1. 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)

2. 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

3. 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

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:
Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers
Nonlinear Systems of Algebraic Equations – Page # 11

12. 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

13. 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

14. Examples Increase L/G Decrease K
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