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Feedback Systems in Reactors: Water Quality Modeling and Gauss-Jordan Method

This lecture explores the principles of feedback systems in reactors focused on water quality modeling. It discusses key concepts such as steady-state conditions in lakes, system parameters, and matrix equations for solving unknowns related to loadings. Emphasizing the Gauss-Jordan method, the lecture provides a step-by-step process for computing matrix inverses and applying linear algebra techniques. Real-world applications, including the use of Excel's MINVERSE function for concentration calculations, are highlighted, aiming to enhance understanding and practical skills in this area.

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Feedback Systems in Reactors: Water Quality Modeling and Gauss-Jordan Method

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  1. Lecture 6: Feedback Systems of Reactors CE 498/698 and ERS 685 Principles of Water Quality Modeling Lecture 6

  2. W1 W2 Q01c0 Q12c1 Q23c2 Q21c2 k2V2c2 k1V1c1 Feedback Lecture 6

  3. 1 W1 2 W2 Q01c0 Q12c1 Q23c2 Q21c2 Lake 1: k2V2c2 k1V1c1 Lake 2: Lecture 6

  4. Steady-state: Lake 1: and Lake 2: Lecture 6

  5. system parameters unknowns loadings LINEAR ALGEBRAIC EQUATIONS Matrix algebra Lecture 6

  6. Lecture 6

  7. 33 identity matrix: augmented matrix: Gauss-Jordan method To compute the matrix inverse Identity matrix Lecture 6

  8. Gauss-Jordan method To compute the matrix inverse 1) Normalize 2) Elimination Lecture 6

  9. Divide by a11 Gauss-Jordan method 1) Normalize Lecture 6

  10. Gauss-Jordan method 2) Elimination Lecture 6

  11. Gauss-Jordan method 2) Elimination Lecture 6

  12. Gauss-Jordan method 1) Normalization Lecture 6

  13. Matrix inverse Gauss-Jordan method Lecture 6

  14. augmented matrix Gauss-Jordan method example Lecture 6

  15. Divide by 3 (normalize) Gauss-Jordan method example Lecture 6

  16. Divide by 7.003 (normalize) Gauss-Jordan method example Lecture 6

  17. Gauss-Jordan method example Lecture 6

  18. Gauss-Jordan method example Lecture 6

  19. Gauss-Jordan method • Can also be used to solve for concentrations Lecture 6

  20. Excel - MINVERSE • Enter your [A] matrix • Block an area the same size • Type =MINVERSE(block location of [A]matrix)and press CNTL+SHIFT+ENTER Lecture 6

  21. Multiply both sides by [A]-1 Definitions of identity matrix We want to solve for {C} Lecture 6

  22. Homework Problem 6.2(a) • Use both Gauss-Jordan method and Excel MINVERSE function Lecture 6

  23. Unit change in loading of reactor 2 Response of reactor 1 {C} = response {W} = forcing functions [A]-1 = parameters {response} =[interactions]{forcing functions} Lecture 6

  24. Matrix Multiplication (Box 6.1) # columns in matrix 1 = # rows in matrix 2 Lecture 6

  25. Terminology SUPERDIAGONAL Effects of d/s loadings on u/s reactors DIAGONAL Effect of direct loading SUBDIAGONAL Effects of u/s loadings on d/s reactors Lecture 6

  26. where Time-variable response for two reactors Lecture 6

  27. ’s are functions of ’s c’s are coefficients that depend on eigenvalues and initial concentrations where f= fast eigenvalue s = slow eigenvalue f >>s Time-variable response for two reactors General solution if c1=c10 at t = 0 see formulason page 111 Lecture 6

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