Lecture 5: Feedback Systems of Reactors
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Transcript Lecture 5: Feedback Systems of Reactors
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
1
Feedback
W1
Q01c0
Q12c1
W2
Q23c2
Q21c2
k1V1c1
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
k2V2c2
2
W1
Q01c0
W2
Q12c1
1
Q21c2
Q23c2
2
k1V1c1
k2V2c2
W1 Q01c0
Lake 1:
V1
dc1
W1 Q12 c1 k1V1c1 Q21c2
dt
Lake 2:
V2
dc2
W2 Q12 c1 Q23c2 k 2V2 c2 Q21c2
dt
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
3
Lake 1:
dc1
V1
W1 Q12 c1 k1V1c1 Q21c2
dt
Lake 2:
V2
dc2
W2 Q12 c1 Q23c2 k 2V2 c2 Q21c2
dt
dc
Steady-state:
0
dt
a11c1 a12c2 W1
and
a11 Q12 k1V1
a21 Q12
a22 Q21 k 2V2 Q23
a12 Q21
CE 498/698 and ERS 685
(Spring 2004)
a21c1 a22c2 W2
Lecture 6
4
system parameters unknowns loadings
a11c1 a12 c2 a13c3 W1
a21c1 a22 c2 a23c3 W2
A C W
a31c1 a32 c2 a33c3 W3
LINEAR ALGEBRAIC EQUATIONS
Matrix algebra
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
5
a11c1 a12 c2 a13c3 W1
a21c1 a22 c2 a23c3 W2
A C W
a31c1 a32 c2 a33c3 W3
a11
A a21
a31
a12
a22
a32
CE 498/698 and ERS 685
(Spring 2004)
a13
a23
a33
c1
C c2
c
3
Lecture 6
W1
W W2
W
3
6
Gauss-Jordan method
To compute the matrix inverse
Identity matrix
1 0 0
33 identity matrix: I 0 1 0
0 0 1
I C C
augmented matrix:
CE 498/698 and ERS 685
(Spring 2004)
a11
a
21
a31
Lecture 6
a12
a22
a32
a13 1 0 0
a23 0 1 0
a33 0 0 1
7
Gauss-Jordan method
To compute the matrix inverse
1) Normalize
2) Elimination
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
8
Gauss-Jordan method
1) Normalize
a11
a
21
a31
1
a21
a31
a22
a32
a13 1 0 0
a23 0 1 0
a33 0 0 1
a12
a11
a22
a32
a13
a11
a23
a33
a12
CE 498/698 and ERS 685
(Spring 2004)
1
a11
0
0
Divide by a11
0 0
1 0
0 1
Lecture 6
9
Gauss-Jordan method
2) Elimination
1
a21
a31
a12
a11
a22
a32
a13
a11
a23
a33
1
a11
0
0
1
a21 a21
a31
CE 498/698 and ERS 685
(Spring 2004)
0 0
1 0
0 1
a21
a12
a11
a
a22 12 a21
a11
a32
Lecture 6
a12
a21
a11
a13
1
a21
a21 0 0
a11
a11
a13
1
0 0
a11
a11
a
1
a23 13 a21 a21 1 0
a11
a11
a33
0
0 1
10
Gauss-Jordan method
2) Elimination
a12
1
a11
0 a a12 a
22
21
a11
a12
a31
0 a32
a11
CE 498/698 and ERS 685
(Spring 2004)
a13
1
0 0
a11
a11
a13
1
a23 a21 a21 1 0
a11
a11
a13
1
a33 a31 a31 0 1
a11
a11
Lecture 6
11
Gauss-Jordan method
1) Normalization
a12
1
a11
0 a a12 a
22
21
a11
a12
a31
0 a32
a11
CE 498/698 and ERS 685
(Spring 2004)
a13
1
0 0
a11
a11
a13
1
a23 a21 a21 1 0
a11
a11
a13
1
a33 a31 a31 0 1
a11
a11
Lecture 6
12
Gauss-Jordan method
1 0 0 a111
1
0
1
0
a
21
1
0 0 1 a31
a121
1
a22
1
a32
a131
1
a23
1
a33
Matrix inverse
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
13
Gauss-Jordan method
example
3c1 0.1c2 0.2c3 7.85
0.1c1 7c2 0.3c3 19.3
0.3c1 0.2c2 10c3 71.4
augmented
matrix
CE 498/698 and ERS 685
