mass balance, phase equilibrium and energy balance equations, and the unknown variables that are calculated are the temperature T,, and the liquid and vapor component flows 1, and vd, for all stages. The liquid and vapor component flows are computed in an inner loop, in which the temperature and pressure of each stage is held fixed, by solving the mass balance and phase equilibrium equations in a stage-to-stage manner. The temperature profile, on the other hand, is adjusted in an outer loop using the energy balance relationships. The reader is referred to the papers by Sujata (1961) and Burningham and Otto (1967) for the details of traditional sum-rates methods. The remainder of this section is concerned with a presentation of the salient features of a sum-rates method in the context of the analysis given in Sridhar and Lucia (1989). belong to a class of methods known as equation-tearing algorithms, and they are still readily available in commerical process simulators such as ASPEN Plus and Design II for simulating multistage separation processes, for those who prefer to use them. While many modifications have been suggested over the years, guidelines for their applicability have remained steadfast. Bubble point methods are traditionally recommended for narrow boiling mixtures; sum-rates methods are usually applied to wide boiling mixtures. Intermediate boiling mixtures, on the other hand, represent something of a dichotomy (see for example, Friday and Smith, 1964). In a recent manuscript, Sridhar and Lucia (1989) provide a rigorous analysis of multistage separation processes involving homogeneous binary mixtures. The main objectives of this paper are to illustrate that insights from this analysis can be used to develop a modified sum-rates algorithm and to demonstrate that this modified method is an improvement over existing algorithms. Bubble point and sum-rates 2. SUM-RATES methods 2. I. The ALGORITHMS that for relatively high values of Aos> where Aos is the difference between the bubble and dew point temperatures of the primary feed, sum-rates algorithms will solve separation pro____ __ It is widely tTo whom accepted all correspondence should inner X”‘+ 1= G(Xk), (I) where XT = (VT, ~13,. , I~~_)‘. The Jacobian of this fixed-point iteration is given by: be addressed. 0 M2 M,(I G’ loop Our calculational procedure for the inner loop involves decoupling the stages of the process, and traversing the separator by solving the mass balance and phase equilibrium equations for each stage simultaneously. Single-stage perturbation relationships derived by Sridhar and Lucia are then used to reassemble the column model as a fixed-point iteration of the form: - JQf 2) 0.. .o o...o M3 = . M,,~(‘--M,,~-,)...(/-MM,) M ,,,,(I--,,<_,)...(I-MM,) 901 ... Mn.-I M,JI-M,~\ ,) matrix 902 L. N. SR~DHAR and A. LUCIA where IU, = (V’GY and V’G,? energy + V2Gt)-‘V2GF, are the Hessian function associated for with and where matrices the V2Gj of the Gibbs liquid and vapor free where G’ is defined r is given by: r2 It is more convenient to express the fixed-point iteration given by equation (1) in the form: G(X) = 0, t2 r3 r= = X - (3) the corresponding is given Jacobian where the vector - M2 tl t s-M3t 2-M,(I-M* )t , r, . (10) = t, - M4 tJ - M4 (I - M,)t * - M2 - (3) M.0 where (2) phases stage i. F(X) by equation matrix of equation - M3 )(I It , :iI by: F’(X) = 2 - i G’(X). (4) and where While many exist, we choices prefer of approximations to use Newton inner loop, in which the equation: Xk + to F’(X) acceleration the iterates of are computed the tk = using - pm ‘F(p), (5) this, Jacobian G’(X) is calculated using equation (2). terms The temperature adjusted profile in an outer equations and loop results for using the separator the energy - Hi)?. (11) tridiagonal for approximation the For example, outer of can be the con- = 1, _ , rzr, the diagonal are given by: fork of the Jacobian loop matrix z= HZ($)+ zfg) The outer loop 2.2. a matrix structed. where + V’G:)-I(&’ k With ’= p (V*G: is k balance in the iteration: +H:(g)+L@ ) Tk+’ = Th + ATk, (6) where (Q-C?), and where Qz, . . . , QJT Q = (Q,, is Furthermore, a vector of stage heat duty specifications and vector of heat duties that satisfies 0 is a similar the single-stage energy to use equation (7), (12) -HI+,&p_,(* ). the balance equations. Jacobian structed. Jacobian In order matrix, [aQ,/aT,], In traditional sum-rates matrix