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© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2: Waste Sludge Incineration:
Steady-State Nonlinear DR and Detection of Gross Errors Through Analysis
*Flowsheet taken from Felder & Rousseau “Elementary Principles of Chemical Processes”, page 502.
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
Dryer Incinerator
Boiler
Sludge F1
Sat. Vap. F2
Conc. Sludge F3 Waste Gas F14
Sat. Liq. F4
Sat. Vap. F5
(p=4 bar)
Waste Gas F8
Boiler Feed Water F6
#6 Fuel Oil F7
Air F10
Cool Water F16Hot Water F15
Preheated Air F9
Natural Gas F11
Hot Water F17
Cool Water F18
Air F13
Preheated Air F12
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
Incinerator Problem Engineering Problem
Accounting ProblemManagement Problem
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
Stream # Flow Rate (kg/day)
Temperature (ºC)
Mass Fraction Solid
1 23800 23 0.36
2 17100 N/A N/A
3 ? N/A ?
4 ? N/A N/A
5 ? N/A N/A
6 30300 N/A N/A
7 3500 N/A N/A
8 ? N/A N/A
9 ? 127 N/A
Table 2.1a: List of measured and unmeasured variables
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
Stream # Flow Rate (kg/day)
Temperature (ºC)
Mass Fraction Solid
10 60000 24 N/A
11 1000 N/A N/A
12 ? 124 N/A
13 160000 26 N/A
14 ? N/A N/A
15 14600 135 N/A
16 14500 35 N/A
17 35600 135 N/A
18 35700 39 N/A
Table 2.1b: List of measured and unmeasured variables
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
Gross Errors?
Need to change the problem from
NONLINEAR to BILINEAR!!
See the end of this module.
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
#6 Fuel Oil:
87% C, 10% H, 0.84% S, and 2.16% inert (weight percent)
HHV = 3.75 x 104 kJ/kg
Natural Gas:
90% CH4, 10% C2H6 (mole percent)
Boiler:
Efficiency = 62%
25% Excess Air
Sludge:
Cp of solids = 2.5 kJ/kg·ºC
Liquid ~ Water
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
Dryer:Efficiency = 55%Pressure at 1 bar on sludge side, and 4 bar on steam sideAll steam and condensate flows are saturated
Incinerator:Sludge must enter at higher than 75% consistencyHHV of concentrated sludge = 19000 kJ/kg dry solids195 SCM natural gas/tonne wet sludge2.5 SCM air/10000 kJ of sludge HHV for complete combustion100% Excess Air for sludge and natural gas
Standard Deviation:Flows 500 kg/day, Temperature 2ºC, Composition
0.03
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
1) Define the y and z matrices for the measured variables and unmeasured variables.
2) Set up the V matrix, scaling down if necessary.3) Fill in the matrix with the measured values and the matrix with
guesses for the unmeasured variables, scaling down appropriately.4) Determine the mass, component, and heat balances for the process, f(
, ) = 0, scaling down parameters where necessary.
5) Determine the Jacobian matrices, Ay and Az, and solve them using the values from the and matrices.
6) Calculate b0 = Ay + Az – f ( , )
7) Carry out QR Factorization of the Az matrix, separating the Q matrix into Q1 (m x n) and Q2 (m x m-n) matrices, and the R matrix into an R1 (n x n) matrix.
0y
0y
0y
y
0y
0z
0z
0z
0z
z
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
8) Calculate = – V(Q2TAy)T[(Q2
TAy)V(Q2TAy)T]-1(Q2
TAy – Q2Tb0)
9) Calculate = R1-1Q1
Tb0 – R1-1Q1
TAy - R1-1R2
(the last term is not necessary if R2 is a zero matrix)
10) Replace , , and b0 with , , and b1.
11) Repeat steps 5 to 10 until the difference between both and , and and are very small.
