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Linear Systems of Equations Iterative and Relaxation Methods Ax = b

Linear Systems of Equations Iterative and Relaxation Methods Ax = b. Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH H önggerberg/ HCI F135 – Z ürich (Switzerland) E-mail: lattuada@chem.ethz.ch

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Linear Systems of Equations Iterative and Relaxation Methods Ax = b

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  1. Linear Systems of EquationsIterative and Relaxation MethodsAx = b Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH Hönggerberg/ HCI F135 – Zürich (Switzerland) E-mail: lattuada@chem.ethz.ch http://www.morbidelli-group.ethz.ch/education/index

  2. Gas/Liquid Adsorption Column • Aim: • To adsorb a dilute component in the gas (G) phase (e.g. ammonia) into the liquid (L) phase • Hypotheses: • Steady state conditions are reached • The column can be described as a series of N equilibrium stages (plates) • The liquid and the gas fluxes are constant along the column For more info on adsorption columns: http://www.cheresources.com/packcolzz.shtml Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 2

  3. Lxn-1 Gyn Adsorption Plate n Lxn Gyn+1 Gas/Liquid Adsorption Column • Mass balance on n-th plate 1 n • Equilibrium condition • Final Mass Balance Legend L = liquid flow rate G = gas flow rate xi = liquid conc. in i-th plate yi = gas conc. in i-th plate Tridiagonal System N Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 3

  4. Gas/Liquid Adsorption Column Final System of Linear Equations: 1 n Where: x0 = liquid solute conc. in the fresh liquid (typically = 0) xN+1 = liquid solute conc. in the exhaust yN+1 = initial (bottom) gas solute conc. y0 = residual (top) gas solute conc. = (1-a) yN+1 a = fraction of adsorbed solute N = number of plates N Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 4

  5. The use of these methods is particularly advantageous if the matrix has a dominant structure (A~ S) Iterative Methods Hypothesis: S is a structured matrix which is particularly easy/fast to factorize/solve (e.g. diagonal, tridiagonal, triangular) and which can be consider as constant. Solution: I proceed iteratively, until the method converges. Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 5

  6. Convergence of Iterative Methods Let us re-write the convergence loop as: If x is the solution, the error at the k-th iteration is: Then, it follows that (remember the properties of norms): Spectral radius of M: max(|l(M)|) < 1 Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 6

  7. Jacobi’s Method 1st case: S is diagonal Detailed procedure: Conditions: matrix A is diagonally dominant  Method of Jacobi For more infos about the Jacobi method: http://math.fullerton.edu/mathews/n2003/gaussseidel/GaussSeidelBib/Links/GaussSeidelBib_lnk_1.html Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 7

  8. Seidel-Gauss Methods 2nd case: S is right (lower) triangular Detailed procedure: Method of Gauss-Seidel For more infos about the Gauss-Seidel method: http://math.fullerton.edu/mathews/n2003/gaussseidel/GaussSeidelBib/Links/GaussSeidelBib_lnk_1.html Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 8

  9. Solution procedure In both cases, one needs to solve: (1) The formal solution is given by: This does not mean that one has to compute the matrix S-1. It means that the right hand side of (1) has to be treated as a known vector. Naming: One need to solve In solving this system, take advantage of the special form of S (diagonal or triangular). Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 9

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