Newton-Raphson method equilibrium calculations

As stated, the most commonly used procedure for temperature and composition calculations is the versatile computer program of Gordon and McBride [4], who use the minimization of the Gibbs free energy technique and a descent Newton-Raphson method to solve the equations iteratively. A similar method for solving the equations when equilibrium constants are used is shown in Ref. [7],... 

SC (simultaneous correction) method. The MESH equations are reduced to a set of N(2C +1) nonlinear equations in the mass flow rates of liquid components ltJ and vapor components and the temperatures 2J. The enthalpies and equilibrium constants Kg are determined by the primary variables lijt vtj, and Tf. The nonlinear equations are solved by the Newton-Raphson method. A convergence criterion is made up of deviations from material, equilibrium, and enthalpy balances simultaneously, and corrections for the next iterations are made automatically. The method is applicable to distillation, absorption and stripping in single and multiple columns. The calculation flowsketch is in Figure 13.19. A brief description of the method also will be given. The availability of computer programs in the open literature was cited earlier in this section. 

Box 7. When r3>e3, corrections to the lrp vip and Ts are calculated from the nonlinear MEH equations by the Newton-Raphson method. In these equations the enthalpies and equilibrium constants usually are nonlinear functions of the temperatures. 

Equilibrium compositions of systems of biochemical reactions can be calculated using the following two programs. The first was written by Fred Krambeck (Mobil Research and Development) and the second was written by Krambeck and Alberty. The Newton-Raphson method is used to iterate to the composition with the lowest possible Gibbs energy or transformed Gibbs energy. 

Equilibrium compositions of systems of chemical reactions or systems of enzyme-catalyzed reactions can only be calculated by iterative methods, like the Newton-Raphson method, and so computer programs are required. These computer programs involve matrix operations for going back and forth between conservation matrices and stoichiometric number matrices. A more global view of biochemical equilibria can be obtained by specifying steady-state concentrations of coenzymes. These are referred to as calculations at the third level to distinguish them from the first level (chemical thermodynamic calculations in terms of species) and the second level (biochemical thermodynamic calculations at specified pH in terms of reactants). 

Calculate the equilibrium composition (component mole fractions) of the reactor contents. [Suggestion Express ATi and Ki in terms of the extents of the two reactions, ii and (See Section 4.6d.) Then use an equation-solving program or a trial-and-error procedure, such as the Newton-Raphson method (Appendix A.2), to solve for 1 and fe, and use the results to determine the equilibrium mole fractions.]... 

Numerical Methods and Data Structure. Both EQ3NR and EQ6 make extensive use of a combined method, using a "continued fraction" based "optimizer" algorithm, followed by the Newton-Raphson method, to make equilibrium calculations. The method uses a set of master or "basis" species to reduce the number of iteration variables. Mass action equations for the non-basis species are substituted into mass balance equations, each of which corresponds to a basis species. 

The equations required to describe this column are developed in the order in which they are solved sequentially in the proposed calculational procedure. On the basis of assumed temperature and L/V profiles, the material balances, the physical equilibrium relationships, and the chemical rate expressions (or chemical equilibrium expressions) are solved for the moles of each component which reacts per stage per unit time and for the component-flow rates. A formulation of the Newton-Raphson method is used. 

The central portion of the algorithm in Figure 11.6 exactly parallels the standard Rachford-Rice procedure. First, we use (11.1.27)-(11.1.29) to compute the mole fractions for all phases, then we compute all fugacity coefficients and all activity coefficients. With those quantities we can obtain new estimates for the Cs and Ks from the phase-equilibrium relations (11.1.15) and (11.1.24). Now we use (11.1.31) and (11.1.32) to calculate values for the Rachford-Rice functions, Fj and F2, and test for convergence. If our convergence criteria are not met at iteration k, then we use the Newton-Raphson method to estimate the unknown L and V at the next iteration (fc + 1). 