0y
0y
0y
0z
1y
1y
1y
1z
1z
ˆN rz
ˆny1
ˆny
ˆnz
1nz
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 1:
1 1
2 2
3 6
4 7
5 10
6 11
7 13
8 15
9 16
10 17
11 18
12 1
13 1
14 9
15 10
16 12
17 13
18 15
19 16
20 17
21 18
y F
y F
y F
y F
y F
y F
y F
y F
y F
y F
y y F
y X
y T
y T
y T
y T
y T
y T
y T
y T
y T
,
1 3
2 4
3 5
4 8
5 9
6 12
7 14
8 1
9 3
10 3
z F
z F
z F
z F
z Fz
z F
z F
z y
z y
z x
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 2:
V(1,1), V(2,2), V(3,3), … , V(11,11) = 0.000025 (after scaling)
V(12,12) = 0.0009 (not necessary to scale)
V(13,13), V(14,14), … , V(21,21) = 0.0004 (after scaling)
Assumed that Covariance = 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 3:
0
0.238
0.171
0.303
0.035
0.600
0.010
1.600
0.146
0.145
0.356
ˆ 0.357
0.36
0.23
1.27
0.24
1.24
0.26
1.35
0.35
1.35
0.39
y
, 0
0.11
0.30
0.30
0.64
0.60ˆ
1.60
1.73
0.64
0.22
0.78
z
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:F3 + F11 + F12 - F14 = 0
F4 - F5 = 0
F15 – F16 = 0
F5 - F6 = 0
Dryer Incinerator
Boiler
Sludge F1Sat. Vap. F2
Conc. Sludge F3 Waste Gas F14
Sat. Liq. F4Sat. Vap. F5
(p=4 bar)
Waste Gas F8
Boiler Feed Water F6
#6 Fuel Oil F7
Air F10
Cool Water F16Hot Water F15
Preheated Air F9
Natural Gas F11
Hot Water F17
Cool Water F18
Air F13
Preheated Air F12
F12 - F13 = 0
F9 - F10 = 0
F7 + F9 - F8 = 0
F17 – F18 = 0
Mass Balances:
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:
Component Balances:
DryerSludge F1
Sat. Vap. F2
Conc. Sludge F3
Solids:
F1x1 - F3x3 = 0
Water:
F1y1 - F3y3 - F2 = 0
Normalization Constraints:
x3 + y3 - 1 = 0
x1 + y1 - 1 = 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:
Balances on Heat Exchangers:
Air F10
Cool Water F16Hot Water F15
Preheated Air F9
Hot Water F17
Cool Water F18
Air F13
Preheated Air F12
1.046F9T9 – 1.046F10T10 + 4.18F16T16 – 4.18F15T15 = 0
1.046F12T12 – 1.046F13T13 + 4.18F18T18 – 4.18F17T17 = 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:
Heat Flow Through Boiler:
(0.62)(HHVOIL)(F7) = (h5)(F5) – (h6)(F6)
23250F7 – 2737.6F5 + 83.9F6 = 0
2.325F7 – 0.27376F5 + 0.00839F6 = 0
Boiler
Sat. Vap. F5
(p=4 bar)
Waste Gas F8
Boiler Feed Water F6
#6 Fuel Oil F7
Preheated Air F9
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:
Air / Oil Ratio in Boiler:
Boiler
Sat. Vap. F5
(p=4 bar)
Waste Gas F8
Boiler Feed Water F6
#6 Fuel Oil F7
Preheated Air F9
For 1 kg of oil:
0.87 kg C 0.0724 kmoles C 0.10 kg H 0.0992 kmoles H 0.0084 kg S 0.000262 kmoles S
O2 required (25% excess):
For C (0.0724)(1.25)(1) For H (0.0992)(1.25)(0.25) For S (0.000262)(1.25)(1)
Total 0.1218 kmoles O2
16.83F7 – F9 = 0
0.5801 kmoles Air
16.83 kg Air
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:
Heat Flow Through Dryer:DryerSludge F1Sat. Vap. F2
Conc. Sludge F3
Sat. Liq. F4Sat. Vap. F5
(p=4 bar)
(x1)(F1)(Cps)(TB-T1) + (y1)(F1)(CpH2O)(TB-T1) + (F2)(LvH2O) – (0.55)(h5-h4)(F5) = 0
250x1F1 – 2.5x1F1T1 + 418y1F1 – 4.18y1F1T1 + 2257F2 – 1173.1F5 = 0
