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

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 H önggerberg/ HCI F135 – Z ürich (Switzerland) E-mail: lattuada@chem.ethz.ch

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

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  1. Zero of a Nonlinear System of Algebraic Equationsf(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

  2. In the defined intervals, at least one zero exists • We are looking for one zero, and notall of them Definition of the Problem 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 Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Systems of Algebraic Equations – Page # 2

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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

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

  11. 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

  12. Mass balance on n-th plate • Mass balance on first n plates or: Gas/Liquid Adsorption Column Lx0 Gy1 Plate 1 Lxn-1 Gyn Plate n 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

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

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

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

  16. 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

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

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