458.308 Process Control & Design
Jong Min Lee
School of Chemical and Biological Engineering Seoul National University
Lecture 3: Dynamic Simulation & Analysis (part 2)
0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time
Tank Level (h)
Linear Approximation
Exact Nonlinear Solution
Deviation Variables for linear/affine systems Heated Mixing Tank
When the volume, flow rate, and inlet temp. are constant and at their steady state values.
u
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dt = F ¯ i
V ¯ T ¯ i T + Q V ¯ ⇢C p
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dy
dt = F ¯ i
V ¯ y + 1
V ¯ ⇢C p u + F ¯ i T ¯ i V ¯
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Define
Perturbations from a steady-state operating condition
Q
<latexit sha1_base64="cmpZw1rFQUqHHTLoYcSFp4ZHF2I=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbRiyURQS9CwYvHBqwttKFstpt26WY37G4KJfSfePGgiFf/iTf/jds2B219MPB4b4aZeVHKmTae9+2U1tY3NrfK25Wd3b39A/fw6EnLTBHaJJJL1Y6wppwJ2jTMcNpOFcVJxGkrGt3P/NaYKs2keDSTlIYJHggWM4KNlXquG5yjOxSgS9SNsMqDac+tejVvDrRK/IJUoUCj5351+5JkCRWGcKx1x/dSE+ZYGUY4nVa6maYpJiM8oB1LBU6oDvP55VN0ZpU+iqWyJQyaq78ncpxoPUki25lgM9TL3kz8z+tkJr4NcybSzFBBFovijCMj0SwG1GeKEsMnlmCimL0VkSFWmBgbVsWG4C+/vEqermq+V/OD62q9XsRRhhM4hQvw4Qbq8AANaAKBMTzDK7w5ufPivDsfi9aSU8wcwx84nz/TfpHW</latexit><latexit sha1_base64="cmpZw1rFQUqHHTLoYcSFp4ZHF2I=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbRiyURQS9CwYvHBqwttKFstpt26WY37G4KJfSfePGgiFf/iTf/jds2B219MPB4b4aZeVHKmTae9+2U1tY3NrfK25Wd3b39A/fw6EnLTBHaJJJL1Y6wppwJ2jTMcNpOFcVJxGkrGt3P/NaYKs2keDSTlIYJHggWM4KNlXquG5yjOxSgS9SNsMqDac+tejVvDrRK/IJUoUCj5351+5JkCRWGcKx1x/dSE+ZYGUY4nVa6maYpJiM8oB1LBU6oDvP55VN0ZpU+iqWyJQyaq78ncpxoPUki25lgM9TL3kz8z+tkJr4NcybSzFBBFovijCMj0SwG1GeKEsMnlmCimL0VkSFWmBgbVsWG4C+/vEqermq+V/OD62q9XsRRhhM4hQvw4Qbq8AANaAKBMTzDK7w5ufPivDsfi9aSU8wcwx84nz/TfpHW</latexit><latexit sha1_base64="cmpZw1rFQUqHHTLoYcSFp4ZHF2I=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbRiyURQS9CwYvHBqwttKFstpt26WY37G4KJfSfePGgiFf/iTf/jds2B219MPB4b4aZeVHKmTae9+2U1tY3NrfK25Wd3b39A/fw6EnLTBHaJJJL1Y6wppwJ2jTMcNpOFcVJxGkrGt3P/NaYKs2keDSTlIYJHggWM4KNlXquG5yjOxSgS9SNsMqDac+tejVvDrRK/IJUoUCj5351+5JkCRWGcKx1x/dSE+ZYGUY4nVa6maYpJiM8oB1LBU6oDvP55VN0ZpU+iqWyJQyaq78ncpxoPUki25lgM9TL3kz8z+tkJr4NcybSzFBBFovijCMj0SwG1GeKEsMnlmCimL0VkSFWmBgbVsWG4C+/vEqermq+V/OD62q9XsRRhhM4hQvw4Qbq8AANaAKBMTzDK7w5ufPivDsfi9aSU8wcwx84nz/TfpHW</latexit><latexit sha1_base64="cmpZw1rFQUqHHTLoYcSFp4ZHF2I=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbRiyURQS9CwYvHBqwttKFstpt26WY37G4KJfSfePGgiFf/iTf/jds2B219MPB4b4aZeVHKmTae9+2U1tY3NrfK25Wd3b39A/fw6EnLTBHaJJJL1Y6wppwJ2jTMcNpOFcVJxGkrGt3P/NaYKs2keDSTlIYJHggWM4KNlXquG5yjOxSgS9SNsMqDac+tejVvDrRK/IJUoUCj5351+5JkCRWGcKx1x/dSE+ZYGUY4nVa6maYpJiM8oB1LBU6oDvP55VN0ZpU+iqWyJQyaq78ncpxoPUki25lgM9TL3kz8z+tkJr4NcybSzFBBFovijCMj0SwG1GeKEsMnlmCimL0VkSFWmBgbVsWG4C+/vEqermq+V/OD62q9XsRRhhM4hQvw4Qbq8AANaAKBMTzDK7w5ufPivDsfi9aSU8wcwx84nz/TfpHW</latexit>0 = Q Q ¯
