Zero of a Nonlinear System of Algebraic Equations fx 0

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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_at_chem.ethz.ch
• http//www.morbidelli-group.ethz.ch/education/inde
  x

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Definition of the Problem

• Definition of the problem
• Research of the zero in a interval (in general
  -8 lt x lt 8)
• Research of the zero within the uncertainty
  interval a,b
• f(a)f(b) lt 0
• Types of algorithms available
• Bisection method
• Substitution algorithms
• Methods based on function approximation

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Example
f(x) 0
f1(x,y) 0
f2(x,y) 0
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Projection on the (x-y) Plane
f(x) 0
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Function Linearization

• First order Taylor expansion
• Matrix form
• Compact form

Newton Method
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Application to Example

• Nonlinear system
• Linearization ((x0, y0) is the starting point)
• Starting point (x0, y0) (0, 0)

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Graphical Interpretation

• Nonlinear system Linearized system
• Linearized system
• in (x0, y0) (0, 0)

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Projection on the (x-y) Plane

• Nonlinear system Linearized system

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Effect of Different Starting Points
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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.ch
eresources.com/packcolzz.shtml
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Gas/Liquid Adsorption Column
Lx0
Gy1
Plate 1
Lxn-1
Gyn
Plate n
Lxn
Gyn1
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
GyN1
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Gas/Liquid Adsorption Column

• Mass balance on
• first n plates

• Equilibrium
• condition

Lx0
Gy1
Plate 1
Lxn-1
Gyn
Plate n
Lxn
Gyn1
Plate N
LxN
GyN1
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Examples
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Nonlinear Gas-Liquid Equilibrium
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The Nonlinear Problem

• Mass balance on n-th plate

• Equilibrium condition (nonlinear)

• Final Equation

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Matlab fsolve Function
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Exercise

• Use the following parameters L 5, G 1, xTop
  0, yBottom 0.05, N 8.
• Use the following function to plot the results
  plot_column (on-line)
• Use a linear equilibrium y 7x (or K 7).
  Solve the linear system of equation as already
  done in class number 2 and plot the result. This
  result will be the starting point for the
  iterations with the Newton method.
• Use the nonlinear equilibrium y 7x-2E4x3.
  Solve the nonlinear system of equations with the
  Matlab function fsolve. Plot the result.
• Calculate (analytically) the Jacobian matrix for
  the previous set of equations. Write a Matlab
  function which receives as input the vector x of
  the liquid concentrations and returns the
  Jacobian matrix and the vector f(x) of the
  residuals. (find online a model for this
  function)
• Compute 10 iterations with the Newton method.
  Does the problem converge to the solution
  computed at point 4?