quasi-Newton methods have been used in process design optimization cal - PDF document

quasi-Newton methods have been used in process design optimization calculations some time. They continue and as more recent hybrid method. While Newton-based fixed-point many different chemical process design application has been multicomponent separation problems, espe- stage distillation. Quite aside this, classical thermodynamics existing nonsymmetric formulae were thermodynamically inconsistent Jacobian and thermodynamically consistent formulae was presented chemical equilibrium consistent quasi-Newton formulae, hybrid method, a hybrid algorithm (Lucia generates matrix approximations that involving nonideal thermodynamic de- by Schubert's method, that matrix does either the Gibbs-Duhem equation. flash problems involving ethanol and n-hexane. Furthermore, in that presented two quasi-Newton formulae in a thermodynamically consistent. same set flash problems, demonstrated that consistent quasi-Newton formula can and greater computational efficiency when manuscript, we first extend nonsymmetric quasi- homogeneity equations must one phase, as liquid-liquid and vapor-liquid-liquid equilibria. Lucia cannot a hybrid optimization method for calculating chemical energy minimization. this we fying the zero-degree homogeneity simultaneously. Next, these new thermodynamically consistent quasi-Newton for- can result tional efficiency when Newton-like methods. make some concluding remarks concerning equation can using partition In the convenient to consider presentation clearer. Consider, for assumed to operate at equations that model this equilibrium equations and the molal flow rate and molal feed flow rate. distribution ratio for the component. Finally, component index, total number in the mixture under consideration, and simply distinguish two dense by some Newton-based fixed-point algorithm. Therefore, in order to describe method and many recent we let Moreover, we and further that the Jacobian matrix be written the computed part Jacobian, can readily available the ap- proximated part, contains terms derivative information that can to obtain (e.g., activity coefficient-component From the derivative information model, while contains terms that More specifically, for equations are activity coefficient and 2 point by minimizing system subject straints, instead model defined by for com- temperature and pressure. Thus, in the the equivalent optimization problem that we subject to clear to the reader that to the constrained optimization problem by Newton's by Eqs. with respect computationally expensive. In this case, we suppose that the derivatives (or matrix) can liquid-liquid flash problem, which in this energy function because thermodynamic quantities satisfy Euler's which for an given by mole numbers, we have . . the number phases. Moreover, lationship holds zero-degree homogeneity must satisfy given by equations from our opinion, great deal about the approximated a linear operator. In the nonsymmetric that the approximated Jacobian matrix nontrivial null space (or index pairs that define the approximated part molal flow and where null space (for example, the number . . . 0, otherwise 2;" = 0 0 (11) ... 0 ... (-) It is well thermodynamics that thermodynamic quantities mole numbers, typically zero or one. total Gibbs energy and homogeneous functions one, whereas enthalpy and . . Symmetric Case. Together, the zero-degree homogeneity equations imply that the approximated part symmetric and that it has this component molal AlChE Journal cifically, Eqs. NEWTONLIKE METHODS some collection model equations, choice that we some Newton-based to solve our nonlinear model equations. concerned with iterative procedure represents some approximation to the that the Jacobian matrix this general framework, existing nonsymmetric Newtonlike methods. example, when evaluated analytically calculated analytically or by finite difference, again evaluated analytically but calculated by by then we have_the hybrid method Lucia and Macchietto formed from that it zero everywhere the does and denote the generalized inverse matrix transposition respectively. Observe that Eq. generalized representation Finally, we to represent the pure within this approximated part differences or analytically, null space. In other zero-degree homogeneity equations On the other hand, the Schubert update (i.e., build approximations to such approxima- thermodynamically inconsistent. This variational calculus problem from which derived contains no no constraint to satisfy homogeneity equations. this, Lucia lowing two quasi-Newton brid method for chemical chemical wT + 5 (s~zi)(zTzi)+(wTsi)+(e~Az)ei wT, (20) i=1 i=1 and the tandem equations and Page 1384 August, 1985 where z = Z%=, zm, w, = si - (sTzi/zTzi)zi, and s, and zi are the vectors formed from s and z, respectively, such they have does. Observe formula defined sparsity conditions zero-degree homoge- equations for each iteration In contrast, the update defined by zero-degree homogeneity conditions each iteration limit. However, neither can be applied to which zero-degree required to several phases each other, such as equilibria. Rather, the updating by Lucia apply the zero-degree low-to-moderate pressure vapor-liquid vapor phase can often be Symmetric Case. For symmetric much the same; only with building approximated part nonsymmetric case, be written calculate phase points for nonideal mixtures by Gibbs mated part zero-degree homogeneity and the Gibbs-Duhem equation. the nonsymmetric finite differences, On the any symmetric to build iterative representations approximations do either the zero-degree homogeneity More specifically, approximation to in the function at inconsistent. This again because no constraints contained variational calculus problem derived that force the and the equation to be Furthermore, the farb-Shanno (BFGS) quasi-Newton formula, mean any Gibbs-Duhem equations each iteration Thus, we include given by There are, stated earlier, might want to independent zero-degree homogeneity equations in several phases simultaneously uid-liquid or, for that matter, formula defined by this, note several zero-degree ho- does not necessarily the updating formula defined to the multiple zero-degree homogeneity conditions, we simply variational problem subject to index pairs approximated part Jacobian and where it . . . mutually