CE 498/698 and ERS 685 (Spring 2004) Lecture 61 Lecture 6: Feedback Systems of Reactors CE 498/698 and ERS 685 Principles of Water Quality Modeling

CE 498/698 and ERS 685 (Spring 2004) Lecture 62 Feedback W1W1 W2W2 Q 01 c 0 Q 12 c 1 Q 23 c 2 Q 21 c 2 k1V1c1k1V1c1 k2V2c2k2V2c2

CE 498/698 and ERS 685 (Spring 2004) Lecture 63 Lake 1: W1W1 W2W2 Q 01 c 0 Q 12 c 1 Q 23 c 2 Q 21 c 2 k1V1c1k1V1c1 k2V2c2k2V2c2 1 2 Lake 2:

CE 498/698 and ERS 685 (Spring 2004) Lecture 64 Lake 1: Lake 2: Steady-state: and

CE 498/698 and ERS 685 (Spring 2004) Lecture 65 system parametersloadingsunknowns LINEAR ALGEBRAIC EQUATIONS Matrix algebra

CE 498/698 and ERS 685 (Spring 2004) Lecture 66

CE 498/698 and ERS 685 (Spring 2004) Lecture 67 Gauss-Jordan method To compute the matrix inverse Identity matrix 3 3 identity matrix: augmented matrix:

CE 498/698 and ERS 685 (Spring 2004) Lecture 68 Gauss-Jordan method To compute the matrix inverse 1) Normalize 2) Elimination

CE 498/698 and ERS 685 (Spring 2004) Lecture 69 Gauss-Jordan method 1) Normalize Divide by a 11

CE 498/698 and ERS 685 (Spring 2004) Lecture 610 Gauss-Jordan method 2) Elimination

CE 498/698 and ERS 685 (Spring 2004) Lecture 611 Gauss-Jordan method 2) Elimination

CE 498/698 and ERS 685 (Spring 2004) Lecture 612 Gauss-Jordan method 1) Normalization

CE 498/698 and ERS 685 (Spring 2004) Lecture 613 Gauss-Jordan method Matrix inverse

CE 498/698 and ERS 685 (Spring 2004) Lecture 614 Gauss-Jordan method example augmented matrix

CE 498/698 and ERS 685 (Spring 2004) Lecture 615 Gauss-Jordan method example Divide by 3 (normalize)

CE 498/698 and ERS 685 (Spring 2004) Lecture 616 Gauss-Jordan method example Divide by (normalize)

CE 498/698 and ERS 685 (Spring 2004) Lecture 617 Gauss-Jordan method example

CE 498/698 and ERS 685 (Spring 2004) Lecture 618 Gauss-Jordan method example

CE 498/698 and ERS 685 (Spring 2004) Lecture 619 Gauss-Jordan method Can also be used to solve for concentrations

CE 498/698 and ERS 685 (Spring 2004) Lecture 620 Excel - MINVERSE 1.Enter your [A] matrix 2.Block an area the same size 3.Type =MINVERSE(block location of [A]matrix) and press CNTL+SHIFT+ENTER

CE 498/698 and ERS 685 (Spring 2004) Lecture 621 We want to solve for {C} Multiply both sides by [A] -1 Definitions of identity matrix

CE 498/698 and ERS 685 (Spring 2004) Lecture 622 Homework Problem 6.2(a) Use both Gauss-Jordan method and Excel MINVERSE function

CE 498/698 and ERS 685 (Spring 2004) Lecture 623 {C} = response {W} = forcing functions [A] -1 = parameters {response} =[interactions]{forcing functions} Response of reactor 1 Unit change in loading of reactor 2

CE 498/698 and ERS 685 (Spring 2004) Lecture 624 Matrix Multiplication (Box 6.1) # columns in matrix 1 = # rows in matrix 2

CE 498/698 and ERS 685 (Spring 2004) Lecture 625 Terminology DIAGONAL Effect of direct loading SUPERDIAGONAL Effects of d/s loadings on u/s reactors SUBDIAGONAL Effects of u/s loadings on d/s reactors

CE 498/698 and ERS 685 (Spring 2004) Lecture 626 Time-variable response for two reactors where

CE 498/698 and ERS 685 (Spring 2004) Lecture 627 Time-variable response for two reactors General solution if c 1 =c 10 at t = 0 where s are functions of s cs are coefficients that depend on eigenvalues and initial concentrations f = fast eigenvalue s = slow eigenvalue f >> s see formulas on page 111