Newton—Raphson method composition

Simultaneous solution by the Newton-Raphson method yields x = 0.9121, y = 0.6328. Accordingly, the fractional compositions are ... 

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],... 

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. 

He showed that positive dX give d that reduce Gibbs free energy. This method is analogous to that of steepest descent, a first-order method for minimization of Gibbs free energy. Ma and Ship-man (11) used Naphtali s method to estimate compositions at equilibrium and the Newton-Raphson method to achieve convergence. 

The MESH equations can be regarded as a large system of interrelated, nonlinear algebraic equations. The mathematical method used to solve these equations as a group is the Newton-Raphson method. The solution gives the steady-state values of the column variables temperatures, flow rates, compositions, etc. A particular rigorous method may not make use of all of the MESH equations in the Newton-Raphson portion of the method. Instead, it may solve the remaining MESH equations by some other means. The methods in Secs. 

For every side product, a withdrawal factor is added to the independent variables. With it, a specification function joins the energy balances in the independent functions. These and the energy balances are the functions of a Newton-Raphson method, and whenever they are calculated (as in filling the Jacobian), a pass must be made through the calculations of the compositions and temperatures. 

Compositions or flows calculated by component material balances in the theta method and 2N Newton-Raphson method. Sees. 4.2.5 and 4.2.6. 

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.]... 

The problems of interest are finding the conditions for onset of vaporization, the bubble-point for the onset of condensation, the dewpoint and the compositions and the relative amounts of vapor and liquid phases at equilibrium under specified conditions of temperature and pressure or enthalpy and pressure. The first cases examined will take the A, to be independent of composition. These problems usually must be solved by iteration, for which the Newton-Raphson method is suitable. The dependence of K on temperature may be represented adequately by... 

Equation (7-54) allows calculation of the residence time required to achieve a given conversion or effluent composition. In the case of a network of reactions, knowing the reaction rates as a function of volumetric concentrations allows solution of the set of often nonlinear algebraic material balance equations using an implicit solver such as the multi variable Newton-Raphson method to determine the CSTR effluent concentration as a function of the residence time. As for batch reactors, for a single reaction all compositions can be expressed in terms of a component conversion or volumetric concentration, and Eq. (7-54) then becomes a single nonlinear algebraic equation solved by the Newton-Raphson method (for more details on this method see the relevant section this handbook). 

A number of formulations of the Newton-Raphson method for the solution of distillation problems has been proposed. A brief description of each of several of these formulations wherein the number of independent variables ranges from N to 2N is given below. (Other formulations of the Newton-Raphson method in which compositions or component-flow rates are among the independent variables are described in Chap. 5.)... 

Another formulation of the Newton-Raphson method was proposed by Newman20 in 1963 in which the total-flow rates of the liquid Lj were taken as the independent variables and the corresponding sets of temperatures needed to satisfy the component-material balances and equilibrium relationships was found by successive application of the Newton-Raphson equations. The compositions and temperatures so obtained were used to solve the enthalpy balances explicitly for a new set of liquid rates. The procedure was then repeated by commencing with this most recent set of liquid rates Lj. ... 

The proposed calculational procedure consists of first making one trial calculation on the reboiled absorber by use of the 2N Newton-Raphson method and then one trial on the distillation column by use of the 0 method. Then the capital 0 method is applied one time to the system in order to place it in overall component-material balance and in agreement with the specified values of the terminal flow rates. To initiate the calculational procedure the composition of any recycle stream which is needed is assumed. After the first trial through the system, the composition of such recycle streams found by the method are used. The steps of the proposed calculational procedure follow. 

It is desired to determine Vv Lx, and their compositions by use of the Newton-Raphson method. Evaluate the functions Fx and Gx and the partial derivatives 3FxfdOx and dGx/dOx. These are two of the four partial derivatives which are needed to make the first trial and calculation by use of the Newton-Raphson method. For the first trial take 0X = 1 and T, = 105°F. 

Highly nonideal solutions are characterized by the fact that the activity coefficients and the partial molar enthalpies are strongly dependent upon composition. In order to compute the partial derivatives of these quantities which are needed in the application of the Newton-Raphson method, it is convenient to choose compositions or component-flow rates as members of the set of independent variables. Numerous choices of the independent variables have been made.6, lf 8 13,15 17 19-20 To demonstrate the formulation of the Newton-Raphson method, the choice of independent variables proposed by Naphtali and Sandholm17 is used. The Almost Band Algorithm may be formulated for other choices of independent variables as shown by Gallun and Holland.7,8 9... 

The calculational procedure is as follows. On the basis of an assumed set of compositions for the stream B2, one trial calculation is made on column 1 by use of the 2N + l formulation of the Newton-Raphson method. Then the equations for the heat exchanger are solved for Thi. 0 and TN2f 0 on the basis of the most recently calculated values for TNl and TN2. For the first trial through the system, an assumed value for TN2 is used in Eq. (7-27) in the calculations for the heat exchanger. Then the value so obtained for TN2t 0 is used in making one trial on column 2 by use of the 6 method. Next, the capital 0 method is applied. 

Where K-values are nonlinear in pressure and temperature and are composition dependent, algorithms such as those in Figs. 7.6 and 7.7 can be employed. For solving (7-19) and (7-21), the Newton-Raphson method is convenient if X-values can be expressed analytically in terms of temperature or pressure. Otherwise, the method of false position can be used. Unfortunately, neither method is guaranteed to converge to the correct solution. A more reliable but tedious numerical method, especially for bubble-point temperature calculations involving strongly nonideal liquid solutions, is Muller s method. ... 

One method would be to use Eq. (3.62) and utilise a Newton-Raphson technique to perform a Gibbs energy minimisation with respect to the composition of either A or B. This has an advantage in that only the integral function need be calculated and it is therefore mathematically simpler. The other is to minimise the difference in potential of A and B in the two phases using the relationships... 

In the SR method, temperatures are the dominant variables and are found by a Newton-Raphson solution of the stage energy balances. Compositions do not have as great an influence in calculating the temperatures as do heat effects or latent heats of vaporization. The component flow rates are found by the tridiagonal matrix method. These are summed to get the total rates, hence the name sum rates. 

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