is approximated matrix using respect to temperature. o&y changes must matrix analytical form inner loop enthalpy with is: as A flowchart in Fig. of 1. The and the mass balance for each the sensitivity temperature is high, wide boiling, such a simplified as in mixtures a poor approximation. In order to obtain a more approximation, the variation of flows accurate liquid and in stage Jacobian are vapor molar G’)-‘r, feasible and (9) is given are decoupled using relation- Newton-like (i.e. the component matrix with solved to (see Sridhar, the inner loop. In particular, the variation in vapor component flows AU with respect to temperature AT is given by: perturbations Au = (/ - remain subproblems flows algorithm methods. Asymmetric trust region methods are used to ensure that the iteration variables for the single- gence is assumed when balance and equilibrium less than 10v5. An analytical expression for respect to temperature. the variation of the vapor profile with respect to temperature was developed by Sridhar and Lucia, and is a direct consequence of Newton acceleration of in the SUM-RATES and phase equilibrium can be to be able to compute the accelerating of the separator methods matrix of obtained method. Newton-like that are not very Jacobian it is essential of to variations of the proposed stages terms easily as a byproduct stage flash calculations In cases where noted, 3. A MODIFIED ships (8) nontraditional are, by Newton’s con- algorithms, the by a tridiagonal in molar That be the Jacobian Convergence of the improve the these, and on 1990 for Aowrates) reliability other, details). of small Conver- the two-norm of the mass equations reaches a value single-stage calculations for all stages results in a new vapor component flowrate profile. Equation (2) is then used to accelerate the inner loop calculations. The single-stage and acceleration calculations are repeated alternately until convergence of the inner loop to a tolerance of lo-’ is obtained. Typically, 3-5 Newton acceleration iterations are required. Equations (6) and (7) are then used to calculate a new temperature profile. The outer loop is assumed to be converged when Tearing algorithms for separation process simulation 903 START specify: aI1 F) 91 J.T~ .PF~ pJ * oi 4 init ie iizs tear va ria ble s T j ,vi i L Solve single -st a ge proble m for st a ge s using N e w t on’s m e t hod all t c I New Vi 4 Ac c e le ra t e T -loop N e w t on’s m e t hod j’s c a lc ula t ions and obt a in a using aTj c NO is T -loop c onve rge d? I Y ES he a t dut y prOfile U SinQ single -st a ge e ne rgy ba la nc e CSlCuk it e Obt a ln ba la nc e new T ] ‘8 by e qua t ions solving using e ne rgy N e w t on’s m e t hod c Y ES ST OP Fig. I. Flowchart for sum-rates algorithm. the two-norm of the energy balance equations falIs below a tolerance of 10’. Initialization of all variables at all levels of computation is done automatically (see Sridhar, 1990) and Newton-like methods are used for all equation-solving tasks. 4. NUMERICAL RESULTS AND DISCUSSION In this section, the numerical performance of the proposed algorithm is compared to that of traditional sum-rates methods on a set of six example problems taken from the literature. The procedures and data given in Prausnitz et nl. (1980) were used to model the physical properties in al1 cases, and all calculations were done on a Gould 9080 computer in doubleprecision arithmetic. 4.1. Numerical results A description of the example problems is given in Table I, and the performance of both algorithms is compared in Table 2. Overall, it was found that the modified algorithm took fewer iterations than the traditional sum-rates algorithm on problems involving wide boiling mixtures. On the other hand, for problems involving intermediate and narrow boiling mixtures, the traditional sum-rates algorithm failed to converge, while the modified sum-rates algorithm converged to the solution. 