0.25x1F1 – 0.0025x1F1T1 + 0.418y1F1 – 0.00418y1F1T1 + 2.257F2 – 1.1731F5 = 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:
NG / Sludge Ratio:IncineratorConc. Sludge F3
Natural Gas F11
195 SCM/tonne wet sludge
8.7054 kmol NG/tonne F3
7.8348 kmol CH4 125.67 kg 0.1257 tonnes
0.8706 kmol C2H6 26.18 kg 0.0262 tonnes
0.1519 tonnes NG/tonne F3
0.1519F3 – F11 = 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:
Air / Sludge + NG Ratio:
Therefore for 1 kg of NG: 0.8275 kg CH4 0.0516 kmoles CH4 0.1725 kg C2H6 0.0057 kmoles C2H6
O2 required (100% excess):
For CH4 (0.0516)(2)(2) For C2H6 (0.0057)(2)(7/2)
Total 0.2464 kmoles O2
1.1728 kmoles Air
32.01 kg Air/kg NG
0.9 kmole CH4 14.44 kg (82.75%) 0.1 kmole C2H6 3.01 kg (17.25%)
Since NG is 10 mol% C2H6 and 90 mol% CH4:
IncineratorConc. Sludge F3
Natural Gas F11
Preheated Air F12
To burn NG:
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:
Air / Sludge + NG Ratio:
IncineratorConc. Sludge F3
Natural Gas F11
Preheated Air F12
To burn sludge:
(2.5 SCM air/10000 kJ)(19000 kJ/kg solid) (x3 kg solid/kg sludge) = 4.75x3 SCM air/kg sludge
0.2121x3 kmol air/kg sludge
6.1496x3 kg air/kg sludge
100% Excess:
12.3x3 kg air/kg sludge
12.3F3x3 + 34.01F11 – F12 = 0
Combining the air needed for the both the sludge and the NG:
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:
Summary of Constraints:
1) 2.325y4 – 0.27376z3 + 0.00839y3 = 0
2) 0.25y12y1 – 0.0025y12y1y13 +0.418z8y1 – 0.00418z8y1y13 + 2.257y2 – 1.1731z3 = 0
3) y1y12 – z1z10 = 0
4) y1z8 – z1z9 – y2 = 0
5) z10 + z9 – 1 = 0
6) y12 + z8 – 1 = 0
7) z2 – z3 = 0
8) z1 + y6 + z6 – z7 = 0
9) z6 – y7 = 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 4:
Summary of Constraints:
10) z5 – y5 = 0
11) z3 – y3 = 0
12) y4 + z5 – z4 = 0
13) 16.83y4 – z5 = 0
14) 0.1519z1 – y6 = 0
15) 12.3z1z10 + 34.01y6 – z6 = 0
16) 1.046z5y14 – 1.046y5y15 + 4.18y9y19 – 4.18y8y18 = 0
17) 1.046z6y16 – 1.046y7y17 + 4.18y11y21 – 4.18y10y20 = 0
18) y8 – y9 = 0
19) y10 – y11 = 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 5:
0 0 0.0084 2.325 0 0 0 0 0 0 0
0.3567 2.257 0 0 0 0 0 0 0 0 0
0.36 0 0 0 0 0 0 0 0 0 0
0.64 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0
0 0 0 16.83 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 34.01
yA
0 0 0 0 0
0 0 0 0 0.251 0 0 5.643 1.463 0 0
0 0 0 0 0 0 0.272 0 0 5.643 1.6302
0 0 0 0 0 0 0 1 1 0 0
0 0 0 0 0 0 0 0 0 1 1
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 5:
0 0 0 0 0 0 0 0 0 0
0.0594 0.0009 0 0 0 0 0 0 0 0
0.238 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0.6276 0.6276 0 0 0.6103 0.6061 0 0
0 0 0
0 1.6736 1.6736 0 0 1.4881 1.4923
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 5:
0 0 0.2738 0 0 0 0 0 0 0
0 0 1.1731 0 0 0 0 0.0993 0 0
0.78 0 0 0 0 0 0 0 0 0.11
0.22 0 0 0 0 0 0 0.238 0.11 0
0 0 0 0 0 0 0 0 1 1
0 0 0 0 0 0 0 1 0 0
0 1 1 0 0 0 0 0 0 0
1 0 0 0 0 1 1 0 0 0
0 0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0
0 0 0 1 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0
