Solving Index-2 Separation Problems with APMonitor

D. A. Harney

N. L. Book(a)

(a) Missouri University of Science & Technology

Differentiation Index

• Numerous definitions for index of differential-algebraic equation (tractability index, geometric index, kronecker index, strangeness index, etc.)

• Differentiation index of DAE is the number of times that all or part of the system must be differentiated with respect to time in order to reduce the system to its underlying ODE.

• Many chemical engineering problems can be written in the form

𝒙′ = 𝒇 𝒙 + 𝒃 𝒙 𝒚

𝟎 = 𝒈(𝒙)

Differentiation Index

• Differentiating the algebraic constraint:

𝟎 =𝜕𝒈

𝜕𝒙𝒇 𝒙 + 𝒃 𝒙 𝒚

• and a second time

𝟎 =𝜕𝒈

𝜕𝒙𝒇 𝒙

𝒙

+𝜕𝒈

𝜕𝒙𝒃 𝒙

𝒙

𝒚 𝒇 𝒙 + 𝒃 𝒙 𝒚

+𝜕𝒈

𝜕𝒙𝒃 𝒙 𝒚′

• System has differentiation index-2.

Reactive Flash

Dynamic representation of P-Q reactive flash

𝜏𝑑𝑥𝑖𝑑𝑡= 𝑧𝑖 − 𝑥𝑖 − 𝜙 𝐾𝑖𝑥𝑖 − 𝑥𝑖 − 𝜏 𝑥𝑖 𝜈𝑇,𝑗𝑟𝑗 −

𝑟

𝑗=1

𝜈𝑖,𝑗𝑟𝑗

𝑟

𝑗=1

,

𝑖 = 1. . 𝑛 − 1

𝜏𝐶𝑝𝑑𝑇

𝑑𝑡= ℎ𝑓𝑒𝑒𝑑 + 𝑞 − ℎ𝑙𝑖𝑞 − ℎ𝑙𝑖𝑞𝜏 𝜈𝑇,𝑗𝑟𝑗

𝑟

𝑗=1

+ 𝜙 ℎ𝑙𝑖𝑞 − ℎ𝑣𝑎𝑝

𝐾𝑛 1 − 𝑥𝑖

𝑛−1

𝑖=1

+ 𝐾𝑖𝑥𝑖

𝑛−1

𝑖=1

− 1 = 0

0 = 1 − 𝑙 − 𝜙 + 𝜏 𝜈𝑇,𝑗𝑟𝑗

𝑟

𝑗=1

Ethylene Glycol System

C2H4O + H2O r1 C2H6O2

C2H4O + C2H6O2 r2 C4H10O3

𝑟1 = 3.2022 𝑥 109 exp −

9360.845

𝑇𝑥𝐴𝑥𝐵

𝑟2 = 5.84 𝑥 109 exp −

9360.845

𝑇𝑥𝐴𝑥𝐶

Stability of Steady States

Theorem (Maerz): f(x) ε C2 on an open bounded region D, containing a stationary point x*, f(x*) = 0. Let the matrix pencil (λA – B) be regular with index 2 and all its generalized eigenvalues have negative real parts. Additionally, let the DAE be in Hessenberg form of size 2. Then the DAE is Lyapunov stable at this stationary point.

Hessenberg form of size 2:

𝒙′ = 𝒇 𝒙, 𝒚 𝟎 = 𝒈(𝒙)

(𝜕𝒈/𝜕𝒙) (𝜕𝒇/𝜕𝒚) non-singular

Steady state solutions

0

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0 200 400 600 800

φ

τ

Turning point

unstable

Stable

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T

τ

Turning point

unstable

Stable

Simulation of perturbation from steady state with APMonitor

% option to read data from a CSV file

apm_option(server,app,'nlc.csv_read',1);

% imode = 4, switch to dynamic simulation

apm_option(server,app,'nlc.imode',4);

% nodes = 3, internal nodes in the collocation structure

apm_option(server,app,'nlc.nodes',3);

% time shift

apm_option(server,app,'nlc.time_shift',400);

Simulation of perturbation from steady state with APMonitor τ suddenly switched from 500 to 480 by increasing feedrate.

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φ

t

300

305

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330

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T

t

Simulation of perturbation from steady state with APMonitor τ suddenly switched from 500 to 501 by decreasing feedrate.