<latexit sha1_base64="LOJjPOd90Lv31io6S0kN7n30aYw=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbRiyURQS9CwYvHCqkttKFstpt26e4m7G4KJfSfePGgiFf/iTf/jds2B219MPB4b4aZeVHKmTae9+2U1tY3NrfK25Wd3b39A/fw6EknmSK0SRKeqHaENeVM0qZhhtN2qigWEaetaHQ/81tjqjRLZGAmKQ0FHkgWM4KNlXquG5yjOxSgS9SNsMqDac+tejVvDrRK/IJUoUCj5351+wnJBJWGcKx1x/dSE+ZYGUY4nVa6maYpJiM8oB1LJRZUh/n88ik6s0ofxYmyJQ2aq78nciy0nojIdgpshnrZm4n/eZ3MxLdhzmSaGSrJYlGccWQSNIsB9ZmixPCJJZgoZm9FZIgVJsaGVbEh+Msvr5Knq5rv1fzH62q9XsRRhhM4hQvw4Qbq8AANaAKBMTzDK7w5ufPivDsfi9aSU8wcwx84nz/hcJHf</latexit><latexit sha1_base64="LOJjPOd90Lv31io6S0kN7n30aYw=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbRiyURQS9CwYvHCqkttKFstpt26e4m7G4KJfSfePGgiFf/iTf/jds2B219MPB4b4aZeVHKmTae9+2U1tY3NrfK25Wd3b39A/fw6EknmSK0SRKeqHaENeVM0qZhhtN2qigWEaetaHQ/81tjqjRLZGAmKQ0FHkgWM4KNlXquG5yjOxSgS9SNsMqDac+tejVvDrRK/IJUoUCj5351+wnJBJWGcKx1x/dSE+ZYGUY4nVa6maYpJiM8oB1LJRZUh/n88ik6s0ofxYmyJQ2aq78nciy0nojIdgpshnrZm4n/eZ3MxLdhzmSaGSrJYlGccWQSNIsB9ZmixPCJJZgoZm9FZIgVJsaGVbEh+Msvr5Knq5rv1fzH62q9XsRRhhM4hQvw4Qbq8AANaAKBMTzDK7w5ufPivDsfi9aSU8wcwx84nz/hcJHf</latexit><latexit sha1_base64="LOJjPOd90Lv31io6S0kN7n30aYw=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbRiyURQS9CwYvHCqkttKFstpt26e4m7G4KJfSfePGgiFf/iTf/jds2B219MPB4b4aZeVHKmTae9+2U1tY3NrfK25Wd3b39A/fw6EknmSK0SRKeqHaENeVM0qZhhtN2qigWEaetaHQ/81tjqjRLZGAmKQ0FHkgWM4KNlXquG5yjOxSgS9SNsMqDac+tejVvDrRK/IJUoUCj5351+wnJBJWGcKx1x/dSE+ZYGUY4nVa6maYpJiM8oB1LJRZUh/n88ik6s0ofxYmyJQ2aq78nciy0nojIdgpshnrZm4n/eZ3MxLdhzmSaGSrJYlGccWQSNIsB9ZmixPCJJZgoZm9FZIgVJsaGVbEh+Msvr5Knq5rv1fzH62q9XsRRhhM4hQvw4Qbq8AANaAKBMTzDK7w5ufPivDsfi9aSU8wcwx84nz/hcJHf</latexit><latexit sha1_base64="LOJjPOd90Lv31io6S0kN7n30aYw=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbRiyURQS9CwYvHCqkttKFstpt26e4m7G4KJfSfePGgiFf/iTf/jds2B219MPB4b4aZeVHKmTae9+2U1tY3NrfK25Wd3b39A/fw6EknmSK0SRKeqHaENeVM0qZhhtN2qigWEaetaHQ/81tjqjRLZGAmKQ0FHkgWM4KNlXquG5yjOxSgS9SNsMqDac+tejVvDrRK/IJUoUCj5351+wnJBJWGcKx1x/dSE+ZYGUY4nVa6maYpJiM8oB1LJRZUh/n88ik6s0ofxYmyJQ2aq78nciy0nojIdgpshnrZm4n/eZ3MxLdhzmSaGSrJYlGccWQSNIsB9ZmixPCJJZgoZm9FZIgVJsaGVbEh+Msvr5Knq5rv1fzH62q9XsRRhhM4hQvw4Qbq8AANaAKBMTzDK7w5ufPivDsfi9aSU8wcwx84nz/hcJHf</latexit>T 0 = T T ¯
1
© J.M. Lee 3
Step 1
At a S.S. 0 = d T ¯
dt = F ¯ i
V ¯ T ¯ i T ¯ + Q ¯ V ¯ ⇢C p