(Spring 2004)
3 0.1 0.2 1 0 0
0. 1
7
0. 3 0 1 0
0.3 0.2 10 0 0 1
Lecture 6
14
Gauss-Jordan method
example
Divide by 3
(normalize)
3 0.1 0.2 1 0 0
0. 1
7
0. 3 0 1 0
0.3 0.2 10 0 0 1
1 0.033 0.067 0.333 0 0
0 .1
7
0 .3
0
1 0
10
0
0 1
0.3 0.2
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
15
Gauss-Jordan method
example
1 0.033 0.067 0.333 0 0
0 .1
7
0 .3
0
1 0
10
0
0 1
0.3 0.2
Divide by 7.003
(normalize)
CE 498/698 and ERS 685
(Spring 2004)
1 0.033 0.067 0.333 0 0
0 7.003 0.293 0.033 1 0
0 0.190 10.020 0.100 0 1
Lecture 6
16
Gauss-Jordan method
example
0
0
1 0.033 0.067 0.333
0
1
0.042 0.005 0.143 0
0
1
0 0.190 10.020 0.100
1 0 0.068 0.333 0.005 0
0 1 0.042 0.005 0.143 0
0 0 10.012 0.101 0.027 1
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
17
Gauss-Jordan method
example
1 0 0 0.332 0.005 0.007
0 1 0 0.005 0.143 0.004
0 0 1 0.010 0.003 0.100
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
18
Gauss-Jordan method
• Can also be used to solve for
concentrations
a11
a
21
a31
a12
a22
a32
CE 498/698 and ERS 685
(Spring 2004)
a13 W1
a23 W2
a33 W3
1 0 0 c1
0 1 0 c
2
0 0 1 c3
Lecture 6
19
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
20
A C W
We want to solve for {C}
A A1C A1W
Multiply both sides by [A]-1
A A1 I
I C C
Definitions of identity matrix
C A1W
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
21
Homework Problem 6.2(a)
• Use both Gauss-Jordan method and
Excel MINVERSE function
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
22
{C} = response
{W} = forcing functions
[A]-1 = parameters
{response} =[interactions]{forcing functions}
Unit change in loading of reactor 2
c1 a111W1 a121W2 a131W3
Response of reactor 1
1
1
1
c2 a21
W1 a22
W2 a23
W3
1
1
c3 a31
W1 a32
W2 a331W3
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
23
Matrix Multiplication (Box 6.1)
# columns in matrix 1 = # rows in matrix 2
a11
AB a21
a31
a12
a22
a32
a13 b11 b12
a23 b21 b22
a33 b31 b32
a11b11 a12b21 a13b31
AB a21b11 a22b21 a23b31
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
a11b12 a12b22 a13b32
24
Terminology
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 6
25
Time-variable response for
two reactors
dc1
V1
W1 Q12 c1 k1V1c1 Q21c2
dt
dc2
V2
W2 Q12 c1 Q23c2 k 2V2 c2 Q21c2
dt
Q12
dc1
11
k1
11c1 12 c2
V1
dt
where
dc2
Q21
21c1 22 c2
12
dt
V
1
CE 498/698 and ERS 685
(Spring 2004)
Lecture 6
Q12
21
V2
Q23 Q12
22
k2
V2
26
Time-variable response for
two reactors
General solution if c1=c10 at t = 0
c1 c1 f e
f t
c2 c2 f e
where
c1s e t
f t
s
c2 s e t
s
’s are functions of ’s
c’s are coefficients that depend on
eigenvalues and initial concentrations
f = fast eigenvalue
s = slow eigenvalue
CE 498/698 and ERS 685
(Spring 2004)
f >>s
Lecture 6
see formulas
on page 111
27