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Lecture 6: Feedback Systems of Reactors. CE 498/698 and ERS 685 Principles of Water Quality Modeling. W 1. W 2. Q 01 c 0. Q 12 c 1. Q 23 c 2. Q 21 c 2. k 2 V 2 c 2. k 1 V 1 c 1. Feedback. 1. W 1. 2. W 2. Q 01 c 0. Q 12 c 1. Q 23 c 2. Q 21 c 2. Lake 1:. k 2 V 2 c 2.
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Lecture 6: Feedback Systems of Reactors CE 498/698 and ERS 685 Principles of Water Quality Modeling Lecture 6
W1 W2 Q01c0 Q12c1 Q23c2 Q21c2 k2V2c2 k1V1c1 Feedback Lecture 6
1 W1 2 W2 Q01c0 Q12c1 Q23c2 Q21c2 Lake 1: k2V2c2 k1V1c1 Lake 2: Lecture 6
Steady-state: Lake 1: and Lake 2: Lecture 6
system parameters unknowns loadings LINEAR ALGEBRAIC EQUATIONS Matrix algebra Lecture 6
33 identity matrix: augmented matrix: Gauss-Jordan method To compute the matrix inverse Identity matrix Lecture 6
Gauss-Jordan method To compute the matrix inverse 1) Normalize 2) Elimination Lecture 6
Divide by a11 Gauss-Jordan method 1) Normalize Lecture 6
Gauss-Jordan method 2) Elimination Lecture 6
Gauss-Jordan method 2) Elimination Lecture 6
Gauss-Jordan method 1) Normalization Lecture 6
Matrix inverse Gauss-Jordan method Lecture 6
augmented matrix Gauss-Jordan method example Lecture 6
Divide by 3 (normalize) Gauss-Jordan method example Lecture 6
Divide by 7.003 (normalize) Gauss-Jordan method example Lecture 6
Gauss-Jordan method example Lecture 6
Gauss-Jordan method example Lecture 6
Gauss-Jordan method • Can also be used to solve for concentrations Lecture 6
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
Multiply both sides by [A]-1 Definitions of identity matrix We want to solve for {C} Lecture 6
Homework Problem 6.2(a) • Use both Gauss-Jordan method and Excel MINVERSE function Lecture 6
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
Matrix Multiplication (Box 6.1) # columns in matrix 1 = # rows in matrix 2 Lecture 6
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
where Time-variable response for two reactors Lecture 6
’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