Apps
~~----~~

!!! Multi-component Distillation Column

The distillation column in this example is built from a number of pre-existing model objects. The model objects used in this example include the distillation stage, feed, flash, mixer, splitter, stream lag, and vessel. These basic models are connected to form the multicomponent distillation tower.

* %list list-page% [[Attach:distill.apm| Distillation column model - 5 components, 8 stages]]

* %list list-page% (:html:)<a href="/online/view_pass.php?f=distill.apm">Solve Model Online Through a Web-based Interface</a>(:htmlend:)

----
~~The figure below displays the system response after a step change in the reflux ratio from 3.0 to 1.5. Each trajectory represents the mole fraction of cyclohexane at each tray. The top reflux material becomes less pure (more n-heptane) due to the increased draw from the top of the column.~~
~~!! Distillation Columns~~

%width=~~400px%Attach~~:~~distillation~~.~~png~~

-~~---~~
~~[[Main/PythonApp | See Example Problems for Distillation Column PID and NLC control comparison~~]]
~~Attach:download.jpg ~~[[~~Attach:python_distill_v0.5.5.zip~~ | Distillation Column PID and NLC control comparison]]

Binary Distillation Column~~with 30 trays (cyclohexane~~ n-heptane~~)~~

Distillation column models are generally good test cases for nonlinear model reduction and identification. The concentrations at each stage or tray are highly correlated. The dynamics of the distillation process can be described by a relatively few number of underlying dynamic states. A couple papers have been published with this model as an example application. ~~One in particular is:~~

The figure below displays the system response after a step change in the reflux ratio from 3.0 to 1.5. Each trajectory represents the mole fraction of cyclohexane at each tray. The top reflux material becomes less pure (more n-heptane) due to the increased draw from the top of the column.

## Distillation Column

## Apps.DistillationColumn History

Show minor edits - Show changes to output

Deleted lines 4-13:

!!! Multi-component Distillation Column

The distillation column in this example is built from a number of pre-existing model objects. The model objects used in this example include the distillation stage, feed, flash, mixer, splitter, stream lag, and vessel. These basic models are connected to form the multicomponent distillation tower.

* %list list-page% [[Attach:distill.apm| Distillation column model - 5 components, 8 stages]]

* %list list-page% (:html:)<a href="/online/view_pass.php?f=distill.apm">Solve Model Online Through a Web-based Interface</a>(:htmlend:)

----

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!!! Multi-component Distillation Column

The distillation column in this example is built from a number of pre-existing model objects. The model objects used in this example include the distillation stage, feed, flash, mixer, splitter, stream lag, and vessel. These basic models are connected to form the multicomponent distillation tower.

* %list list-page% [[Attach:distill.apm| Distillation column model - 5 components, 8 stages]]

* %list list-page% (:html:)<a href="/online/view_pass.php?f=distill.apm">Solve Model Online Through a Web-based Interface</a>(:htmlend:)

----

!!! Multi-component Distillation Column

The distillation column in this example is built from a number of pre-existing model objects. The model objects used in this example include the distillation stage, feed, flash, mixer, splitter, stream lag, and vessel. These basic models are connected to form the multicomponent distillation tower.

* %list list-page% [[Attach:distill.apm| Distillation column model - 5 components, 8 stages]]

* %list list-page% (:html:)<a href="/online/view_pass.php?f=distill.apm">Solve Model Online Through a Web-based Interface</a>(:htmlend:)

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to:

The model is available as a Python script with solution provided by [[https://gekko.readthedocs.io/en/latest/|GEKKO]]. The following figure shows a step change in the reflux ratio from 0.7 to 3.0.