0.1519 0 0 0 0 0 0 0 0 0
9.594 0 0 0 0 1
zA
0 0 0 1.353
0 0 0 0 1.3284 0 0 0 0 0
0 0 0 0 0 1.297 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 6:
0 0 0 0 0
0.0000
0.0847
0.0001
0.1281
1.0000
1.0000
0
0.0000
0
ˆ ˆ ˆ ˆ( , ) 0
0
0.0000
0
0
1.0553
0.0347
0.2132
0
0
y zb A y A z f y z
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 7:
1
0 0 0.1749 0 0 0.0000 0.0000 0.0126 0.0003 0.0029
0 0 0.7493 0 0 0.0000 0.0000 0.0423 0.0011 0.0099
0.0806 0 0.0000 0 0 0.0362 0.0343 0.0006 0.0003 0.0027
0.0227 0 0.0000 0 0 0.0102 0.0097 0.2309 0.1035 0.9554
0 0 0 0 0 0 0 0 0.994
Q
3 0.1051
0 0 0 0 0 0 0.0000 0.9709 0.0245 0.2264
0 1.0000 0 0 0 0 0 0 0 0
0.1033 0 0.0000 0 0 0.5532 0.8266 0.0000 0.0000 0.0000
0 0 0 0 0 0.5067 0.3391 0.0020 0.0009 0.0090
0 0 0 0 0.5154 0 0 0 0 0.0000
0 0 0.6387 0 0 0.0000 0.0000 0.0461 0.0012 0
.0108
0 0 0 1.0000 0 0 0 0 0 0
0 0 0 0 0.5154 0 0 0 0 0.0000
0.0157 0 0.0000 0 0 0.0071 0.0067 0.0001 0.0001 0.1547
0.9910 0 0.0000 0 0 0.0610 0.0830 0.0052 0.0024 0.0241
0 0 0 0 0.6847 0 0 0 0 0.0000
0 0 0 0 0 0.6573 0.4399 0.0025 0.0011 0.0117
0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 7:
2
0.1473 0.0977 0.0648 0.0024 0.9469 0.0860 0.1728 0 0
0.4942 0.2036 0.1180 0.0219 0.1140 0.1567 0.3147 0 0
0.5864 0.3539 0.2125 0.0438 0.2713 0.2823 0.5670 0 0
0.0001 0.0211 0.0017 0.1485 0.0021 0.0023 0.0014 0 0
0.0000 0.
Q
0023 0.0002 0.0163 0.0002 0.0003 0.0002 0 0
0.0490 0.0252 0.0113 0.0332 0.0108 0.0150 0.0309 0 0
0 0 0 0 0 0 0 0 0
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0
0.0220 0.3567 0.2968 0.0413 0.0041 0.3942 0.5051 0 0
0.0080 0.
6453 0.3305 0.1029 0.0082 0.4391 0.0725 0 0
0.6201 0.2121 0.1206 0.0263 0.1255 0.1603 0.3218 0 0
0 0 0 0 0 0 0 0 0
0.0029 0.2334 0.7578 0.0372 0.0030 0.3217 0.0262 0 0
0.0007 0.1389 0.0114 0.9777 0.0136 0.0151 0.0090 0 0
0.0477 0.030
5 0.0171 0.0085 0.0222 0.0228 0.0460 0 0
0.0038 0.3100 0.3217 0.0494 0.0039 0.5727 0.0348 0 0
0.0198 0.2515 0.2156 0.0384 0.0140 0.2864 0.4249 0 0
0 0 0 0 0 0 0 1.0000 0
0 0 0 0 0 0 0 0 1.0000
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 7:
1
9.6812 0 0 0 0 0.8877 0.1033 0.0054 0.0025 1.3497
0 1.0000 1.0000 0 0 0 0 0 0 0
0 0 1.5656 0 0 0.0000 0.0000 0.0744 0.0000 0.0000
0 0 0 1.0000 1.0000 0 0 0 0 0
0 0 0 0 1.9403 0 0 0 0 0
0 0 0 0 0 1.9734 0.5532 0.0024 0.0011 0.0785
0 0 0 0 0 0 0.8266 0.0
R
023 0.0011 0.1161
0 0 0 0 0 0 0 1.0300 0.0254 0.0072
0 0 0 0 0 0 0 0 1.0057 0.9911
0 0 0 0 0 0 0 0 0 0.1380
2
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
R
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 8:
1
1 0 2 2 2 2 0 2 0
0.2293
0.1277
0.3147
0.0359
0.6045
0.0154
1.5988
0.1482
0.1482
0.3590
ˆ ˆ ˆ( ) [( ) ( ) ] ( ) 0.3590
0.3798
0.2303
1.2570
0.2530
1.2069
0.2931
1.3627
0.3374
1.3794
0.3605
T T T T TT T
y y y yy y V Q A Q A V Q A Q A y Q b
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Step 9:
1 1
1 1 1 0 1 1 1
0.1017
0.3147
0.3147
0.6404
0.6045ˆ ˆ
1.5988
1.7159
0.6202