A theoretical model for the calculation of the number of theoretical plates using the Newton-Raphson method is presented by Kaibel et al. (31). However, it does not incorporate a constraint on T so that temperature becomes an independent variable. Such an assumption is obviously highly questionable. Nevertheless, this difficulty can be overcome by incorporating such a constraint into the equations. The problem of different plate efficiencies for concentration and reaction equilibrium is, however, considerably more difficult to handle. It would appear that the best approach will be to abandon completely the concept of theoretical plates and efficiencies and develop instead a plate-to-plate calculation method based on real plates. Here the extension of the differential equations for packed columns into difference equations and their subsequent modification to apply to each individual plate offers the best chance of success. 

With these definitions of the system activity coefficients and the equilibrium constants calculated using the tabulated thermodynamic data and equations (3.31), the ten equations, (C.l) - (C.3) and (C.6) - (C.12), may be solved for the ten unknowns with the Newton-Raphson method. 

With these expressions for the activity coefficients, the equilibrium constants and Henry s constant calculated with the fit equations (S.4) and (S.7)> and the fugacity coefficients calculated using Nakamura s method as outlined in Appendix 9.3, the nine unknown concentrations may be determined for set input amounts of HjO and SO2 with the Newton-Raphson method using equations (S.l). (S.2), (S.3), (S.5), (S.6), (S.8), (S.9), (S.IO) and (S.ll) as the system model. An example of such a calculation is shown in Figure 9.10. 

Applying the Newton-Raphson method in a nonlinear finite element system will yield results only in the pre-collapse range, but it will fail to give information about the post-collapse response. To circumvent this limitation, a constraint can be added into the finite element system, which relates the load increment and the incremental displacements within each iteration (Fig. 12). This technique allows the calculation of the whole equilibrium path, even beyond the critical limit points. A number of different solution algorithms have been proposed in the literature (Riks 1979 Crisfield 1981 Ramm 1981 Bathe and Dvorkin 1983). 

The study described above for the water-gas shift reaction employed computational methods that could be used for other synthesis gas operations. The critical point calculation procedure of Heidemann and Khalil (14) proved to be adaptable to the mixtures involved. In the case of one reaction, it was possible to find conditions under which a critical mixture was at chemical reaction equilibrium by using a one dimensional Newton-Raphson procedures along the critical line defined by varying reaction extents. In the case of more than one independent chemical reaction, a Newton-Raphson procedure in the several reaction extents would be a candidate as an approach to satisfying the several equilibrium constant equations, (25). 

The NONLIN module is responsible for intializing the concentration vector, C(t), for l i NRCT. Here NRCT is the number of reactants. If there are no equilibrium reactions, then C i) is set to IC i), the initial concentration vector, for 1 < f < NRCT. If equilibrium reactions do exist, then the type (2) equations (with derivatives set to zero) and the Type (1) and Type (3) equations are all solved simultaneously for the equilibrium concentrations of all reactants. Because the equilibrium equations are generally nonlinear, the Newton-Raphson iteration method is used to solve these equations. Also, since there is no symbol manipulation capability in the current version of CRAMS, numerical differentiation is used to calculate the required partial derivatives. That is, the rate expressions cannot at this time be automatically differentiated by analytical methods. A three point differentiation formula is used 27) ... 

C.M. Bethke (52) has shown that significant numerical advantages in such calculations can be realized by switching into the basis set a mineral species that is in partial equilibrium with the aqueous phase. This avoids expansion of the size of the Jacobian matrix and reduces computation time. A method based on this concept is being developed for use in the 3270 version of EQ3/6. The concept appears to show promise for improvement of the "optimizer" algorithm as well as the Newton-Raphson one. 

The vapor-liquid equilibrium is assumed ideal. Column pressure P is optimized for each case. With pressure P and tray hquid compositions x j known at each point in time on each tray, the temperature T and the vapor compositions y j can be calculated. This is a bubblepoint calculation and can be solved by a Newton-Raphson iterative convergence method. 

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