SOLVING ALGEBRAIC EQUATIONS

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SOLVING ALGEBRAIC EQUATIONS
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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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Simultaneous Linear Algebraic Equations

• a11x1 a12x2 a13x3 c1
• a21x1 a22x2 a23x3 c2
• a31x1 a32x2 a33x3 c3
• Objective is to determine the values of x1, x2
  and x3

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Example O2 Enrichment Mixer-Heater (FR, 10.1 a)
Q (kJ)
n1 (kg O2) 40 ?C
n4 (kg O2) n5 (kg N2) 50 ?C
MIXER-HEATER
n2 (kg O2) n3 (kg N2) 25 ?C
Process Description Consider a mixer-heater in
which stream of oxygen and nitrogen at 25 ?C is
blended (mixed) with pure oxygen stream at 40?C
and heated to 50 ?C.
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Simultaneous Linear Algebraic Equations

• Gauss elimination
• Gauss-Jordan
• Gauss-Seidel
• Jacobi
• LU Decomposition

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Matrix Form Representation of System of Linear
Eqns

• We are interested in finding solution to a system
  of ne equations (f1, f2, fne) in ne state
  variables (xS1, xS2, xSne).
• The set of function is to
  be solved such that
• given a set of specified input or design
  variables, , which is a
• vector of design variables ("df"
  represents "degrees of freedom")

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

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

Let us apply this to the Mixer Heater Problem
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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?

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Independency of Equations

• Another implicit assumption made in deriving the
  solution to 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 ?

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Nonlinear Algebraic Equations

• Beattie-Bridgeman
• PV4 RTV3 bV2 gV d 0
• Fanning friction factor

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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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Nonlinear Algebraic Equations

• Bracketing Methods
• Graphical
• Bisection
• Incremental search
• Open Methods
• Newton-Raphson
• Secant
• Polynomials

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Illustration of Graphical Method
Example Hydrogen Storage For Fuel Cell Powered
Vehicles
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Graphical Method is useful to illustrate
(3) ill-conditioned system
(2) infinitely many solutions
(1) no solution
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• Solve the following system
• 10x1 7x2 8x3 7x4 32
• 7x1 5x2 6x3 5x4 23
• 8x1 6x2 10x3 9x4 33
• 7x1 5x2 9x3 10x4 31