%width=550px%Attach:distillation_gekko.png

(:source lang=python:)

from gekko import GEKKO

import numpy as np

import matplotlib.pyplot as plt

# Initialize Model

m = GEKKO()

# Define constants

#Reflux Ratio

rr=m.Param(value=0.7)

# Feed flowrate (mol/min)

Feed=m.Const(value=2)

# Mole fraction of feed

x_Feed=m.Const(value=.5)

#Relative volatility = (yA/xA)/(yB/xB) = KA/KB = alpha(A,B)

vol=m.Const(value=1.6)

# Total molar holdup on each tray

atray=m.Const(value=.25)

# Total molar holdup in condenser

acond=m.Const(value=.5)

# Total molar holdup in reboiler

areb=m.Const(value=.1)

# mole fraction of component A

x=[]

for i in range(32):

x.append(m.Var(.3))

# Define intermediates

# Distillate flowrate (mol/min)

D=m.Intermediate(.5*Feed)

# Liquid flowrate in rectification section (mol/min)

L=m.Intermediate(rr*D)

# Vapor Flowrate in column (mol/min)

V=m.Intermediate(L+D)

# Liquid flowrate in stripping section (mol/min)

FL=m.Intermediate(Feed+L)

# vapor mole fraction of Component A

# From the equilibrium assumption and mole balances

# 1) vol = (yA/xA) / (yB/xB)

# 2) xA + xB = 1

# 3) yA + yB = 1

y=[]

for i in range(32):

y.append(m.Intermediate(x[i]*vol/(1+(vol-1)*x[i])))

# condenser

m.Equation(acond*x[0].dt()==V*(y[1]-x[0]))

# 15 column stages

n=1

for i in range(15):

m.Equation(atray * x[n].dt() ==L*(x[n-1]-x[n]) - V*(y[n]-y[n+1]))

n=n+1

# feed tray

m.Equation(atray * x[16].dt() == Feed*x_Feed + L*x[15] - FL*x[16] - V*(y[16]-y[17]))

# 14 column stages

n=17

for i in range(14):

m.Equation(atray * x[n].dt() == FL*(x[n-1]-x[n]) - V*(y[n]-y[n+1]))

n=n+1

# reboiler

m.Equation(areb * x[31].dt() == FL*x[30] - (Feed-D)*x[31] - V*y[31])

# steady state solution

m.solve()

print(x) # with RR=0.7

# switch to dynamic simulation

m.options.imode=4

nt = 61

m.time=np.linspace(0,60,61)

# step change in reflux ratio

rr_step = np.ones(nt) * 0.7

rr_step[10:] = 3.0

rr.value=rr_step

m.solve()

plt.subplot(2,1,1)

plt.plot(m.time,x[0].value,'r--',label='Condenser')

plt.plot(m.time,x[5].value,'b:',label='Tray 5')

plt.plot(m.time,x[10].value,'g--',label='Tray 10')

plt.plot(m.time,x[15].value,'r-',label='Tray 15')

plt.plot(m.time,x[20].value,'y-',label='Tray 20')

plt.plot(m.time,x[25].value,'b-',label='Tray 25')

plt.plot(m.time,x[31].value,'k-',label='Reboiler')

plt.ylabel('Composition')

plt.legend(loc='best')

plt.subplot(2,1,2)

plt.plot(m.time,rr.value,'r.-',label='Reflux Ratio')

plt.ylabel('Reflux Ratio')

plt.legend(loc='best')

plt.xlabel('Time (min)')

plt.show()

(:sourceend:)

The model is also available in the APMonitor Modeling Language. The figure below displays the system response after a step change in the reflux ratio from 3.0 to 1.5. Each trajectory represents the mole fraction of cyclohexane at each tray. The top reflux material becomes less pure (more n-heptane) due to the increased draw from the top of the column.

%width=550px%Attach:distillation_gekko.png

(:source lang=python:)

from gekko import GEKKO

import numpy as np

import matplotlib.pyplot as plt

# Initialize Model

m = GEKKO()

# Define constants

#Reflux Ratio

rr=m.Param(value=0.7)

# Feed flowrate (mol/min)

Feed=m.Const(value=2)

# Mole fraction of feed

x_Feed=m.Const(value=.5)

#Relative volatility = (yA/xA)/(yB/xB) = KA/KB = alpha(A,B)

vol=m.Const(value=1.6)

# Total molar holdup on each tray

atray=m.Const(value=.25)

# Total molar holdup in condenser

acond=m.Const(value=.5)

# Total molar holdup in reboiler

areb=m.Const(value=.1)

# mole fraction of component A

x=[]

for i in range(32):

x.append(m.Var(.3))

# Define intermediates

# Distillate flowrate (mol/min)

D=m.Intermediate(.5*Feed)

# Liquid flowrate in rectification section (mol/min)

L=m.Intermediate(rr*D)