orthogonal. solution to this problem problem wT j=l where The extended form updating formula defined an iterated (Dennis and and ()()()(33) Observe that the updating formula defined corresponds to only satisfy secant condition limit, unlike one defined by Eq. the secant condition each iteration. however, satisfy zero-degree homogeneity sparsity condi- each iteration. zero-degree homogeneity each iteration convenient to update the Hessian blockwise. updating formula defined by associated diagonal computed part defined by Eq. the appropriate associated with that the update defined by corrections to each diagonal block Hessian. Because each iteration In this section, we present some results for the ther- modynamically consistent quasi-Newton previous section. In particular, we various single-stage, flash problems involving different multicomponent this for both compare the approximated by finite-difference and by existing quasi- formulae. Finally, in all examples liquid phase activity coefficient, while vapor phase the null Nonsymmetrlc Case. example problems feed mixture n-heptane, and and water feed component flow rates for water were kmol/s, respectively. an accuracy by Newton's finite difference two thermodynamically defined by and Schubert's method. presented here, well as others, were initiated 2 9(27) 3 6(24) 15(18) 72(75) each column are: Iterations (Rigorous thermodynamic calculations). liquid and for dew point bubble point calculations equal to for all and the dogleg strategy barring a few exceptions, important to note that tended to zero, which frequently, the quasi-Newton correction either Eq. or Eqs. usually inordinately large. part, we the fact that the the ()(can become small and because this zero component rate can take consistent with zero-degree homogeneity condition. Newton's method does these circumstances. to the numerical results, perhaps the this point our study that nonsymmetric formulae that account for zero-degree homogeneity properties can provide improvements tional efficiency. have observed this in many Sometimes these consistent quasi-Newton not provide They take iterations as Schubert's formula to reach desired accuracy. Rarely, they require more iterations. We believe this to the fact that we have the nonsymmetric case, which natural setting for in- zero-degree homogeneity Gibbs-Duhem equations quasi-Newton approximations, very encouraging. illustrate this, we present nu- results for following isothermal flash problems by minimizing the feed mixture equimolar mixture ethanol, acetone, chloroform, methanol, rates for were each those for methanol, ethanol, and acetone were each using various methods for approxi- Gibbs function and the Hessian. In we approximated Hessian by finite difference, the the thermodynamically by Eqs. each problem, were terminated the Kuhn-Tucker shown in three starting liquid phase 0.55, 0.50, vapor phase equilibrium with liquid feed for examples other starting points. Again, results were qualitatively unchanged. For the indicate that a hybrid proach with the approximated part built by formula can compete favorably with Newton's method and the by Gibbs mization. This clearly superior both quasi-Newton for constrained either the to build function and hybrid approach that secant information to update the approximated part updates to very well, likely because the approximated proach with the thermodynamically consistent quasi-Newton fact that balance constraint manifold. This has calculations. However, it cannot account for all Full Full PSB PSB 16(16) 16(16) except finite-difference. pairs in Approximation columns are: Function gradient calls (Rigorous thermodynamic AND WATER Approximation to Full Full except finite-difference. pairs in Approximation columns are: Function and gradient (Rigorous thermodynamic calculations) F =failure. havior. When updated by quasi-Newton methods information, the numerical performance can deteriorate. the fact that the that the from the zero-degree homogeneity we have argued that phase equilibria it gener- ates matrix approximations satisfy various constraints such as zero-degree homogeneity existing quasi-Newton for- the other hand, do quasi-Newton formulae. nonsymmetric case, we suggested two new quasi-Newton formulae that any number These updates were single-stage, isothermal flash calculations involving different multicomponent consistent quasi-Newton formulae can efficiency when to existing quasi-Newton methods. Unfortunately, results were observed to dynamically consistent updates. the symmetric zero-degree homogeneity and the Duhem equation. Single-stage, isothermal flash problems for several mixtures, this system subject the numerical encouraging and symmetric and thermodynamically consistent quasi-Newton used in optimization algorithm, can compete favorably with formula and results for nonsymmetric case improved, perhaps markedly, by developing nonsymmetric formulae that satisfy both the minimum calculations seem to support this. However, to this, we must split Jacobian matrix dif- partition symmetry. In particular, diagonal matrix form, then for problems involving chemical turns out to this, consider illustrative example. defined by 7(49) 14(14) 23(23) 12(84) 28(28) 18(18) methods, except in Approximation gradient calls thermodynamic calculations). in order Gibbs-Duhem equations each phase, we partition-symmetric ma- note that namically consistent quasi-Newton problems in- carries over optimization since themselves homogeneous. this context approximating, among numbers, it the thermodynamic . . complicated, these constraints as well same form in this term that partially based supported by National Science under Grant Jacobian or computed part Jacobian or unit vector vector function model equations, molal flow the rnth energy, standard state free en- distribution ratio vector orthogonal to the in function or gradient from one contained in constraint mani- molal flow Greek Letters component liquid phase activity coefficient iteration counter ideal solution standard state Nonlinear Simultaneous an Algorithm Sparse Nonlinear Quasi-Newton Methods,” Goldfarb, D., Macchietto, “New Approach to Approximation Involving Physical Derivatives in Cornell Univ., Algorithm for Unconstrained Nonlinear Programming Fast Algorithm for Nonlinearly Constrained Optimization Lecture Notes in Mathematics, and Liquid-Liquid Equilibria, and Symmetric Subject to