0.1475
0.8525
T T
yz R Q b R Q A y
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Steps 10 & 11:
0.2294
0.1277
0.3147
0.0359
0.6045
0.0155
1.5988
0.1482
0.1482
0.3590
ˆ 0.3590
0.3803
0.2303
1.2571
0.2529
1.2070
0.2930
1.3626
0.3375
1.3793
0.3606
y
0.1017
0.3147
0.3147
0.6404
0.6045ˆ
1.5988
1.7160
0.6197
0.1424
0.8576
z
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Results:
VariablesRaw
MeasurementReconciled
MeasurementAdjustment
(%)F1 23800 22940 -3.61
F2 17100 12770 -25.32
F3 ? 10170 N/A
F4 ? 31470 N/A
F5 ? 31470 N/A
F6 30300 31470 3.86
F7 3500 3590 2.57
F8 ? 64040 N/A
F9 ? 60450 N/A
F10 60000 60450 0.75
Table 2.2a: Measured and reconciled data
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Results:
VariablesRaw
MeasurementReconciled
MeasurementAdjustment
(%)F11 1000 1550 55.00
F12 ? 159880 N/A
F13 160000 159880 -0.08
F14 ? 171600 N/A
F15 14600 14820 1.51
F16 14500 14820 2.21
F17 35600 35900 0.84
F18 35700 35900 0.56
x1 0.36 0.3803 5.64
x3 ? 0.8576 N/A
Table 2.2b: Measured and reconciled data
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Results:
VariablesRaw
MeasurementReconciled
MeasurementAdjustment
(%)y1 ? 0.6197 N/A
y3 ? 0.1424 N/A
T1 23 23.03 0.13
T9 127 125.71 -1.02
T10 24 25.29 5.38
T12 124 120.70 -2.66
T13 26 29.30 12.69
T15 135 136.26 0.93
T16 35 33.75 -3.57
T17 135 137.93 2.17
T18 39 36.06 -7.54
Table 2.2c: Measured and reconciled data
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Interpretation:
Dryer IncineratorF1 = 23800
F2 = 17100
F3 = 23800 – 17100 = 6700
F11 = 6700(0.1519) = 1018
Raw Measurements:
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Interpretation:
Dryer IncineratorF1 = 22940
F2 = 12770
F3 = 22940 – 12770 = 10170
F11 = 10170(0.1519) = 1545
!Reconciled Measurements:
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Interpretation:
Control natural gas feed based on
incinerator temperature
Incorrect measurement of vapor flow from dryer (F2)
Insufficient natural gas fed to incinerator (F11)
Incinerator temperature too low
Install more measurement devices
throughout the process
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
MATLAB code used to solve Case Study #1:
f=[0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0];V=[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0];
for I=1:11,V(I,I)=0.005^2;end;
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
for I=12,V(I,I)=0.03^2;end;
for I=13:21,V(I,I)=0.02^2;end;
y=[0.238; 0.171; 0.303; 0.035; 0.600; 0.010; 1.600; 0.146; 0.145; 0.356; 0.357; 0.36; 0.23; 1.27; 0.24; 1.24; 0.26; 1.35; 0.35; 1.35; 0.39];
z=[0.11; 0.3; 0.3; 0.64; 0.6; 1.6; 1.73; 0.64; 0.22; 0.78];
flag=1;
while flag>0,SSEy=0;SSEz=0;
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