# Vapor Flowrate in column (mol/min)

V=m.Intermediate(L+D)

# Liquid flowrate in stripping section (mol/min)

FL=m.Intermediate(Feed+L)

# vapor mole fraction of Component A

# From the equilibrium assumption and mole balances

# 1) vol = (yA/xA) / (yB/xB)

# 2) xA + xB = 1

# 3) yA + yB = 1

y=[]

for i in range(32):

y.append(m.Intermediate(x[i]*vol/(1+(vol-1)*x[i])))

# condenser

m.Equation(acond*x[0].dt()==V*(y[1]-x[0]))

# 15 column stages

n=1

for i in range(15):

m.Equation(atray * x[n].dt() ==L*(x[n-1]-x[n]) - V*(y[n]-y[n+1]))

n=n+1

# feed tray

m.Equation(atray * x[16].dt() == Feed*x_Feed + L*x[15] - FL*x[16] - V*(y[16]-y[17]))

# 14 column stages

n=17

for i in range(14):

m.Equation(atray * x[n].dt() == FL*(x[n-1]-x[n]) - V*(y[n]-y[n+1]))

n=n+1

# reboiler

m.Equation(areb * x[31].dt() == FL*x[30] - (Feed-D)*x[31] - V*y[31])

# steady state solution

m.solve()

print(x) # with RR=0.7

# switch to dynamic simulation

m.options.imode=4

nt = 61

m.time=np.linspace(0,60,61)

# step change in reflux ratio

rr_step = np.ones(nt) * 0.7

rr_step[10:] = 3.0

rr.value=rr_step

m.solve()

plt.subplot(2,1,1)

plt.plot(m.time,x[0].value,'r--',label='Condenser')

plt.plot(m.time,x[5].value,'b:',label='Tray 5')

plt.plot(m.time,x[10].value,'g--',label='Tray 10')

plt.plot(m.time,x[15].value,'r-',label='Tray 15')

plt.plot(m.time,x[20].value,'y-',label='Tray 20')

plt.plot(m.time,x[25].value,'b-',label='Tray 25')

plt.plot(m.time,x[31].value,'k-',label='Reboiler')

plt.ylabel('Composition')

plt.legend(loc='best')

plt.subplot(2,1,2)

plt.plot(m.time,rr.value,'r.-',label='Reflux Ratio')

plt.ylabel('Reflux Ratio')

plt.legend(loc='best')

plt.xlabel('Time (min)')

plt.show()

(:sourceend:)

The model is also available in the APMonitor Modeling Language. The figure below displays the system response after a step change in the reflux ratio from 3.0 to 1.5. Each trajectory represents the mole fraction of cyclohexane at each tray. The top reflux material becomes less pure (more n-heptane) due to the increased draw from the top of the column.

Changed lines 1-5 from:

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to:

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<iframe width="560" height="315" src="https://www.youtube.com/embed/a6eIEeCrJdU" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>

(:htmlend:)

<iframe width="560" height="315" src="https://www.youtube.com/embed/a6eIEeCrJdU" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>

(:htmlend:)

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* %list list-blogroll% [[Main/PythonApp | Download Example Problems for Distillation Column]]

Changed line 16 from:

to:

[[Main/PythonApp | See Example Problems for Distillation Column PID and NLC control comparison]]

Changed lines 14-18 from:

This distillation column is a separation of cyclohexane and n-heptane. The two components are separated over 30 theoretical trays. In general, distillation column models are generally good test cases for nonlinear model reduction and identification. The concentrations at each stage or tray are highly correlated. The dynamics of the distillation process can be described by a relatively few number of underlying dynamic states. This model was published in:

to:

This distillation column is a separation of cyclohexane and n-heptane. The two components are separated over 30 theoretical trays. In general, distillation column models are generally good test cases for nonlinear model reduction and identification. The concentrations at each stage or tray are highly correlated. The dynamics of the distillation process can be described by a relatively few number of underlying dynamic states.

