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where for any stagey, the term for the main diagonal is m, = -(1 + LJKij V;) = - (1 + Ay)

for the lower diagonal, the term for stage j is the absorption factor Al}. given by Eq. (4.6), and for the upper diagonal, the terms for all stages are unity.

For a stage with a side product, the term for the main diagonal is modified to include the ratio of the side product total flow rate to the stage flow rate corresponding to its phase (see Sec. 4.1.1). For a stage p with a vapor side product, Wv , the term is miP = -(1 + LP/KtP VP + WVJVP)

For a stage p with a liquid side product, WLp, the term for the main diagonal, is mtP = -[1 + LpjKip Vp (1 + WuJLp)]

The total distillate rate replaces the vapor rate in the terms for a condenser, For a partial condenser, the terms for the middle and lower diagonals are mn = -(1 + Lx!Kl{ D) An = LjKn D

For a total condenser, the terms include just the reflux ratio:

For a reboiler, as stage N, the term for the middle diagonal is

The absorption factor in Eq. (4.6) is used to get the liquid rates after the vapor rates are calculated in the solution of the component balances described above. The equations can just as well be rearranged to calculate the component liquid rates with the component vapor rates found from Eq. (4,5).

Calculating the total flow rates. Rearrangement of the total material balance equations parallels that of the component balance equations above. To create the tridiagonal matrix, the equations are rearranged to allow for the total vapor rates to be used as the independent variables. Ratios of the total liquid and vapor rates become the coefficients of the vapor rate terms that go into the tridiagonal matrix.

{Lj.l/VJ.1)Vj_l - (1 + LJVj) V,- + VJ + 1 = 0 (4.42)

In matrix form, the total material balance equations are expressed as:

The new liquid rates are found by multiplying the new total vapor rate by the ratio of the old liquid and vapor rates:

Few of the rigorous methods use this means to calculate the total fiow rates. The tridiagonal total material balance matrix [Eqs. (4.42) to (4.44)] can still be used to bring the column back into a good material balance before the next calculation of the component material balances. It can also be used to get a complete initial column flow rate profile once some initial estimates have been entered.

The numerical method for solving the tridiagonal matrix. There are solution techniques geared specifically for tridiagonal matrices. One method commonly used is the recurrence formulas of the Thomas algorithm. In recurrence formulas, the calculation (forward elimination) will begin at the upper left corner (top stage) of the tridiagonal and proceed down the main diagonal. A set of factors will be generated for each row of the matrix. Then a calculation (back substitution) is made starting at the lower right corner (last stage) of the matrix and proceeding up the diagonal to get the values of the independent variables (component or total vapor rates).

In calculation methods like the original Thomas algorithm, where the factors of a stage are calculated from the factors of a previous stage, a buildup of computer truncation errors can occur (from subtraction operations). Boston and Sullivan (25) presented a modified Thomas algorithm with the subtraction steps eliminated. This modified version has been shown to help some columns that have difficulty in solving the component balances, especially those with multiple feeds and side streams. The modified version has also been shown to help wide-boiling mixtures that have components with /f-values greater than unity in one part of the column and less than unity in another part of the column.

When the component balances are solved by recurrence formulas, the flow rate of a component on the extremes of the mixture boiling range often approaches zero. This leads to computer underflow and divide-by-zero errors. Protection against such errors should be included in the recurrence calculations to trap such errors and set the component flow rate to zero on that stage.

The shape of a tridiagonal matrix permits easy storage in three vectors instead of one large JV x JV matrix. Since this storage is consistent for all tridiagonal matrices, they can all be handled in the same manner a"nd there are a number of subroutines available specifically for these. Press et al. (121), the IMSL package from IMSL, in Houston, Texas, and others have tridiagonal matrix solution routines.

4.2.5 Bubble-point (BP) methods

The BP methods use a form of the equilibrium equation and summation equation to calculate the stage temperatures. The first BP method, by Wang and Henke (24), included the first presentation of the tridiagonal method to calculate the component flow rates or compositions. These are used to calculate the temperatures by solving the bubble-point equation but this temperature calculation can be prone to failure.

An alternative is to use the tridiagonal method for calculating compositions, but to calculate the new temperatures directly, without iterating on the bubble-point equation. These new temperatures are approximate but as long as the internal compositions are properly corrected during each column trial, the temperature profile will continue to move toward the solution. This is the basis of the theta method of Holland (7, 9, 26). With either alternative, the energy balances are used to find the total flow rates.

The BP methods generally work best for narrow-boiling, ideal or nearly ideal systems, where composition has a greater effect on temperature than the latent heat of vaporization.

The theta method. This method has been primarily applied to the Thiele-Geddes equations but a form of the theta method equation has also been applied to the equations of the Lewis-Matheson method. The main independent variable of the method is a convergence promoter, theta (or 0). The convergence promoter 0 is used to force an overall component and total material balance and to adjust the compositions on each stage. These new compositions are then used to calculate new stage temperatures by an approximation of the dew- or bubble-point equation called the Kb method. The power of the Kb method is that it directly calculates a new temperature without the sort of failures that occur when iteratively solving the bubble- or dew-point equations.

The theta function. This function begins with the component balance for a conventional column:

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