4.2. Discussion of a sample problem Consider a problem, such as Example 5, involving a mixture which can be classified as intermediate boiling. As seen in Table 2, the traditional sum-rates method failed on this problem while the modified sum-rates algorithm converged in six iterations. This marked difference in numerical performance is due to significant differences in the Jacobian matrix approximations [aQ,/aTj] for the two algorithms. Note that the initial Jacobian matrix for the traditional sum-rates algorithm, in which only the changes 904 L. N. Table Problem No. (Reference) I (Shinohara er al., 2 (Shinohara et al., 1. Problem NO. of stages PEssllre 6 0.5 I75 1972) 1972) descriptions duty specifications (MJ h-‘) Heat (MPa) 6 A. LUCIA and SRIDHAR Feed specifications FIowrates (kmol h-‘) I vapor 1 liquid, stage 1, 305.22 K; stage 6, 288.55 o.oc, 1644.11 0.0 nc, 51.439 O.OC, 166.19 0.0 nc, 23.74 0.0 C, 94.96 533.0 nc,, 0 Q, = 0 0.5175 Qs= --I same K as I Q,=O,j#3 3 (Henley and 465-66 pp. Seader, 6 1981) 4 1.013 5 (Shinohara 6 (Shinohara in enthalpy sidered, is: et al., 1972) with $,,;2.Oli3 ;#1,8 I liquid C,H, stage 2, 50 176.15 Q, = 2.615 C,H, Q, = -1.994 Q2 = 8,371 Q, = 4.935 liquid stage 2, nc, 40 Q* = 4.452 “C,, Q, Q2 Q, Q, Q, Q, Q, Qs liquid stage 4, 348.45 MeOH 40 H,O 60 4.013 et OZ., 1972) I.013 respect to temperature 5316.78 r I liquid, 1 vapor stage 1. 305.22 K; stage OC, 160 OnC,25 oc, 370 0.78 nC, 5 oc, 240 164.17nC,,0 Q, = 0 2.76 = = = = = = = = K K 50 I 343.44 K 30 I -66.69 -2.214 0.265 I .023 -0.34 0.2259 0.566 81.67 K are con-5712.66 0 0 -81.2938 17,313 - 1977.8 0 0 - 11,600.3 13,536.4 - 532.389 0 0 - 12,338.6 13,276.9 L 6. 313.55 (13) On the other hand, the initial Jacobian matrix for the proposed algorithm, in which changes in total flow with respect to temperature are also included, is: - 11,388.57 aQi 29,888.7 aT,= 0 - 18,500 19,656.9 - 3033.64 1 O 0 - 16,124.l 16,961.2 1 [II L It is easily seen that the Jacobian matrix approximation for the traditional sum-rates algorithm differs significantly from the one for the modified sum-rates method. Furthermore, our experience shows that the corresponding temperature step obtained from Table 2. Numerical results Iterations Problem No. I 2 3 4 5 6 0 7289.57 - 2770.3 1 r Modified sum-rates ~___ 5 5 6 12 6 26 Traditional sum-rates 9 9 7 F F F 0 -3532.82 0 (14) equation (7) by the traditional sum-rates algorithm is often a poor one, and that a series of such steps frequently causes the temperatures of some of the stages in the separator to leave the two-phase region, and that this usually results in the failure of traditional sum-rates algorithms. In contrast, the modified sumrates method results in good temperature steps and usually converges in relatively few iterations. 5. CONCLUSIONS Rigorous mathematical analysis was used to develop a modified sum-rates method for simulating separation processes. This modified multistage Tearing algorithms for separation process simulation algorithm was shown to be more reliable for solving problems involving intermediate and narrow boiling mixtures and more efficient for problems involving wide boiling mixtures than traditional sum-rates methods. Newton’s method was used to solve the appropriate model equations at all levels of computation, including the initialization strategy for the single-stage flash calculations, and asymmetric trust region methods were used to guarantee feasible iterates for related bubble point, dew point and flash calculations. No reliabilitv difficulties were exnerienced and, consequently, the proposed sum-rates method performed very well on the example problems tested. Acknon,ledgement-This work was supported by the Office of Basic Energy Sciences, U.S. Department of Energy. under Grant No. DE-FG02-86ERL3552. NOMENCLATURE G i. G” = Total Gibbs free energy of liquid phase, of vapor phase. G’ = Jacobian matrix of fixed-point iteration HL, NV = Molar enthalpy of liquid phase, of vapor phase I?, R” = Partial molar enthalpy of liquid phase, of vapor phase L = Total liquid Rowrate Q = Input heat duty to any stage T = Temperature V = Total vapor flowrate Subscriprs i = Component index .i, k = Stage index 905 Superscripts k = Iteration counter L, V = Liquid, vapor Greek lerter A = Perturbation in any variable. REFERENCES Burningham D. W. and F. D. Otto, Which computer Process. 46, 163-170 design for absorbers. Hydrocarbon (1967). Friday j. R. and B. D. Smith, An analysis of the equilibrium stage separations problem-formulation and convergence. AIChE JI 10, 698-707 (1964). Henley E. J. and J. D. Seader, EquiIibr&-Stage Separation Operarions in Chemical Engineering. Wiley, New York (1981). Prausnitz J. M., T. F. Andersen, E. A. Grens, C. A. Eckert, R. Hsieh and J. P. O’Connell, Computer Calculations .for .Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria. 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