Zero of a Nonlinear Function f(x) = 0

1. Zero of a Nonlinear Functionf(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 in a interval (in general: -∞ < x < ∞) • Research of the zero within the uncertainty interval [a,b] f(a)f(b) < 0 Types of algorithms available: • Bisection method • Substitution algorithms • Methods based on function approximation Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 2

3. After n iterations, the uncertainty interval is reduced by 2n times • Final precision can be predicted a priori • Function characteristics are not used to compute the zero and speed up the solution Bisection Method x2 x0 x1 • Define starting uncertainty interval • Compute x0 = mean(x1, x2) • Compute f(x0) • Define new uncertainty interval • Iterate 2 → 4 Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 3

4. x0 = 0.9 y=x x0 = 1.1 y=x2 • It needs simple functions • It often diverges (even with linear functions) • It requires a preliminary study to assure convergence Substitution Methods Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 4

5. It has a second order convergence • Convergence is not assured even when uncertainty interval is known • It is necessary to know f´(x). If derivative is numerical, secant method is more convenient Newton Method Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 5

6. x0 = 1.8 x1 = 1.7 Secant Newton • It does not require the computation of the first order derivative • Convergence is not assured even when uncertainty interval is known • It has a convergence order of 1.618 < 2 Secant Method Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 6

7. CSTR Multiple Steady States CSTR Mass Balance Reaction: Rate of reaction: Mass balance: IN: OUT: Steady state: F C0 V C F C Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 7

8. CSTR Multiple Steady States CSTR Heat Balance Steady state conc.: Heat balance: IN: OUT: Steady state: F C0 V C T0 T F C T Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 8

9. F C0 V C T0 T F C T Example of Non-Isothermal CSTR CSTR Multiple Steady State Operating Points Data: F = 1 L/min V = 50 L rcp = 1 Kcal/L/K UA = 0.1 Kcal/min/K k0 = 2.6E20 1/min DE = 30 Kcal/mol DH = -20 Kcal/mol C0 = 2 mol/L T0 = 280 K Tj = 278 K Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 9

10. T0 = 290 T0 = 275 T0 = 295 T0 = 280 T0 = 300 T0 = 285 rcp = 2.5 UA = 10 Possible Conditions Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 10

11. Q(T) = Qin(T) – Qout(T) Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Iterative and Relaxation Methods for Linear Systems – Page # 11