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The calculation begins with the energy balance of the condenser to get its duty and continues stage to stage down through the column.

Next trial. With these new total flow rates and using the new temperatures, the component flow rates can be recalculated using the tri-diagonal method and a new column trial can be initiated. Note again that the theta method is different from other methods in that the total product rates and reflux ratio are specified while the condenser and reboiler duties are calculated.

Summary of the theta method procedure

1. Set initial stage temperatures, T,'s, and vapor rates, V/s. Initial liquid rates are found using the tridiagonal method for the total material balances. The theta method requires the distillate rate, sidestream product rates, and reflux ratio to be specified.

2. Based on the most recent set of temperatures and total flow rates, calculate the component vapor rates using the tridiagonal matrix method. Find the component liquid rates by ltJ = Ay i>y.

3. Calculate the 6 (or 6's for a complex column) and p- s such that the theta function (or functions) is very nearly zero.

4. Correct the component flow rates and the compositions for each stage using the p/s developed in step 3.

5. With these corrected compositions, calculate new stage temperatures, Tj's, using the Kb method.

6. Calculate the total liquid rates from the constant composition method, using the most recent set of compositions and temperatures. Total vapor rates are found from the material balance for each stage.

7. Convergence is achieved when the 6's are very nearly unity and all other solution criteria (see Sec. 4.2.3) have been met. If not solved, return to the component balances of step 2.

The theta method has found many applications. The FRAKB routines of FLOWTRAN use the theta method. Portions of the theta method, such as the Kb method for temperatures, appear in other rigorous methods. Poor initial temperature and flow rate estimates do not greatly hinder the approach to solution and the calculation is relatively rapid. It has been shown to work well for the systems that the BP methods are meant for, i.e., narrow- and middle-boiling ranges, ideal or nearly ideal mixtures. It has solved columns with as many as 200 stages without great difficulty.

4.2.6 Numerical methods—the Newton-Raphson technique

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 Sees.

4.2.7 to 4.2.13 make some use of the Newton-Raphson method. This section reviews the Newton-Raphson technique itself. Detailed discussion of the Newton-Raphson method and its variations can be found in Holland's text (8).

Each MESH equation is dependent on more than one MESH variable. The MESH equations are represented as a set of functions, fv f2,...) fn, with a corresponding set of independent variables, x2,..., xn. The Newton-Raphson method is a matrix method in which the partial derivatives or change of each function with respect to each variable are placed in a square n x n matrix called the Jacobian.

Mi

Ml..

..Mi.

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