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2
1 2 d(T T ¯ )
dt = F ¯ i
V ¯ T T ¯ + (Q Q) ¯ V ¯ ⇢C p
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Step 2
dT 0
dt = F ¯ i
V ¯ T 0 + 1
V ¯ ⇢C p Q 0
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dy 0
dt = F ¯ i
V ¯ y 0 + 1
V ¯ ⇢C p u 0
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standard form of 1st-order process
⌧ p : time constant k p : process gain
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V ¯ F ¯ i
dy 0
dt + y 0 = 1
F ¯ i ⇢C p u 0
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Linearization of Nonlinear Models
M.B. and E.B. models are generally nonlinear, while commonly used control strategies are based on linear systems theory
Taylor Series Expansion
One state variable (no input) dx
dt = f (x)
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f is a nonlinear function
f (x) = f (¯ x) + @ f
@x x=¯ x · (x x) + ¯ · · ·
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dx
dt ⇡ f (¯ x) + @ f
@x x=¯ x (x x) ¯
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© J.M. Lee 5
Since dx
dt = d(x x) ¯ dt
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d(x x) ¯
dt ⇡ @ f
@ x x ¯ (x x) ¯
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dx 0
dt = ax 0
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One state and one input dx
dt = f (x, u)
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f (x, u) = f (¯ x, u) + ¯ @f
@ x u=¯ x=¯ x u (x x) + ¯ @ f
@ u x=¯ u=¯ x u (u u) + ¯ · · ·
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d(x x) ¯
dt ⇡ @ f
@ x ss (x x) + ¯ @ f
@ u ss (u u) ¯
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dx 0
dt = ax 0 + bu 0
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Two states and two inputs
dx 1
dt = f 1 (x 1 , x 2 , u 1 , u 2 ) dx 2
dt = f 2 (x 1 , x 2 , u 1 , u 2 )
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dx 0 1
dt = @ f 1
@x 1 ss x 0 1 + @ f 1
@ x 2 ss x 0 2 + @ f 1
@ u 1 ss u 0 1 + @ f 1
@ u 2 ss u 0 2
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dx 0 2
dt = @ f 2
@x 1 ss x 0 1 + @ f 2
@ x 2 ss x 0 2 + @ f 2
@ u 1 ss u 0 1 + @ f 2
@ u 2 ss u 0 2
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© J.M. Lee 7
For the CSTR model with a single, second-order reaction below, find a linear model that is valid near ( ¯
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dC A
dt = F
V (C A,i C A ) kC A 2
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k = 1 [L/mol · min]
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V
