Tridiagonal matrix algorithm

From the model definition, it can be seen that Lq = 0 and V/v+1, = 0- The above equation is expanded for any component i on stages 1 to A  

The equation matrix is tridiagonal, that is, all its elements are zero except the three middle diagonals. This matrix lends itself to a direct solution algorithm consisting of forward elimination followed by backward substitution. For simplicity the component subscript i is dropped since the matrix is solved for one component at a time. 

To start the forward elimination, write the stage 1 equation and divided it by B  

EXAMPLE 13.1 COMPONENT FLOW RATES BY THE TRIDIAGONAL MATRIX METHOD 

The liquid rate from stage 1 is calculated from the distillate rate and reflux ratio i, = 2V, =(2)(48) = 96 kmol/h 

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EQUATION-TEARING PROCEDURES USING THE TRIDIAGONAL-MATRIX ALGORITHM... 

Tridiagonal-Matrix Algorithm Both the BP and the SR equation-tearing methods compute hqnid-phase mole fractions in the same way by first developing linear matrix equations in a manner shown by Amundson andPontinen [Jnd. ng. Ch m., 50, 730 (1958)]. Equations (13-69) and (13-68) are combinedto eliminate yjj and yij + i (however, the vector yj stiU remains imphcitly in K j) ... 

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. 

Equation (7.30) is solved with a tridiagonal matrix algorithm as described in Patankar (1980). First reform equation (7.29) into... 

Equation (140) can be solved either using an iteration method such as the Gauss-Siedel method or a direct method such as tridiagonal-matrix algorithm (TDMA). Other methods, such as the Alternating-Direction Implicit (ADI) method combine iteration with the TDMA. 

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