MESH equations and variables

MESH equations are divided and grouped or partitioned and paired with MESH variables to be solved in a series of steps. The Sc methods attempt to solve all of the MESH equations and variables together. Additional classes are... 

Vickery and Taylor (81) used a Naphtali-Sandholm method containing all of the MESH equations and variables [M2C + 3) equations] with the variables represented by x. H is the Jacobian from the Naphtali-Sandholm method solution of the known problem, G(x) = 0, This is numerically integrated from t = 0 to t - 1, finding a H, at each Step and updating H when the solution is reached at each step, With Hj. and H, known, dxjdt is solved, and with step size t, a new set of values for the independent variables x is found by Euler s rule... 

In Chapter 1 it was shown that the number of independent variables for any problem is equal to the difference between the total number of variables and the number of linking equations and other relationships. Examples of the application of this formal procedure for determining the number of independent variables in separation process calculations are given by Gilliland and Reed (1942) and Kwauk (1956). For a multistage, multicomponent column, there will be a set of material and enthalpy balance equations and equilibrium relationships for each stage (the MESH equations) and for the reboiler and condenser, for each component. 

The residual vector F has a component for every equation at every mesh point, and there is a dependent variable for every residual equation at every mesh point. The dependent variable and the residual vectors are arranged similarly as... 

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. 

A brief description of this procedure is abstracted from the fuller treatment of Henley and Seader (1981). The MESH equations (13.182)-(13.186) in terms of mol fractions are transformed into equations with molal flow rates of individual components in the liquid phase ltJ and vapor phase vif as the primary variables. The relations between the transformed variables are in this list ... 

The rigorous methods thus convert a column to a group of variables and equations. The equations were first referred io as the MESH equations by Wang and Henke (24). The MESH variahles are often referred to as state variables. These are... 

Produced from the manipulation of the Jacobian are the changes in the variables, i,e., the Ax vector. The variables for the next trial are calculated from x + = x + s Ax (i.e,, . + T = xlk + sk Ajc1jA, etc,). The s scalar is generated to ensure that the norm of functions improve between trial k + 1 and trial k. Usually, s = 1 but may have to be smaller on some trials. The Newton-Raphson method assumes that the curves of the independent functions are close to linear and the slopes will point toward the answers. The MESH equations can be far from linear and the full predicted steps, Ax, can take the next trial well off the curves. The s scaler helps give an improved step search or prevente overstepping the solution. Holland (8) and Broyden (119) present formulas for getting s. ... 

Boston (75) added a middle loop to allow for column specifications and constraints. The arrangement of equations in the inner loop, where the solution of the MESH variables occurs, allows for only a few control or specified variables, such as fixed reflux ratio and product rates, The middle loop adjusts the control variables to meet the specifications. The middle loop can be built as an optimization method with process specification equations and economic objectives and constraints. 

The quantities for stage j that appear in these equations are summarized in Table 13-9. The total number of variables appearing in these equations is 3c + 10. Note that the K values and endialpies that also appear in the MESH equations are not included in the table of variables, nor are equations for their estimation included in the list of equations. Thermodynamic properties are functions of temperature, pressure, and composition, quantities that do appear in the table of variables. 

This equation is familiar to us from bubble point calculations. In this formulation of the MESH equations, the vapor-phase mole fractions no longer are independent variables but are denned by Eq. (13-52). This formulation of the MESH equations has been used in quite a number of algorithms. It is less useful if vapor-phase nonideality is important (and, therefore, the K values depend on the vapor-phase composition). 

When the radial variation of temperature must be taken into account, the problem assumes an entirely different character. Each of the equations is now a partial differential equation, and both radial and axial profiles must be calculated a mesh or network of radial and axial lines is set up, and the temperature and composition are calculated for each intersection. A great deal of work has been done on the formulation of difference equations for solving the related diffusion or heat-conduction equations most of this has been directed towards the case in which there is only one dependent variable and in which the source is a linear function of that variable. Although the results obtained for one dependent variable are only partially applicable to the multiple-variable problem,... 

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... 

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