Ay=[0 0 0.00839 2.3250 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0.25*y(12)-0.0025*y(12)*y(13) +0.418*z(8)-0.00418*z(8)*y(13) 2.257 0 0 0 0 0 0 0 0 0 0.25*y(1)-0.0025*y(1)*y(13) -0.0025*y(12)*y(1)-0.00418*z(8)*y(1) 0 0 0 0 0 0 0 0;y(12) 0 0 0 0 0 0 0 0 0 0 y(1) 0 0 0 0 0 0 0 0 0;z(8) -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 16.83 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 34.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 -1.046*y(15) 0 0 -4.18 *y(18) 4.18*y(19) 0 0 0 0 1.046*z(5) -1.046*y(5) 0 0 -4.18*y(8) 4.18*y(9) 0 0;0 0 0 0 0 0 -1.046*y(17) 0 0 -4.18*y(20) 4.18*y(21) 0 0 0 0 1.046*z(6) -1.046*y(7) 0 0 -4.18*y(10) 4.18*y(11);0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0];
Az=[0 0 -0.27376 0 0 0 0 0 0 0; 0 0 -1.1731 0 0 0 0 0.418*y(1)-0.00418*y(1)*y(13) 0 0; -z(10) 0 0 0 0 0 0 0 0 -z(1); -z(9) 0 0 0 0 0 0 y(1) -z(1) 0; 0 0 0 0 0 0 0 0 1 1; 0 0 0 0 0 0 0 1 0 0; 0 1 -1 0 0 0 0 0 0 0; 1 0 0 0 0 1 -1 0 0 0; 0 0 0 0 0 1 0 0 0 0; 0 0 0 0 1 0 0 0 0 0; 0 0 1 0 0 0 0 0 0 0; 0 0 0 -1 1 0 0 0 0 0; 0 0 0 0 -1 0 0 0 0 0; 0.1519 0 0 0 0 0 0 0 0 0; 12.3*z(10) 0 0 0 0 -1 0 0 0 12.3*z(1); 0 0 0 0 1.046*y(14) 0 0 0 0 0; 0 0 0 0 0 1.046*y(16) 0 0 0 0;0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0];
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
f(1)=2.3250*y(4)-0.27376*z(3)+0.00839*y(3);f(2)=0.25*y(12)*y(1)-0.0025*y(12)*y(1)*y(13)+0.418*z(8)*y(1)-0.00418*z(8)*y(1)*y(13) +2.257*y(2)-
1.1731*z(3);f(3)=y(1)*y(12)-z(1)*z(10);f(4)=y(1)*z(8)-z(1)*z(9)-y(2);f(5)=z(10)+z(9)-1;f(6)=y(12)+z(8)-1;f(7)=z(2)-z(3);f(8)=z(1)+y(6)+z(6)-z(7);f(9)=z(6)-y(7);f(10)=z(5)-y(5);f(11)=z(3)-y(3);f(12)=y(4)+z(5)-z(4);f(13)=16.83*y(4)-z(5);f(14)=0.1519*z(1)-y(6);f(15)=12.3*z(1)*z(10)+34.01*y(6)-z(6);f(16)=1.046*z(5)*y(14)-1.046*y(5)*y(15)+4.18*y(9)*y(19)-4.18*y(8)*y(18);
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
f(17)=1.046*z(6)*y(16)-1.046*y(7)*y(17)+4.18*y(11)*y(21)-4.18*y(10)*y(20);f(18)=y(8)-y(9);f(19)=y(10)-y(11);b=Ay*y+Az*z-f;[Q,R]=qr(Az); for I=1:19,for J=1:10,Q1(I,J)=Q(I,J);end;end; for I=1:19,for J=11:19,Q2(I,J-10)=Q(I,J);end;end;
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
for I=1:10,for J=1:10,R1(I,J)=R(I,J);end;end; yhat=y-V*(Q2'*Ay)'*inv((Q2'*Ay)*V*(Q2'*Ay)')*(Q2'*Ay*y-Q2'*b);zhat=inv(R1)*Q1'*b-inv(R1)*Q1'*Ay*yhat; for I=1:21,SSEy=SSEy+(y(I)-yhat(I))^2;end; for I=1:10,SSEz=SSEz+(z(I)-zhat(I))^2;end;
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
if and(SSEy<1.0e-6,SSEz<1.0e-6),
flag=-1;
end;
y=yhat;
z=zhat;
end
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2Bilinear Method with Gross Error Detection
y12 = x1F1 = 0.0857
y13 = T1F1 = 0.0547
y14 = T10F10 = 0.1440
y15 = T13F13 = 0.4160
y16 = T15F15 = 0.1971
z8 = y1F1
z9 = y3F3
z10 = x3F3
z11 = T9F9
z12 = T12F12
z13 = T1F1y1
y17 = T16F16 = 0.0507
y18 = T17F17 = 0.4806
y19 = T18F18 = 0.1392
y20 = T1F1x1 = 0.0197
Altered Bilinear Variables:
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
1) 2.325y4 – 0.27376z3 + 0.00839y3 = 0
2) 0.25y12 – 0.0025y20 + 0.418z8 – 0.00418z13 + 2.257y2 – 1.1731z3 = 0
3) y12 – z10 = 0
4) z8 – z9 – y2 = 0
5) z10 + z9 – z1 = 0
6) y12 + z8 – y1 = 0
7) z2 – z3 = 0
