Linear Systems of Equations Iterative and Relaxation Methods Ax = b Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH Hönggerberg/ HCI F135 – Zürich (Switzerland) Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH Hönggerberg/ HCI F135 – Zürich (Switzerland)

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 2 Gas/Liquid Adsorption Column For more info on adsorption columns: Aim: To adsorb a dilute component in the gas (G) phase (e.g. ammonia) into the liquid (L) phase Hypotheses: 1.Steady state conditions are reached 2.The column can be described as a series of N equilibrium stages (plates) 3.The liquid and the gas fluxes are constant along the column

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

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

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 5 Iterative Methods The use of these methods is particularly advantageous if the matrix has a dominant structure (A ~ S) 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 # 6 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): 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): Convergence of Iterative Methods Spectral radius of M: max(| (M)|) < 1

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 7 1 st case: S is diagonal Detailed procedure: Conditions: matrix A is diagonally dominant  1 st case: S is diagonal Detailed procedure: Conditions: matrix A is diagonally dominant  Jacobi’s Method Method of Jacobi For more infos about the Jacobi method:

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 8 2 nd case: S is right (lower) triangular Detailed procedure: 2 nd case: S is right (lower) triangular Detailed procedure: Seidel-Gauss Methods Method of Gauss-Seidel For more infos about the Gauss-Seidel method:

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). 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