Solution of Nonlinear Functions

1. Solution of Nonlinear Functions Daniel BaurETH Zurich, Institut für Chemie- und BioingenieurwissenschaftenETH Hönggerberg / HCI F128 – ZürichE-Mail: daniel.baur@chem.ethz.chhttp://www.morbidelli-group.ethz.ch/education/index Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

2. Zero of a Nonlinear Function • Problem definition: • Find the solution of the equation f(x) = 0for scalar valued f and x; Look for the solution either in • An interval, generally –∞ < x < ∞ • In the uncertainty interval [a, b], where f(a)f(b) < 0 • Types of algorithms available: • Bisection method • Substitution methods • Methods based on function approximation • Assumptions: • In the defined intervals, at least one solution exists • We are looking for one solution, not all of them Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

3. Bisection Method • Define starting interval [a,b] (check that f(a)f(b) < 0) • Compute x0 = mean(a, b) • Redefine the interval • Set either a = x0 or b = x0 so that f(a)f(b) < 0 is still fulfilled • Iterate 2 and 3 until the requestedprecision is reached • Advantages • After n iterations, the interval is reducedby 2n • Final precision can be predicted a priori • Disadvantages • Function characteristics are not usedto speed up the algorithm b a a x0 Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature

4. Substitution Methods • Formulate the function in a way that can be iterated, e.g. • Disadvantages • Needs simple functions • Often diverges(even with linear functions) • Requires a preliminarystudy to assureconvergence Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

5. Newton Method • The Newton method is based on Taylor expansion • Advantages • It has 2nd order convergence • Disadvantages • Convergence is not guaranteedeven if the uncertainty intervalis known • If the derivative must becalculated numerically, the secant method is more convenient x1 x0 Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

6. x0 = 1.8 x1 = 1.7 Secant Newton Secant Method • The Secant is based on the same principles as the Newton method, but it approximates the derivative numerically • Advantages • Does not require the analyticalfirst order derivative • Disadvantages • Convergence is not assuredeven if the uncertainty intervalis known • Convergence order is only 1.618 Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

7. How does Matlab do it? Nonlinear Functions • fzero finds the zero of a scalar valued function; It uses a combination of bisection, secant, and inverse quadratic interpolation methods • roots finds all the roots of a polynomial function;It computes the eigenvalues of the companion matrix, which correspond to the roots of the polynomial • A = diag(ones(n-1,1),-1);A(1,:) = -c(2:n+1)./c(1);eig(A); • Where c is the vector of the polynomial coefficients Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

8. Matlab Syntax Hints • x = fzero(@fun, x0); • @fun is a function taking as input a scalar x, returning as output the scalar function value f(x) • x0 is either an initial guess (if it has length 1) or an uncertainty interval (if it has length 2, then f(x0(1))f(x0(2)) < 0 must be fulfilled) • x = roots(c); • c is a vector containing the polynomial coefficients in the order Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

9. Exercise: CSTR Multiple Steady States • Consider a CSTR where a reaction takes place • We assume the following • V = const., i.e. Qin = Qout = const. • Perfect coolant behavior, i.e. TC,in = TC,out = const. • Constant density and heat capacity of the reaction mixture • Constant reaction enthalpy • Constant feed, i.e. cA,in = const., Tin = const. Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

10. CSTR Mass and Energy Balances • The mass and energy balances read • With the T-dependent reaction rate constant Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

11. Dimensionless Mass and Energy Balances • If we define • We get a dimensionless form Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

12. CSTR Temperature Equilibrium • The steady state concentration of A reads • The temperature in steady state is therefore given by Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

13. Assignment • Plot the total heat flow from and to the reactor vs. the dimensionless reactor temperature • Use α = 49.46; κ0 = 2.17e20; KC = 0.83; η = 0.33 and θC = 0.9. • Which are the attracting (stable) and which are the repelling (unstable) solutions? • What happens to the system when you change the coolant temperature? • Try θC = 0.8 and θC= 1. • How many different steady states do you get in each case? Are the solutions stable or unstable (attracting or repelling)? Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations

14. Assignment (Continued) • Implement and use the bisection method to find the three steady state temperatures of the CSTR, using the parameters from task 1. • Implement and use the secant method to find the three steady state temperature of the CSTR. • The secant method uses two starting guesses; provide a θstart,1 and from this, calculate θstart,2 = (1+ε)*θstart,1. Suggest a value for ε (not too small). • In what range of θstart can you converge to the intermediate solution? • If we look at the Newton Method as a perfect Secant Method (analytical gradient!), we can find a maximum possible range where the intermediate solution can be found. How would you do that? Note the directions of the analytical gradients! Daniel Baur / Numerical Methods for Chemical Engineers / Nonlinear Equations