The MESH Equations

For every component, C in number, on every stage, N in number, there are material, equilibrium, and energy balances, and the requirement that the mol fractions of liquid and vapor phases on each tray sum to unity. The four sets of these equations are  

M equations—Material balance for each component (C equations for each stage)  

In order to simplify these equations, the liquid rate at each stage is eliminated with the substitutions 

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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 ... 

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... 

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 ... 

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... 

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. ... 

The method of Gallun and Holland is the broadest application of the MESH equations in a global Newton method and may solve the widest range of columns. Formulations by Gallun and Holland (40) for distillation columns included adding the total material balance to give freedom in specifications or to substitute these for the equilibrium equations for more ideal mixtures. 

The homotopy methods can be divided into two general classes, mathematical homotopies and physical or parametric homotopies. The mathematical homotopies are conventions without a physical relationship to the MESH equations and this occasionally causes problems. The physical homotopies have a basis in the MESH equations and these will be emphasized. Taylor, Wayburn, and Vickery (80) state that the physical homotopies should outperform the mathematical homotopies and are easier to implement. 

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... 

At t = 1, the values of x should yield the difficult solution F(x) = 0. This technique resembles the relaxation method, but only requires modifying the independent MESH functions to get the derivatives with respect to one term t. This is a purely mathematical approach and Ellis et al. (78) state that it can give negative flow rates at intermediate values of t, something that if-value and enthalpy routines may not tolerate. An alternative is a homotopy function that is rooted in the MESH equations themselves. 

The examples tested by Taylor et al. (80) for the efficiency homotopy were for moderate- or narrow-boiling mixtures. No wide-boiling mixtures were tested. Since the temperature profiles at the intermediate values of E yy will be flat and not broad, the homotopy may be best for the moderate- and narrow-boiling systems. Most of the mixtures were nonideal and the efficiency homotopy should lessen the effect of nonideal If-values where E yy acts as a damper on the if-values. The efficiency homotopy does not work for purity specifications because the purity will not be satisfied in solutions of early values of E yy-Vickery and Taylor (81) presented a thermodynamic homotopy where ideal If-values and enthalpies were used for the initial solution of the global Newton method and then slowly converted to the actual If-values and enthalpies using the homotopy parameter t, The homotopy functions were embedded in the If-value and enthalpy routines, freeing from having to modify the MESH equations. The If-values and enthalpies used are the homotopy functions ... 

The total mass transfer rates are added to an expanded set of the MESH equations called the MERQ equations. The new MERQ acronym stands for... 

Tne MESH Equations (the 2c + 3 Formulation) The equations that model equilibrium stages often are referred to as the MESH equations. The M equations are the material balance equations, E stands for equilibrium equations, S stands for mole fraction summation equations, and H refers to the heat or enthalpy balance equations. 

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. 

The 2c + 1 Formulation An alternative form of the MESH equations is used in many algorithms. In this variation, we make use of the component flow defined by... 

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). 

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