CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 1

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 6 2

FeedbackW1 W2

Q01c0 Q12c1 Q23c2

Q21c2

k1V1c1 k2V2c2

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 3

22111111211

1 cQcVkcQWdtdc

V Lake 1:

W1 W2Q01c0 Q12c1 Q23c2

Q21c2

k1V1c1 k2V2c2

1 2

22122222311222

2 cQcVkcQcQWdt

dcV Lake 2:

0011 cQW

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 4

22111111211

1 cQcVkcQWdtdc

V Lake 1:

22122222311222

2 cQcVkcQcQWdt

dcV Lake 2:

Steady-state: 0dtdc

1212111 Wcaca 2222121 Wcaca and

111211 VkQa

2112 Qa 1221 Qa

23222122 QVkQa

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 5

1313212111 Wcacaca

2323222121 Wcacaca

3333232131 Wcacaca

system parameters loadingsunknowns

LINEAR ALGEBRAIC EQUATIONS

Matrix algebra

WCA

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 6

1313212111 Wcacaca

2323222121 Wcacaca

3333232131 Wcacaca WCA

333231

232221

131211

aaa

aaa

aaa

A

3

2

1

c

c

c

C

3

2

1

W

W

W

W

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 7

Gauss-Jordan methodTo compute the matrix inverse

Identity matrix

33 identity matrix:

100

010

001

I

CCI

augmented matrix:

100

010

001

333231

232221

131211

aaa

aaa

aaa

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 8

Gauss-Jordan methodTo compute the matrix inverse

1) Normalize2) Elimination

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 9

Gauss-Jordan method

100

010

001

333231

232221

131211

aaa

aaa

aaa

1) Normalize

100

010

001

111

333231

232221

11

13

11

12

a

aaa

aaaa

a

aa

Divide by a11

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 10

Gauss-Jordan method2) Elimination

100

010

001

111

333231

232221

11

13

11

12

a

aaa

aaaa

a

aa

00

121

1121

11

1321

11

1221 a

aa

a

aa

aa

a

100

011

001

1

2111

11

333231

2111

132321

11

12222121

11

13

11

12

aa

a

aaa

aa

aaa

aa

aaa

a

a

aa

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 11

Gauss-Jordan method2) Elimination

101

011

001

0

0

1

3111

2111

11

3111

133331

11

1232

2111

132321

11

1222

11

13

11

12

aa

aa

a

aa

aaa

aa

a

aa

aaa

aa

a

a

a

aa

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 12

Gauss-Jordan method1) Normalization

101

011

001

0

0

1

3111

2111

11

3111

133331

11

1232

2111

132321

11

1222

11

13

11

12

aa

aa

a

aa

aaa

aa

a

aa

aaa

aa

a

a

a

aa

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 13

Gauss-Jordan method

Matrix inverse

133

132

131

123

122

121

113

112

111

100

010

001

aaa

aaa

aaa

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 14

Gauss-Jordan method example

4.71102.03.0

3.193.071.0

85.72.01.03

321

321

321

ccc

ccc

ccc

100

010

001

102.03.0

3.071.0

2.01.03augmentedmatrix

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 15

Gauss-Jordan method example

100

010

001

102.03.0

3.071.0

2.01.03

100

010

00333.0

102.03.0

3.071.0

067.0033.01

Divide by 3(normalize)

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 16

Gauss-Jordan method example

100

010

00333.0

102.03.0

3.071.0

067.0033.01

10100.0

01033.0

00333.0

020.10190.00

293.0003.70

067.0033.01

Divide by 7.003(normalize)

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 17

10100.0

0143.0005.0

00333.0

020.10190.00

042.010

067.0033.01

Gauss-Jordan method example

1027.0101.0

0143.0005.0

0005.0333.0

012.1000

042.010

068.001

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 18

Gauss-Jordan method example

100.0003.0010.0

004.0143.0005.0

007.0005.0332.0

100

010

001

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 19

Gauss-Jordan method

• Can also be used to solve for concentrations

3

2

1

333231

232221

131211

W

W

W

aaa

aaa

aaa

3

2

1

100

010

001

c

c

c

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 20

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 6 21

WCA We want to solve for {C}

WACAA 11 Multiply both sides by [A]-1

WAC 1

IAA 1

CCI Definitions of identity matrix

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 22

Homework Problem 6.2(a)

• Use both Gauss-Jordan method and Excel MINVERSE function

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 23

{C} = response{W} = forcing functions[A]-1 = parameters

{response} =[interactions]{forcing functions}

31

3321

3211

313

31

2321

2211

212

31

1321

1211

111

WaWaWac

WaWaWac

WaWaWac

Response of reactor 1

Unit change in loading of reactor 2

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 24

Matrix Multiplication (Box 6.1)# columns in matrix 1 = # rows in matrix 2

3231

2221

1211

333231

232221

131211

bb

bb

bb

aaa

aaa

aaa

BA

312321221121

321322121211311321121111

bababa

babababababa

BA

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 25

Terminology

DIAGONAL

Effect of direct loading

SUPERDIAGONALEffects of d/s loadingson u/s reactors

SUBDIAGONALEffects of u/s loadingson d/s reactors

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 26

Time-variable response for two reactors

22111111211

1 cQcVkcQWdtdc

V

22122222311222

2 cQcVkcQcQWdt

dcV

2221212

2121111

ccdt

dc

ccdtdc

11

1211 k

VQ

1

2112 V

Q

2

1221 V

Q

22

122322 k

V

QQ where

CE 498/698 and ERS 685 (Spring 2004)

Lecture 6 27

Time-variable response for two reactors

General solution if c1=c10 at t = 0 t

st

fsf ececc 111

ts

tf

sf ececc 222

where ’s are functions of ’sc’s are coefficients that depend on eigenvalues and initial concentrations

f = fast eigenvalues = slow eigenvalue

f >>ssee formulason page 111