Attach:download.jpg [[Attach:python_distill_v0.5.5.zip | Distillation Column PID and NLC control comparison]]

This model was published in:

Attach:download.jpg [[Attach:python_distill_v0.5.5.zip | Distillation Column PID and NLC control comparison]]

This model was published in:

Changed lines 9-10 from:

* %list list-page% ~~[[~~/online/view_pass.php?f=distill.apm~~|~~Solve ~~model online]]~~

to:

* %list list-page% (:html:)<a href="/online/view_pass.php?f=distill.apm">Solve Model Online Through a Web-based Interface</a>(:htmlend:)

Changed line 9 from:

* %list list-page% [[~~https://apmonitor.ath.cx~~/online/view_pass.php?f=distill.apm|Solve model online]]

to:

* %list list-page% [[/online/view_pass.php?f=distill.apm|Solve model online]]

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(:html:)<font size=~~1~~><pre>

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Changed lines 31-32 from:

! Hahn, J. and T.F. Edgar, An improved method for ~~nonlinear model reduction using balancing of~~

! empirical gramians, Computers and Chemical Engineering, 26, pp. 1379-1397, (2002)

! empirical gramians, Computers and Chemical

to:

! Hahn, J. and T.F. Edgar, An improved method for

! nonlinear model reduction using balancing of

! empirical gramians, Computers and Chemical

! Engineering, 26, pp. 1379-1397, (2002)

! nonlinear model reduction using balancing of

! empirical gramians, Computers and Chemical

! Engineering, 26, pp. 1379-1397, (2002)

Changed lines 9-10 from:

* %list list-page% [https://apmonitor.ath.cx/online/view_pass.php?f=distill.apm|~~ ~~Solve model online]

to:

* %list list-page% [[https://apmonitor.ath.cx/online/view_pass.php?f=distill.apm|Solve model online]]

Added line 9:

* %list list-page% [https://apmonitor.ath.cx/online/view_pass.php?f=distill.apm| Solve model online]

Added lines 26-128:

! APMonitor Modeling Language

! https://www.apmonitor.com

! Binary Distillation Column from

!

! Hahn, J. and T.F. Edgar, An improved method for nonlinear model reduction using balancing of

! empirical gramians, Computers and Chemical Engineering, 26, pp. 1379-1397, (2002)

!

! Liquid mole fraction of component A

! at reflux ratio = 2

!

! Condenser 0.93541941614016

! Tray 1 0.90052553715795

! Tray 2 0.86229645132283

! . 0.82169940277993

! . 0.77999079584355

! . 0.73857168629759

! . 0.69880490932694

! . 0.66184253445732

! . 0.62850777645505

! . 0.59925269993058

! . 0.57418567956453

! . 0.55314422743545

! . 0.53578454439850

! . 0.52166550959767

! . 0.51031495114413

! . 0.50127509227528

! . 0.49412891686784

! . 0.48544992019184

! . 0.47420248108803

! . 0.45980349896163

! . 0.44164297270225

! . 0.41919109776836

! . 0.39205549194059

! . 0.36024592617390

! . 0.32407993023343

! . 0.28467681591738

! . 0.24320921343484

! . 0.20181568276528

! . 0.16177269003094

! Tray 29 0.12514970961746

! Tray 30 0.09245832612765

! Reboiler 0.06458317697321

Model binary

Parameters

! reflux ratio

rr = 0.7

! Feed Flowrate (mol/min)

Feed = 2.0 ! 24.0/60.0

! Mole Fraction of Feed

x_Feed = 0.5

! Relative Volatility = (yA/xA)/(yB/xB) = KA/KB = alpha(A,B)

vol=1.6

! Total Molar Holdup in the Condenser

atray=0.25

! Total Molar Holdup on each Tray

acond=0.5

! Total Molar Holdup in the Reboiler

areb=0.1

End Parameters

Variables

! mole fraction of component A

x[1:32] = 0.3

End Variables

Intermediates

! Distillate Flowrate (mol/min)

D=0.5*Feed

! Flowrate of the Liquid in the Rectification Section (mol/min)

L=rr*D

! Vapor Flowrate in the Column (mol/min)

V=L+D

! Flowrate of the Liquid in the Stripping Section (mol/min)

FL=Feed+L

! Vapor Mole Fractions of Component A

! From the equilibrium assumption and mole balances

! 1) vol = (yA/xA) / (yB/xB)

! 2) xA + xB = 1

! 3) yA + yB = 1

y[1:32] = x[1:32]*vol/(1+(vol-1)*x[1:32])