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-0.3

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-0.1

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t, min

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T, °

K

t, min

Reactive Distillation

Luyben & Yu Base Case : A+B C

– Infinite reflux ratio (product C is the heavier than reactants)

– Total reboiler with no sub-cooling

– Liquid sensible heat effects neglected

– Feed rate of A and B :12.63 and 12.82 mol/s

– 16 stages with reactive stages (holdup = 1000 mol) from stage 2 to 10.

– Target 98 mole% of component C in bottoms stream

A B C

Vapor pressure Aj 12.34 11.65 10.96

Vapor pressure Bj 3862 3862 3862

Heat of formation, kcal/mol 0 0 -10

Heat of vaporization, kcal/mol 6.944 6.944 6.944

Feed rate to column, mol/s 12.63 12.82 0

Feed stage location 10 2 ---

Steady state solutions as function of reactive stage holdup and boilup rate

0.82

0.84

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0.98

1

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xC -

bo

tto

ms

Holdup, mol

H=1000

0.88

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0.92

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0.96

0.98

1

0 20 40 60 80 100 120 140 160

xC -

bo

tto

ms

Boilup, mol/s

Vn = 62.47 mol/s

H = 1000 mol

Stability of Reactive Distillation Column Steady States (evaluated directly from index-2 system)

0.88

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0.98

1

0 20 40 60 80 100 120 140 160

xC -

bo

tto

ms

Boilup, mol/s

0.88

0.9

0.92

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0.98

1

50 70 90 110 130 150 170 190

xC -

bo

tto

ms

Boilup, mol/s

H = 800 mol

H = 1000 mol

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45

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0.86

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1

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xC_b

tms

t

xC16

Vn

Steady State Transitions (add pure ‘C’ to column bottoms. H = 800 mol, Vn = 70 mol/s)