8) z1 + y6 + z6 – z7 = 0
9) z6 – y7 = 0
Altered Bilinear Constraints:
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2Altered Bilinear Constraints:
10) z5 – y5 = 0
11) z3 – y3 = 0
12) y4 + z5 – z4 = 0
13) 16.83y4 – z5 = 0
14) 0.1519z1 – y6 = 0
15) 12.3z10 + 34.01y6 – z6 = 0
16) 1.046z11 – 1.046y14 + 4.18y17 – 4.18y16 = 0
17) 1.046z12 – 1.046y15 + 4.18y19 – 4.18y18 = 0
18) y8 – y9 = 0
19) y10 – y11 = 0
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2Variables
Raw Measurement
Reconciled Measurement
Adjustment (%)
F1 23800 24370 2.39
F2 17100 16530 -3.33
F3 ? 7840 N/A
F4 ? 31030 N/A
F5 ? 31030 N/A
F6 30300 31030 2.41
F7 3500 3540 1.14
F8 ? 63160 N/A
F9 ? 59620 N/A
F10 60000 59620 -0.63
Table 2.3a: Measured and reconciled data using a bilinear approach
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
VariablesRaw
MeasurementReconciled
MeasurementAdjustment
(%)F11 1000 1190 19.00
F12 ? 159880 N/A
F13 160000 159880 -0.13
F14 ? 168910 N/A
F15 14600 14550 -0.34
F16 14500 14550 0.34
F17 35600 35650 0.14
F18 35700 35650 -0.14
x1 0.36 0.3984 10.67
x3 ? 1.2387 N/A
Case Study #2
Table 2.3b: Measured and reconciled data using a bilinear approach
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
VariablesRaw
MeasurementReconciled
MeasurementAdjustment
(%)y1 ? 0.6016 N/A
y3 ? -0.2387 N/A
T1 23 22.45 -2.39
T9 127 122.29 -3.71
T10 24 24.15 0.62
T12 124 111.35 -10.20
T13 26 26.02 0.08
T15 135 135.46 0.34
T16 35 34.85 -0.43
T17 135 134.81 -0.14
T18 39 39.05 0.13
Case Study #2
Table 2.3c: Measured and reconciled data using a bilinear approach
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
592.12)6(0463.12 95.02
NO GROSS ERROR?!?
Need to impose inequality constraints on mass fractions
IMPOSSIBLE WITH BILINEAR DR!
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
0349.0mb
Calculating the bias in measurement y2:
Adjusting the measurement:
y2 = 0.171 - 0.0349 = 0.1361
Faulty measurement is known from previous analysis(bilinear DR can’t properly detect error).
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2Variables
Raw Measurement
Reconciled Measurement
F1 23800 23800
F2 17100 13610
F3 ? 10190
F4 ? 31030
F5 ? 31030
F6 30300 31030
F7 3500 3540
F8 ? 63160
F9 ? 59620
F10 60000 59620
Table 2.4a: Bilinear reconciliation without bias
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2Variables
Raw Measurement
Reconciled Measurement
F11 1000 1550
F12 ? 159980
F13 160000 159980
F14 ? 171720
F15 14600 14550
F16 14500 14550
F17 35600 35650
F18 35700 35650
x1 0.36 0.3667
x3 ? 0.8566
Table 2.4b: Bilinear reconciliation without bias
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2Variables
Raw Measurement
Reconciled Measurement
y1 ? 0.6333
y3 ? 0.1434
T1 23 22.98
T9 127 122.29
T10 24 24.15
T12 124 111.28
T13 26 26.00
T15 135 135.46
T16 35 34.85
T17 135 134.81
T18 39 39.05
Table 2.4c: Bilinear reconciliation without bias
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
NONLINEAR DR!!!