End Intermediates

Equations

! condenser

acond * $x[1] = V*(y[2]-x[1])

! 15 column stages

atray * $x[2:16] = L*(x[1:15]-x[2:16]) - V*(y[2:16]-y[3:17])

! feed tray

atray * $x[17] = Feed*x_Feed + L*x[16] - FL*x[17] - V*(y[17]-y[18])

! 14 column stages

atray * $x[18:31] = FL*(x[17:30]-x[18:31]) - V*(y[18:31]-y[19:32])

! reboiler

areb * $x[32] = FL*x[31] - (Feed-D)*x[32] - V*y[32]

End Equations

End Model

! https://www.apmonitor.com

! Binary Distillation Column from

!

! Hahn, J. and T.F. Edgar, An improved method for nonlinear model reduction using balancing of

! empirical gramians, Computers and Chemical Engineering, 26, pp. 1379-1397, (2002)

!

! Liquid mole fraction of component A

! at reflux ratio = 2

!

! Condenser 0.93541941614016

! Tray 1 0.90052553715795

! Tray 2 0.86229645132283

! . 0.82169940277993

! . 0.77999079584355

! . 0.73857168629759

! . 0.69880490932694

! . 0.66184253445732

! . 0.62850777645505

! . 0.59925269993058

! . 0.57418567956453

! . 0.55314422743545

! . 0.53578454439850

! . 0.52166550959767

! . 0.51031495114413

! . 0.50127509227528

! . 0.49412891686784

! . 0.48544992019184

! . 0.47420248108803

! . 0.45980349896163

! . 0.44164297270225

! . 0.41919109776836

! . 0.39205549194059

! . 0.36024592617390

! . 0.32407993023343

! . 0.28467681591738

! . 0.24320921343484

! . 0.20181568276528

! . 0.16177269003094

! Tray 29 0.12514970961746

! Tray 30 0.09245832612765

! Reboiler 0.06458317697321

Model binary

Parameters

! reflux ratio

rr = 0.7

! Feed Flowrate (mol/min)

Feed = 2.0 ! 24.0/60.0

! Mole Fraction of Feed

x_Feed = 0.5

! Relative Volatility = (yA/xA)/(yB/xB) = KA/KB = alpha(A,B)

vol=1.6

! Total Molar Holdup in the Condenser

atray=0.25

! Total Molar Holdup on each Tray

acond=0.5

! Total Molar Holdup in the Reboiler

areb=0.1

End Parameters

Variables

! mole fraction of component A

x[1:32] = 0.3

End Variables

Intermediates

! Distillate Flowrate (mol/min)

D=0.5*Feed

! Flowrate of the Liquid in the Rectification Section (mol/min)

L=rr*D

! Vapor Flowrate in the Column (mol/min)

V=L+D

! Flowrate of the Liquid in the Stripping Section (mol/min)

FL=Feed+L

! Vapor Mole Fractions of Component A

! From the equilibrium assumption and mole balances

! 1) vol = (yA/xA) / (yB/xB)

! 2) xA + xB = 1

! 3) yA + yB = 1

y[1:32] = x[1:32]*vol/(1+(vol-1)*x[1:32])

End Intermediates

Equations

! condenser

acond * $x[1] = V*(y[2]-x[1])

! 15 column stages

atray * $x[2:16] = L*(x[1:15]-x[2:16]) - V*(y[2:16]-y[3:17])

! feed tray

atray * $x[17] = Feed*x_Feed + L*x[16] - FL*x[17] - V*(y[17]-y[18])

! 14 column stages

atray * $x[18:31] = FL*(x[17:30]-x[18:31]) - V*(y[18:31]-y[19:32])

! reboiler

areb * $x[32] = FL*x[31] - (Feed-D)*x[32] - V*y[32]

End Equations

End Model

Changed lines 21-28 from:

Attach:reflux_step.~~gif~~

to:

Attach:reflux_step.gif

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</pre></font>

(:htmlend:)

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* %list list-page% [[Attach:distill.apm| Distillation column model - 5 components, ~~7~~ stages]]

to:

* %list list-page% [[Attach:distill.apm| Distillation column model - 5 components, 8 stages]]

