Zero of a Nonlinear System of Algebraic Equations f(x) = 0 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

Definition of the Problem Research of the zero (x) in a interval Research of the zero within the uncertainty interval [a,b] Types of algorithms available: Substitution algorithms Methods based on functions approximation In the defined intervals, at least one zero exists We are looking for one zero, and not all of them Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 2

Substitution method It needs simple functions It often diverges (even with linear functions) It requires a preliminary study to assure convergence Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 3

Example f(x) = 0 z=f1(x,y) = 0 z=f2(x,y) = 0 Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 4

Projection on the (x-y) Plane f(x) = 0 Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 5

Function Linearization First order Taylor expansion: Matrix form: Compact form: Newton Method Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 6

Application to Example Nonlinear system: Linearization: ((x0, y0) is the starting point) Starting point: (x0, y0) = (0, 0) Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 7

Graphical Interpretation Nonlinear system: Linearized system: Linearized system in (x0, y0) = (0, 0) : Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 8

Projection on the (x-y) Plane Nonlinear system: Linearized system: Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 9

Effect of Different Starting Points Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 10

Gas/Liquid Adsorption Column Aim: To adsorb a dilute component in the gas (G) phase (e.g. ammonia) in 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 Nonlinear Systems of Algebraic Equations – Page # 11

Gas/Liquid Adsorption Column Lx0 Gy1 Mass balance on n-th plate Plate 1 Mass balance on first n plates Lxn-1 Gyn Plate n or: Lxn Gyn+1 Plate N Legend L = liquid flow rate G = gas flow rate xi = liquid conc. in i-th plate yi = gas conc. in i-th plate LxN GyN+1 Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 12

Gas/Liquid Adsorption Column Mass balance on first n plates Equilibrium condition Lx0 Gy1 Plate 1 y x y1 y3 x2 x3 Lxn-1 Gyn Plate n Lxn Gyn+1 y2 Plate N Target residual concentration x1 LxN GyN+1 Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 13

Examples Increase L/G Decrease K Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 14

Nonlinear Gas-Liquid Equilibrium Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 15

The Nonlinear Problem Mass balance on n-th plate Equilibrium condition (nonlinear) Final Equation Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 16

Matlab fsolve Function Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 17