CHEE 321 CHEMICAL REACTION ENGINEERING

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CHEE 222 CHEMICAL PROCESS DYNAMICS AND NUMERICAL
METHODS
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Module 2. Lumped Parameter Steady State System

• This module deals with various forms of
  algebraic equations that arise for
  lumped-parameter system operating under steady
  state. The following topics will be covered
• System of Linear Equations
• Solution Method
• Matrix Inversion
• Cramers Method
• Computer-based solution (Matrix inversion in
  Matlab)
• Non-linear Single-Variable System
• Iterative Solution
• Bisection Method
• Newtons Method
• Convergence criteria
• Computer-based solution (Matlab fzero, roots)
• Non-linear Multivariable System
• Linearization Introduction to Jacobians
• Newton-Raphson Method

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System of Linear Equations
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Module 2.1 System of Linear Equations

• Review of Matrix Form Representation of System of
  Linear Equations
• Solution Method
• Matrix Inversion
• Adjoint of a Matrix
• Determinant
• Cramers Rule
• Linearity of System of Equations
• Independency of Equations

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Computer based solution method

• Solve the following set of equations
• 2x1 3x2 - 4x3 3 0 (1)
• - 4x2 2x3 2 0 (2)
• x1 - x2 5x3 9 0 (3)

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Matrix Form Representation of System of Linear
Eqns

• We are interested in finding solution to a system
  of n equations (f1, f2, fn) in n state variables
  (x1, x2, xn).
• The set of functions is to be
  solved such that

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Matrix Form Representation of System of Linear
Eqns (cont.)

• The system of equations
  can be expressed as
• where,

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Solution of a System of Equations Matrix
Inversion

• The solution of the following system of equations
• can be written as follows
• The objective is to find the inverse of Matrix A,
  i.e. A-1

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Inverse of a Matrix

• For a square matrix A
• Accordingly,
• How can we calculate the adjoint of a Matrix?

Determinant of A
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Solution of System of Equation Cramers Rule

• The following system of equations
• can be also solved by Cramers Rule, which is
  given as follows
• where, Ai, represents the matrix obtained by
  replacing coefficients associated with ith state
  variable in the A matrix with constants, b.

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Linearity of a System of Equations

• Although not explicitly specified, the solution
  method discussed in previous slide applies to
  linear system of equations.
• What does linearity of a system of equation
  refer to?

Notes to be provided in class
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Independency of Equations

• Another implicit assumption made in deriving the
  solution for linear system of equations is that
  the equations are independent.
• What does independency of equations mean?
• How can we find whether a system of equation is
  independent of each other ?

Notes to be provided in class
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Computer based solution method

• Solve the following set of equations
• 2x1 3x2 - 4x3 3 0 (1)
• - 4x2 2x3 2 0 (2)
• x1 - x2 5x3 9 0 (3)

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Single-variable Non-Linear Equations
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Module 2.2 Solution of Single-Variable
Non-Linear Equation

• Introduction to Non-Linear Equations
• Examples
• Discussion on Non-Linearity
• Solution Methods for Single-Variable Non-Linear
  Equations
• Graphical
• Interval Halving or Bisection Method
• Newtons Method
• Initial Guess General Discussion
• Solution using MATLAB
• Roots function
• Fzero function

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Example Hydrogen Storage For Fuel Cell Powered
Vehicles
Calculate the volume of tank required to store 4
kg of Hydrogen at ambient temperature of
15C. Additional Information Maximum rated
pressure of tank is 5,000 psi
Let us use van der Waals Equation of State
For H2 a 0.0247 J-m3/mol b 2.65x10-5 m3/mol
Source of Pictures www.hydrogensafety.info/procee
dings/ ToddSuckow-NHA-CaFCP-9-04.pdf
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Methods for Solving Single-Variable Non-Linear
Equation

• Graphical Method ?
• Interval Halving or Bisection Method
• False Position (Regular Falsi) Method
• Newtons Method ??

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Illustration of Graphical Method
Example Hydrogen Storage For Fuel Cell Powered
Vehicles
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Newtons Method

• Recall, that our objective is to find the
  solution (ZEROS) of a non-linear function, say,
  f(x).
• We start with Taylor series expansion of the
  function, f(x), around the vicinity of expected
  root, say x x0
• If we consider only the first-order terms, the
  above equation can be simplified as follows
• We use this equation to estimate the value of x
  that will bring us closer to the solution, i.e.
  zero of the function or f(x) 0.

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Newtons Method (Cont.)

• To find a suitable estimate for the solution, we
  set f( x)0, in the following eqn.
• Accordingly, after re-arranging the equation we
  get
• where,
• is the first derivative
  of the function with respect to (w.r.t.) x and
  evaluated at xx0

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Graphical Illustration of Newtons Method
Notes to be provided in class
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Convergence Criteria for Iterative Processes

• Newtons method falls in the category of
  iterative processes of solution method.
• Iterative procedures for numerical solution of a
  function, by definition, require that we stop the
  iteration at some point when we believe that an
  acceptable solution has been reached.
• We say that the iterative procedure has converged
  when the iterated solution is a close enough to
  the actual solution. That is, the solution is
  within a specified level of tolerance
• The simplest criterion for convergence could be
  as follows

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Absolute and Relative Tolerance

• Absolute Tolerance (ea) Relates to the absolute
  difference between two successive iterated
  solution
• Relative Tolerance er Relates to the change
  between two successively iterated solution as a
  fraction of last solution.
• Most procedures employ a combination of the two
  tolerances

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Initial Guess General Consideration
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SOLVING single-variable non-linear equation by
MATLAB
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MATLAB Function roots

• The ROOTS function allows you to calculate the
  roots of a polynomial function.
• Let us say we want to find the roots of the
  following function
• MATLAB Command
• First, we represent the polynomial function by a
  row vector containing the coefficients. For the
  given example, this can be done as follows
• gtgt f 1 -5 -1 2
• Next, we can type the following command to obtain
  the roots of the equation
• gtgt x roots (f)

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MATLAB Function fzero

• fzero function finds zeros of the function of a
  single-variable
• MATLAB Command
• gtgt x fzero(fun,x0)
• tries to find a zero of an function defined by
  fun near x0, if x0 is a scalar.
• fun is an M-file function
• The value x returned by fzero is near a point
  where fun changes sign, or NaN if the search
  fails. In this case, the search terminates when
  the search interval is expanded until an Inf,
  NaN, or complex value is found.

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M-file Script File
Output of this file
name of the function should be the same as file
name
Output of this file

• function ffun(x)
• fx3-5x2-x2
• end

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Multi-variable Non-Linear Equations
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Multi-variable Non-Linear Equations

• Consider a system of n equations in n unknowns
• The objective is to find a set of solutions (x)
  such that

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Solution of Multi-variable Non-Linear Equations

• The solution for multi-variable Non-Linear
  Equations can be achieved by extension of the
  Newtons Method we applied to single-variable
  non-linear system.
• Basic Principle It involves expansion of each of
  the function fi(x) as a Taylor series around a
  given set of initial guess, x0

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Taylor Series Expansion of Multi-variable Function
Notes to be provided in class
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The Jacobian Matrix
Notes to be provided in class