Changed line 13 from:

This distillation column is a separation of cyclohexane and n-heptane. The two components are separated over 30 theoretical trays. In general, distillation column models are generally good test cases for nonlinear model reduction and identification. The concentrations at each stage or tray are highly correlated. The dynamics of the distillation process can be described by a relatively few number of underlying dynamic states. This model was published in~~ ~~

to:

This distillation column is a separation of cyclohexane and n-heptane. The two components are separated over 30 theoretical trays. In general, distillation column models are generally good test cases for nonlinear model reduction and identification. The concentrations at each stage or tray are highly correlated. The dynamics of the distillation process can be described by a relatively few number of underlying dynamic states. This model was published in:

Added lines 4-5:

!!! Multi-component Distillation Column

Changed lines 11-14 from:

Binary Distillation Column

Distillation column models are generally good test cases for nonlinear model reduction and identification. The concentrations at each stage or tray are highly correlated. The dynamics of the distillation process can be described by a relatively few number of underlying dynamic states. A couple papers have been published with this model as an example application

to:

!!! Binary Distillation Column

This distillation column is a separation of cyclohexane and n-heptane. The two components are separated over 30 theoretical trays. In general, distillation column models are generally good test cases for nonlinear model reduction and identification. The concentrations at each stage or tray are highly correlated. The dynamics of the distillation process can be described by a relatively few number of underlying dynamic states. This model was published in

This distillation column is a separation of cyclohexane and n-heptane. The two components are separated over 30 theoretical trays. In general, distillation column models are generally good test cases for nonlinear model reduction and identification. The concentrations at each stage or tray are highly correlated. The dynamics of the distillation process can be described by a relatively few number of underlying dynamic states. This model was published in

Added lines 4-5:

The distillation column in this example is built from a number of pre-existing model objects. The model objects used in this example include the distillation stage, feed, flash, mixer, splitter, stream lag, and vessel. These basic models are connected to form the multicomponent distillation tower.

Added lines 10-15:

Binary Distillation Column with 30 trays (cyclohexane n-heptane)

Distillation column models are generally good test cases for nonlinear model reduction and identification. The concentrations at each stage or tray are highly correlated. The dynamics of the distillation process can be described by a relatively few number of underlying dynamic states. A couple papers have been published with this model as an example application. One in particular is:

Hahn, J. and T.F. Edgar, An improved method for nonlinear model reduction using balancing of empirical gramians, Computers and Chemical Engineering, 26, pp. 1379-1397, (2002)

Distillation column models are generally good test cases for nonlinear model reduction and identification. The concentrations at each stage or tray are highly correlated. The dynamics of the distillation process can be described by a relatively few number of underlying dynamic states. A couple papers have been published with this model as an example application. One in particular is:

Hahn, J. and T.F. Edgar, An improved method for nonlinear model reduction using balancing of empirical gramians, Computers and Chemical Engineering, 26, pp. 1379-1397, (2002)

Added lines 17-18:

The figure below displays the system response after a step change in the reflux ratio from 3.0 to 1.5. Each trajectory represents the mole fraction of cyclohexane at each tray. The top reflux material becomes less pure (more n-heptane) due to the increased draw from the top of the column.

Changed lines 1-5 from:

!! Distillation ~~Column~~

* %list list-~~page~~%~~ ~~[[Attach:distill.apm| Distillation column model - 5 ~~component~~, 7 stages]]

~~* %list list~~-~~page~~%~~ ~~[[Attach:binary.apm| Binary distillation column model - 30 stages]]

* %list list

to:

!! Distillation Columns

----

* %list list-page% [[Attach:distill.apm| Distillation column model - 5 components, 7 stages]]

----

* %list list-page% [[Attach:binary.apm| Binary distillation column model - 30 stages]]

Attach:reflux_step.gif

----

* %list list-page% [[Attach:distill.apm| Distillation column model - 5 components, 7 stages]]

----

* %list list-page% [[Attach:binary.apm| Binary distillation column model - 30 stages]]

Attach:reflux_step.gif

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* %list list-page% [[Attach:~~distill~~.apm| Binary distillation column model - 30 stages]]

to:

* %list list-page% [[Attach:binary.apm| Binary distillation column model - 30 stages]]