Nonlinear equations vapor-liquid equilibria

In the book, Vapor-Liquid Equilibrium Data Collection, Gmehling and colleagues (1981), nonlinear regression has been applied to develop several different vapor-liquid equilibria relations suitable for correlating numerous data systems. As an example, p versus xx data for the system water (1) and 1,4 dioxane (2) at 20.00°C are listed in Table El2.3. The Antoine equation coefficients for each component are also shown in Table E12.3. A12 and A21 were calculated by Gmehling and colleaques using the Nelder-Mead simplex method (see Section 6.1.4) to be 2.0656 and 1.6993, respectively. The vapor phase mole fractions, total pressure, and the deviation between predicted and experimental values of the total p... 

Chapters 2-5 deal with chemical engineering problems that are expressed as algebraic equations - usually sets of nonlinear equations, perhaps thousands of them to be solved together. In Chapter 2 you can study equations of state that are more complicated than the perfect gas law. This is especially important because the equation of state provides the thermodynamic basis for not only volume, but also fugacity (phase equilibrium) and enthalpy (departure from ideal gas enthalpy). Chapter 3 covers vapor-liquid equilibrium, and Chapter 4 covers chemical reaction equilibrium. All these topics are combined in simple process simulation in Chapter 5. This means that you must solve many equations together. These four chapters make extensive use of programming languages in Excel and MATE AB. 

Consider one mole per hour of a stream consisting of n volatile liquids with known compositions, and Xf i to be continuously separated into vapor and liquid streams at a given temperature and pressure (Reklaitis, 1983). It is desired to determine the steady-state flow rates of the vapor stream and of the liquid stream and their compositions. Let /f be the vapor-liquid equilibrium constants, Ki = yJXi, where X and are the liquid and vapor fractions, respectively. Ki is calculated from Raoult s law, A" = Pi(T)IP, where is the vapor pressure obtained from the Antoine equation. The flow rate of vapor stream, V, is obtained by solving the following nonlinear equation resulting out of the material balance on each of the species ... 

The vapor-liquid equilibrium temperature for specified pressure and liquid composition is found as the solution to Eqs. 10.1-2 or, if the system is ideal, as the solution to Eq..10.1-4. However, since the temperature appears only implicitly in these equations through the species vapor pressures, and since there is a nonlinear relationship between the vapor pressure and temperature (cf. the Clausius-Clapeyron equation, Eq. 

Simultaneously, conditions of thermal (equality of temperatures), mechanical (equality of pressures) and phase equilibrium (equality of chemical potentials of both components in both phases) must be satisfied. The constraints are defined by a set of equations nonlinear both in model parameters and in fliermodynamic quantities. When correlating vapor-liquid equilibrium data at low and moderate pressures, liquid phase is described in terms of a G -model and vapor phase is either considered as an ideal gas or its nonideality is expressed by an equation of state, e. g. the virial expansion limited to the second virial coefficient (Chap. 1.6). 

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. 

SC (simultaneous correction) method. The MESH equations are reduced to a set of A(2C + 1) nonlinear equations in the mass flow rates of liquid components ly and vapor components Vij and the temperatures 7. The enthalpies and equilibrium constants Ky are determined by the primary variables lij, Vij, and Tj. The nonlinear equations are solved by the... 

The MESH equations constitute a nonlinear and strongly coupled system of algebraic equations since the equilibrium ratios Ki j and the enthalpies and are complex functions of temperature and concentrations. The system (5.2-71) is numerically solved by the iterative Newton-Raphson algorithm. Commercial software packages (e.g., ASPEN, HYSYS, CHEMCAD) contain both the mathematical solver and the required system properties, such as vapor liquid equilibria and enthalpies. 

Even when the nonlinear-flash equations are properly solved and convergence is achieved, there is no guarantee that the solution obtained is a true solution. The equilibrium condition given by the equality of chemical potentials or fugacities is a necessary but not a sufficient condition. However, for gas-liquid equilibria, the true solution is nearly always obtained from the equality of chemical potentials. For liquid-liquid and vapor-liquid-liquid and higher equilibria calculations, the equality of chemical potentials alone may lead to a... 

The nonlinear algebraic equations that describe a steady-state distiUalion column consist of component balances, energy balances, and vapor-liquid phase equilibrium relationships. These equations are nonlinear, particularly those describing the phase equilibrium of azeotropic systems. Unlike a linear set of algebraic equations that have one unique solution, a nonlinear set can give multiple solutions therefore, the possibility of multiple steady states exists in azeotropic distillation. 

Algorithms for computation of compositions in vapor and liquid phases at equilibrium usually solve the nonlinear algebraic equations... 

For calculation of compositions in vapor and liquid phases at equilibrium without reaction, when solving nonlinear algebraic equations involving K-values, we recommend the use of volatility and enthalpy parameters, as defined by Boston and Britt (13). 

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