<latexit sha1_base64="GIS+kETyRPfmZWuTjjjyqFxF1lk=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSyCp5KIYC9CwYsHDxXsB6ShbLabdulmE3Ynaon1n3jxoIhXf4k3/43bNgdtfTDweG+GmXlBIrgGx/m2Ciura+sbxc3S1vbO7p5d3m/pOFWUNWksYtUJiGaCS9YEDoJ1EsVIFAjWDkaXU799x5TmsbyFccL8iAwkDzklYKSeXW7hC+w+Ya8L7AGy64nfsytO1ZkBLxM3JxWUo9Gzv7r9mKYRk0AF0dpznQT8jCjgVLBJqZtqlhA6IgPmGSpJxLSfzU6f4GOj9HEYK1MS8Ez9PZGRSOtxFJjOiMBQL3pT8T/PSyGs+RmXSQpM0vmiMBUYYjzNAfe5YhTE2BBCFTe3YjokilAwaZVMCO7iy8ukdVp1nap7c1ap1/I4iugQHaET5KJzVEdXqIGaiKJ79Ixe0Zv1aL1Y79bHvLVg5TMH6A+szx/SEpMK</latexit><latexit sha1_base64="GIS+kETyRPfmZWuTjjjyqFxF1lk=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSyCp5KIYC9CwYsHDxXsB6ShbLabdulmE3Ynaon1n3jxoIhXf4k3/43bNgdtfTDweG+GmXlBIrgGx/m2Ciura+sbxc3S1vbO7p5d3m/pOFWUNWksYtUJiGaCS9YEDoJ1EsVIFAjWDkaXU799x5TmsbyFccL8iAwkDzklYKSeXW7hC+w+Ya8L7AGy64nfsytO1ZkBLxM3JxWUo9Gzv7r9mKYRk0AF0dpznQT8jCjgVLBJqZtqlhA6IgPmGSpJxLSfzU6f4GOj9HEYK1MS8Ez9PZGRSOtxFJjOiMBQL3pT8T/PSyGs+RmXSQpM0vmiMBUYYjzNAfe5YhTE2BBCFTe3YjokilAwaZVMCO7iy8ukdVp1nap7c1ap1/I4iugQHaET5KJzVEdXqIGaiKJ79Ixe0Zv1aL1Y79bHvLVg5TMH6A+szx/SEpMK</latexit><latexit sha1_base64="GIS+kETyRPfmZWuTjjjyqFxF1lk=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSyCp5KIYC9CwYsHDxXsB6ShbLabdulmE3Ynaon1n3jxoIhXf4k3/43bNgdtfTDweG+GmXlBIrgGx/m2Ciura+sbxc3S1vbO7p5d3m/pOFWUNWksYtUJiGaCS9YEDoJ1EsVIFAjWDkaXU799x5TmsbyFccL8iAwkDzklYKSeXW7hC+w+Ya8L7AGy64nfsytO1ZkBLxM3JxWUo9Gzv7r9mKYRk0AF0dpznQT8jCjgVLBJqZtqlhA6IgPmGSpJxLSfzU6f4GOj9HEYK1MS8Ez9PZGRSOtxFJjOiMBQL3pT8T/PSyGs+RmXSQpM0vmiMBUYYjzNAfe5YhTE2BBCFTe3YjokilAwaZVMCO7iy8ukdVp1nap7c1ap1/I4iugQHaET5KJzVEdXqIGaiKJ79Ixe0Zv1aL1Y79bHvLVg5TMH6A+szx/SEpMK</latexit><latexit sha1_base64="GIS+kETyRPfmZWuTjjjyqFxF1lk=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSyCp5KIYC9CwYsHDxXsB6ShbLabdulmE3Ynaon1n3jxoIhXf4k3/43bNgdtfTDweG+GmXlBIrgGx/m2Ciura+sbxc3S1vbO7p5d3m/pOFWUNWksYtUJiGaCS9YEDoJ1EsVIFAjWDkaXU799x5TmsbyFccL8iAwkDzklYKSeXW7hC+w+Ya8L7AGy64nfsytO1ZkBLxM3JxWUo9Gzv7r9mKYRk0AF0dpznQT8jCjgVLBJqZtqlhA6IgPmGSpJxLSfzU6f4GOj9HEYK1MS8Ez9PZGRSOtxFJjOiMBQL3pT8T/PSyGs+RmXSQpM0vmiMBUYYjzNAfe5YhTE2BBCFTe3YjokilAwaZVMCO7iy8ukdVp1nap7c1ap1/I4iugQHaET5KJzVEdXqIGaiKJ79Ixe0Zv1aL1Y79bHvLVg5TMH6A+szx/SEpMK</latexit>= 1 [L] C A,i = 2 [mol/L]
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dC A
dt = dC A 0
dt = f ( ¯ C A , F ¯ ) + @ f
@ C A C ¯
A , F ¯
C A C ¯ A + @ f
@ F C ¯
A , F ¯
F F ¯
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0
@ f
@ C A C ¯
A , F ¯
= F
V 2kC A
ss
= 1
1 2 · 1 · 1 = 3 min 1
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@ f
@ F C ¯
A , F ¯
= C A,i C A
V ss = 2 1
1 = 1 mol L 2
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