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SOLVING ALGEBRAIC EQUATIONS

SOLVING ALGEBRAIC EQUATIONS. System of Linear Equations. Review of Matrix Form Representation of System of Linear Equations Solution Method Matrix Inversion Adjoint of a Matrix Determinant Cramer’s Rule Linearity of System of Equations Independency of Equations.

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SOLVING ALGEBRAIC EQUATIONS

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  1. SOLVING ALGEBRAIC EQUATIONS

  2. System of Linear Equations • Review of Matrix Form Representation of System of Linear Equations • Solution Method • Matrix Inversion • Adjoint of a Matrix • Determinant • Cramer’s Rule • Linearity of System of Equations • Independency of Equations

  3. 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

  4. Example: O2 Enrichment Mixer-Heater (F&R, 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.

  5. Simultaneous Linear Algebraic Equations • Gauss elimination • Gauss-Jordan • Gauss-Seidel • Jacobi • LU Decomposition

  6. 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")

  7. 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

  8. 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

  9. Inverse of a Matrix For a square matrix A Accordingly, How can we calculate the adjoint of a Matrix? Determinant of A

  10. Solution of System of Equation: Cramer’s Rule The following system of equations can be also solved by Cramer’s 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.

  11. 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?

  12. 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 ?

  13. Nonlinear Algebraic Equations • Beattie-Bridgeman • PV4 – RTV3 – bV2 – gV – d = 0 • Fanning friction factor

  14. 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/proceedings/ ToddSuckow-NHA-CaFCP-9-04.pdf

  15. Nonlinear Algebraic Equations • Bracketing Methods • Graphical • Bisection • Incremental search • Open Methods • Newton-Raphson • Secant • Polynomials

  16. Illustration of Graphical Method Example: Hydrogen Storage For Fuel Cell Powered Vehicles

  17. x2 x2 x2 x1 x1 x1 Graphical Method is useful to illustrate: (3) ill-conditioned system (2) infinitely many solutions (1) no solution

  18. Solve the following system 10x1 + 7x2 + 8x3 + 7x4 = 32 7x1 + 5x2 + 6x3 + 5x4 = 23 8x1 + 6x2 + 10x3 + 9x4 = 33 7x1 + 5x2 + 9x3 + 10x4 = 31

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