1. Zero of a Nonlinear Function f(x) = 0
Marco Lattuada
Swiss Federal Institute of Technology - ETH
Institut für Chemie und Bioingenieurwissenschaften
ETH Hönggerberg/ HCI F135 – Zürich (Switzerland)

2. 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
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
Iterative and Relaxation Methods for Linear Systems – Page # 2

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

4. Substitution Methods It needs simple functions
x0 = 1.1
x0 = 0.9
y=x2
y=x
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
Iterative and Relaxation Methods for Linear Systems – Page # 4

5. Newton Method 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
Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers
Iterative and Relaxation Methods for Linear Systems – Page # 5

6. Secant Method
x0 = 1.8
x1 = 1.7
Secant
Newton
Convergence is not assured even when uncertainty interval is known
It has a convergence order of < 2
It does not require the computation of the first order derivative
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. 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 F C0 V C T0 T
Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers
Iterative and Relaxation Methods for Linear Systems – Page # 9

10. Possible Conditions T0 = 290 T0 = 275 T0 = 295 T0 = 280 T0 = 300
rcp = 2.5
UA = 10
T0 = 275
T0 = 280
T0 = 285
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