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2
y=[0.238; 0.171; 0.303; 0.035; 0.600; 0.010; 1.600; 0.146; 0.145; 0.356; 0.357; 0.0857; 0.0547; 0.144; 0.416; 0.1971; 0.0507; 0.4806; 0.1392; 0.0197];
V=[0.000025 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0.000025 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0.000025 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0.000025 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0.000025 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0.000025 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0.000025 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0.000025 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0.000025 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0.000025 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0.000025 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0.00002 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0.00002 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0.0001 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.001 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00005 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00001 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00001 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00005 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000004];
MATLAB code used for bilinear DR:
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2Ay=[0 0 0.00839 2.325 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 2.257 0 0 0 0 0 0 0 0 0 0.25 0 0 0 0 0 0 0 -0.0025;0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0;0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;-1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 16.83 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 34.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 -1.046 0 -4.18 4.18 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.046 0 0 -4.18 4.18 0;0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0];
Az=[0 0 -0.27376 0 0 0 0 0 0 0 0 0 0; 0 0 -1.1731 0 0 0 0 0.418 0 0 0 0 -0.00418;0 0 0 0 0 0 0 0 0 -1 0 0 0;0 0 0 0 0 0 0 1 -1 0 0 0 0;-1 0 0 0 0 0 0 0 1 1 0 0 0;0 0 0 0 0 0 0 1 0 0 0 0 0;0 1 -1 0 0 0 0 0 0 0 0 0 0;1 0 0 0 0 1 -1 0 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0 0 0 0;0 0 0 -1 1 0 0 0 0 0 0 0 0;0 0 0 0 -1 0 0 0 0 0 0 0 0;0.1519 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 -1 0 0 0 12.3 0 0 0;0 0 0 0 0 0 0 0 0 0 1.046 0 0;0 0 0 0 0 0 0 0 0 0 0 1.046 0;0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 0 0 0 0 0 0 0 0 0];
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2[Q,R]=qr(Az);
for I=1:19,
for J=1:13,
Q1(I,J)=Q(I,J);
end;
end;
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2for I=1:19,
for J=14:19,
Q2(I,J-13)=Q(I,J);
end;
end;
for I=1:13,
for J=1:13,
R1(I,J)=R(I,J);
end;
end;
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2G=Q2'*Ay;
yhat=y-V*G'*inv(G*V*G')*G*y;
zhat=-inv(Az'*Az)*Az'*(Ay*yhat);
yhat2=yhat;
yhat2(12)=yhat2(12)/yhat2(1);
yhat2(13)=yhat2(13)/yhat2(1);
yhat2(14)=yhat2(14)/yhat2(5);
yhat2(15)=yhat2(15)/yhat2(7);
yhat2(16)=yhat2(16)/yhat2(8);
yhat2(17)=yhat2(17)/yhat2(9);
yhat2(18)=yhat2(18)/yhat2(10);
yhat2(19)=yhat2(19)/yhat2(11);
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2zhat2=zhat;
zhat2(8)=zhat2(8)/yhat2(1);
zhat2(9)=zhat2(9)/zhat2(1);
zhat2(10)=zhat2(10)/zhat2(1);
zhat2(11)=zhat2(11)/zhat2(5);
zhat2(12)=zhat2(12)/zhat2(6);
Vr=G*V*G';
r=G*y;
tau=r'*inv(Vr)*r;
Id=[1 0 0 0 0 0;0 1 0 0 0 0;0 0 1 0 0 0;0 0 0 1 0 0;0 0 0 0 1 0;0 0 0 0 0 1];
© Universidad de Guanajuato, Mexico © University of Ottawa, Canada, 2004
Case Study #2for I=1:20,
for J=1:6,
Ac(J,1)=G(J,I);
end;
Vri=inv(Vr)*(Id-Ac*inv(Ac'*inv(Vr)*Ac)*Ac'*inv(Vr));
O(I)=r'*Vri*r;
end;
B=[0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0];
P=G*B;
m=inv(P'*inv(G*V*G')*P)*P'*inv(G*V*G')*G*y;