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xC_b

tms

Feed C, mol/s

0.465

0.47

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0.495

0.5

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Un

reac

ted

A, m

ole

s/s

Feed C, mol/s

How to transition from low purity steady state at H=800 mol and Vn = 62.47 mol/s

Vn = 70 mol/s

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1

0 500 1000 1500 2000 2500

xC -

bo

tto

ms

Holdup, mol

Stable

Unstable

H=800

0.82

0.84

0.86

0.88

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0.92

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0.98

1

0 2000 4000 6000 8000 10000

xC -

bo

tto

ms

Holdup, mol

H=1000

Vn = 62.47 mol/s

0.88

0.9

0.92

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0.98

1

50 70 90 110 130 150 170 190

xC -

bo

tto

ms

Boilup, mol/s

H = 800 mol

Dynamic Simulation (using APMonitor) Constant liquid holdup

𝐻𝑗𝑑𝑥𝐴,𝑗

𝑑𝑡= 𝑓𝐴,𝑗 − 𝑉𝑗𝑦𝐴,𝑗 − 𝐿𝑗𝑥𝐴,𝑗 + 𝐿𝑗−1𝑥𝐴,𝑗−1 + 𝑉𝑗+1𝑦𝐴,𝑗+1 − 𝐻𝑗𝑟𝑗 , 𝑗 = 2. . 15

𝐻𝑗𝑑𝑥𝐵,𝑗

𝑑𝑡= 𝑓𝐵,𝑗 − 𝑉𝑗𝑦𝐵,𝑗 − 𝐿𝑗𝑥𝐵,𝑗 + 𝐿𝑗−1𝑥𝐵,𝑗−1 + 𝑉𝑗+1𝑦𝐵,𝑗+1 − 𝐻𝑗𝑟𝑗 , 𝑗 = 2. . 15

𝐻𝑗𝐶𝑝,𝑎𝑣𝑔,𝑗𝑑𝑇𝑗

𝑑𝑡= 𝑓𝑇,𝑗ℎ𝑓,𝑗 − ℎ𝑙,𝑗𝐿𝑗 − ℎ𝑣,𝑗𝑉𝑗 + ℎ𝑙,𝑗−1𝐿𝑗−1 + ℎ𝑣,𝑗+1𝑉𝑗+1, 𝑗 = 2. . 15

𝑓𝑇,𝑗 − 𝑉𝑗 − 𝐿𝑗 + 𝜈𝑇𝐻𝑗𝑟𝑗 + 𝐿𝑗−1 + 𝑉𝑗+1 = 0, 𝑗 = 2. . 15

𝐾𝑖,𝑗𝑥𝑖,𝑗 − 𝑦𝑖,𝑗 = 0, 𝑖 = 𝐴, 𝐵, 𝐶, 𝑗 = 1. . 16

𝑥𝑖,𝑗 =

3

𝑖=1

𝑦𝑖,𝑗 = 1

3

𝑖=1

, 𝑗 = 1. . 16

−𝐿1 + 𝑉2 = 0

−𝑉16 − 𝐿16 + 𝐿15 = 0 −𝐿1𝑥𝑖,1 + 𝑉2𝑦𝑖,2 = 0 , 𝑖 = 𝐴, 𝐵

−𝐿16𝑥𝑖,16 − 𝑉16𝑦𝑖,16 + 𝐿15𝑥𝑖,15 = 0 , 𝑖 = 𝐴, 𝐵

Dynamic Simulation (using APMonitor) Starting from steady state at H=800 mol, Vn=70 mol/s

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t, sec

xC16

Feed

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Dynamic Simulation (using APMonitor) Increase time at low feed rate. Slow down feed ramps.

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Feed

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xC10

Feed

Dynamic Simulation (using APMonitor) – Adjust boilup rate Steady state transition not possible.

H = 800 mol 100% = 70 mol/s boilup

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t, sec

xC9

Boilup

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Boilup

Dynamic Simulation – Introduce Holdup as dynamic variable

𝑑 𝐻𝑗𝑥𝐴,𝑗

𝑑𝑡= 𝑓𝐴,𝑗− 𝑉𝑗𝑦𝐴,𝑗 − 𝐿𝑗𝑥𝐴,𝑗 + 𝐿𝑗−1𝑥𝐴,𝑗−1 + 𝑉𝑗+1𝑦𝐴,𝑗+1 −𝐻𝑗𝑟𝑗, 𝑗 = 2. . 15

𝑑 𝐻𝑗𝑥𝐵,𝑗

𝑑𝑡= 𝑓𝐵,𝑗− 𝑉𝑗𝑦𝐵,𝑗 − 𝐿𝑗𝑥𝐵,𝑗 + 𝐿𝑗−1𝑥𝐵,𝑗−1 + 𝑉𝑗+1𝑦𝐵,𝑗+1 − 𝐻𝑗𝑟𝑗, 𝑗 = 2. . 15

𝐶𝑝,𝑎𝑣𝑔,𝑗𝑑 𝐻𝑗 𝑇𝑗 − 𝑇0

𝑑𝑡= 𝑓𝑇,𝑗ℎ𝑓,𝑗 − ℎ𝑙,𝑗𝐿𝑗 − ℎ𝑣,𝑗𝑉𝑗 + ℎ𝑙,𝑗−1𝐿𝑗−1 + ℎ𝑣,𝑗+1𝑉𝑗+1, 𝑗 = 2. . 15

𝑑𝐻𝑗

𝑑𝑡= 𝑓𝑇,𝑗− 𝑉𝑗 − 𝐿𝑗 + 𝜈𝑇𝐻𝑗𝑟𝑗 + 𝐿𝑗−1 + 𝑉𝑗+1, 𝑗 = 2. . 15

𝐾𝑖,𝑗𝑥𝑖,𝑗 − 𝑦𝑖,𝑗 = 0, 𝑖 = 𝐴, 𝐵, 𝐶, 𝑗 = 1. . 16

𝑥𝑖,𝑗 =

3

𝑖=1

𝑦𝑖,𝑗 = 1

3

𝑖=1

, 𝑗 = 1. . 16

−𝐿1 + 𝑉2 = 0

−𝑉16 − 𝐿16 + 𝐿15 = 0 −𝐿1𝑥𝑖,1 + 𝑉2𝑦𝑖,2 = 0 , 𝑖 = 𝐴, 𝐵

−𝐿16𝑥𝑖,16 − 𝑉16𝑦𝑖,16 + 𝐿15𝑥𝑖,15 = 0 , 𝑖 = 𝐴, 𝐵

Dynamic Simulation – Introduce Holdup as dynamic variable

0.88

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1

50 60 70 80 90 100

xC_b

tms

Boilup, mol/s

Stable

Unstable

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100%

120%

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xC16

Feed

Conclusions

• Differential algebraic systems defining reactive flash and reactive distillation systems were in Hessenberg form of size 2.

• Stability of steady states can be calculated at each point on the bifurcation path.

• APMonitor can reliably traverse the orbit of these index-2 DAE IVP.

• Index reductions such as that proposed by Kumar et al. (2009) (Comput. Chem. Engng., 33, p. 1336) to solve for intermediate vapor flows not required.

• Theses dynamic simulations are a powerful tool for checking viability of various steady state transitions.