MESH-equations

In general, values and molal enthalpies in these MESH equations are complex imphcit functions of stage temperature, stage pressure, and equilibrium mole fractions ... 

Compute values of Xi j by solving Eqs. (13-75) through (13-79) by the tridiagonal-matrix algorithm once for each component. Unless all mesh equations are converged, X, Xi j 1 for each stage J. 

In the inner-loop calculation sequence, component flow rates are computed from the MESH equations by the tridiagonal matrix method. The resulting bottoms-product flow rate deviates somewhat from the specified value of 50 lb mol/h. However, by modifying the component stripping factors with a base stripping factor, S, in (13-109) of 1,1863, the error in the bottoms flow rate is reduced to 0,73 percent. 

Another implementation of homotopy-continuation methods is the use of problem-dependent homotopies that exploit some physical aspect of the problem. Vickeiy and Taylor [AIChE J., 32, 547 (1986)] utilized thermodynamic homotopies for K values and enthalpies to gradually move these properties from ideal to ac tual values so as to solve the MESH equations when veiy nonideal hquid solutions were involved. Taylor, Wayburn, and Vickeiy [I. Chem. E. Symp. Sen No. 104, B305 (1987)] used a pseudo-Murphree efficiency homotopy to move the solution of the MESH equations from a low efficiency, where httle separation occurs, to a higher and more reasonable efficiency. 

To simulate the RD system, the MESH model is used, which is assumed that each plate is in vapor-liquid equilibrium. The MESH equations are as follows ... 

These four equations are the so-called MESH equations for the stage Material balance, Equilibrium, Summation and Heat (energy) balance, equations. MESH equations can be written for each stage, and for the reboiler and condenser. The solution of this set of equations forms the basis of the rigorous methods that have been developed for the analysis for staged separation processes. 

The complexity of multicomponent distillation calculations can be appreciated by considering a typical problem. The normal procedure is to solve the MESH equations (Section 11.3.1) stage-by-stage, from the top and bottom of the column toward the feed point. For such a calculation to be exact, the compositions obtained from both the bottom-up and top-down calculations must mesh at the feed point and match the feed composition. But the calculated compositions will depend on the compositions assumed for the top and bottom products at the commencement of the calculations. Though it is possible to... 

The third module is the simulator itself. As stated earlier, each of simulators from the different software vendors uses the core code from Berkeley to iterate solutions of the circuit using mesh equations. 

Three other variables occurring in the MESH equations are functions of more fundamental variables, namely,... 

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 ... 

MESH) equations which are solved for the whole column, decanter included and taking into account the liquid-liquid phase split. Numerical treatment of the Differential Algebraic Equation (DAE) system and discrete events handling is performed with DISCo, a numerical package for hybrid systems with a DAE solver based on Gear s method. The column technical features and operating conditions are shown in Table 4. A sequence of two operational batch steps, namely... 

MESH equations. All of the equations used to describe the steady-state operation of a distillation column (Sec. 4.1.2). MESH stands for ... 

Rigorous method The mathematical method used to solve the MESH equations. 

Solution A solution is reached when all of the MESH equations are satisfied. 

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... 

The energy balances are not solved in the same manner as the component or total material balances. With some solution methods, they are simultaneously solved with other MESH equations to get the independent cc umn variables in others they are used in a more limited manner to get a new set of total flow rates or stage temperatures. 

This can be used in the MESH equations to account for stage nonideality, This vaporization efficiency is applied to the equilibrium constant Ktj and appears as the product E Ky. The vaporization efficiency does solve a computational problem in placing an efficiency in the MESH equations. As shown by Lockett (105), a major disadvantage of the vaporization efficiency is that it does vary with composition. Near the top of a high-purity column, as yLj + x and x(j approach unity, Ejj also approaches unity, and so a vaporization efficiency does not ti uly reflect stage nonidealities. 

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... 

Errors in the MESH equations of Sec. 4.1.2 should be small, including the stage energy total material and component balances and summation equation should be small. The physical solution criteria above should take precedence over any mathematical criteria, such as having Newton-Raphson functions approach zero (Sec. 4.2.6). 

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. 

Each MESH equation is dependent on more than one MESH variable. The MESH equations are represented as a set of functions, ft, f2, ..., ft, with a corresponding set of independent variables, r1( xn. The Newton-Raphson method is a matrix method in which the partial derivatives or change of each function with respect to each vain-able are placed in a square n x n matrix called the Jacobian. 

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. ... 

---