Dynamics of forced and

(1)

Dynamics of forced and

unsteady-state processes

Davide Manca

Lesson 3 of “Dynamics and Control of Chemical Processes” – Master Degree in Chemical Engineering

(2)

Forced unsteady-state reactors

• Forced unsteady state (FUS) reactors allow reaching higher conversions than conventional reactors.

• Two alternatives:

 Reverse flow reactors, RFR;

 Network of reactors: simulated moving bed reactors, SMBR.

• Within a SMBR network, the simulated moving bed is accomplished by periodically switching the feed inlet from one reactor to the following one.

• APPLICATION: Methanol synthesis (ICI patent).

 Operating temperature: 220-300 °C;

 Pressure: 5-8 MPa;

 CO = 10-20%; CO ₂ = 6-10%; H ₂ = 70-80%.

(3)

Forced unsteady-state reactors

• Catalytic exothermic reactions can be carried out with an autothermal regime.

• FUS reactors are mainly advantageous when either the reactants concentration or the reactions exothermicity are low.

• There is an increase of both conversion and productivity that allows:

 Using smaller reactors,

 Lower amounts of catalyst.

(4)

Methanol synthesis in forced unsteady-state reactors

• The methanol synthesis reaction is:

• The reaction takes place with a reduction of the moles number. Therefore, the reaction is carried out at high pressure.

• In the past, the methanol plants worked at 100 ÷ 600 bar.

• Nowadays, methanol plants work at lower pressures (50 − 80 bar).

• In the low-pressure plants, the following reactions are important too:

2 3

2 _R 90.769 kJ/mol

CO H CH OH H

2 2 2

2 3 2 3 2

CO H CO H O

CO H CH OH H O

(5)

Reverse Flow Reactors

• Valves: V ₁ , V ₂ , V ₃ , V ₄ allow periodically inverting the feed direction in the reactor.

(6)

RFR: first–principles model

Equations #Eq. Eq. type Variables

Gas phase enthalpic

balance 1 Partial

derivative

Solid phase enthalpic

balance (catalyst) 1 Partial

derivative

Gas phase material

balance nComp Partial

derivative

Solid phase material

balance (catalyst) nComp Non-linear algebraic



1...nComp i

, ,



G,i S

G T y

T

 

1...nComp i

,

, ,



S,i G,i

S

G T y y

T

 

1...nComp i

,

, ,



S,i G,i

S

G T y y

T

 

1...nComp i

,

, ,



S,i G,i

S

G T y y

T

2 nComp 2 2 nComp 2

(7)

RFR: methanol synthesis

Reactor diameter Reactor length Void fraction Catalyst mass

Apparent catalyst density Catalyst porosity

Pellet diameter

Inlet temperature

Working pressure

Surface flowrate

(8)

RFR: methanol synthesis

(9)

Temperature profile once the pseudo-stationary condition is reached

Reverse Flow Reactors

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Reverse Flow Reactors

Concentration profile once the pseudo-stationary condition is reached

(11)

Simulated Moving Bed Reactors

Step 1: 1g2g3 Step 2: 2g3g1 Step 3: 3g1g2

1

2

3

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Simulated Moving Bed Reactors

• Kinetic equations corresponding to a dual-site Langmuir- Hinshelwood mechanism, based on three independent reactions: methanol formation from CO, water-gas-shift reaction and methanol formation from CO2:

Velardi S., A. Barresi, D. Manca, D. Fissore, Chem. Eng. J., 99 117–123, 2004

(13)

Simulated Moving Bed Reactors

• Reaction rates for a catalyst based on Cu–Zn–Al mixed oxides

(14)

Simulated Moving Bed Reactors

(15)

Simulated Moving Bed Reactors

• After a suitable number of switches the temperature profile reaches a pseudo- stationary condition.

High switch times Low switch times

(16)

SMBR: the thermal wave

320 300 280 260 240 220 200 180 160 140

120 0 1 2 3

Reactors T

GAS

[° C]

260 s

150 s

170 s

220 s

240 s

250 s

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SMBR: open loop response

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SMBR: open loop response

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Disturbance stop after 195 switches Disturbance stop after 310 switches

There are more periodic stationary conditions

SMBR: open loop response

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The control problem

• FEATURES

 The system may suddenly diverge to unstable operating conditions (even chaotic behavior);

 The network may shut-down;

 The reactors may work in a suboptimal region.

• PROBLEM

 The reactor network should work within an optimal operating range;

 Such a range is often narrow and its identification may be difficult.

• SOLUTION

 A suitable control system must be synthesized and implemented on-line to avoid both shut-down and chaotic behaviors;

 An advanced control system is highly recommended;

 Model based control  Model Predictive Control, MPC.

(21)

Numerical modeling

• The model based approach to the control problem calls for the implementation of a numerical model of the network that will be used for:

1. Identification of the optimal operating conditions;

2. Control purposes, i.e. to predict the future behavior of the system.

• The numerical model is based on a first principles approach:

 The reactors are continuously evolving (they never reach a steady-state condition)  time derivative;

 Each PFR reactor must be described spatially  spatial derivative;

 The reacting system is catalyzed (therefore it is heterogeneous). Consequently, an algebraic term is required  PDAE system.

 The PDAE system is spatially discretized  DAE system.

 A total of 1067 differential and algebraic equations must be solved to

determine the dynamic evolution of the network.

(22)

Mathematical tricks…

 ^





 ¹

1

1 nComp

iComp

G,iComp

G,nComp y

Stoichiometric closure: y

H

2

mol ar fr actio n

0 1 2 3

Reactors 0.936

0.934

0.932

0.930

0.928

0.926

0.924

0.922

0.920

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Numerical solution of the DAE

Boolean matrix that shows the presence indexes of the

differential-algebraic system

Specifically tailored numerical

algorithm for tridiagonal block

systems

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Numerical solution of the DAE

Boolean matrix that shows the presence indexes of the

differential-algebraic system

Specifically tailored numerical

algorithm for banded systems

(25)

BzzDAEFourBlocks

Numerical solution of the DAE

Algebraic equations Algebraic variables

Differential equations Algebraic variables

Differential equations Differential variables

Algebraic equations

Differential variables

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• Simulation time: 4000 s

• Switch time: 40 s

• Spatial discretization nodes: 97

• Number of equations per node: 11

• Total number of DAEs: 1067

• CPU: Intel ^® Pentium IV 2.4 GHz

• RAM: 512 MB

• OS: MS Windows 7 Professional

• Compiler: COMPAQ Visual Fortran 6.1 + MICROSOFT C++ 6.0

Numerical solution of the DAE

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Numerical simulation

CH

3

OH mol ar fr actio n

0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000

0 1 2 3

Reactors

0 1 2 3

T

GAS

[° C]

300 280 260 240 220 200 180 160 140 120

Reactors

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Numerical simulation

0.045 0.040 0.035 0.030 0.025 0.020 0.015 CH

3

OH mol ar fr actio n T

GAS

[° C]

310 300 290 280 270 260 250

0 1000 3000 4000

Switch time = 40 s

Time [s]

0 1000 Time [s] 3000 4000

(29)

Parametric sensitivity study

• Variability interval of the analyzed process variables

Variable Interval

Inlet gas temperature, T _in [K] 300÷593

Inlet gas velocity, v _in [m/s] 0.0189÷0.0231

Switch time, t _c [s] 1÷350

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Parametric sensitivity study

(31)

Parametric sensitivity study

(32)

Parametric sensitivity study

(33)

Need for speed

• THE POINT: to simulate 100 switches of the inlet flow with a switch time of 40 s (total of 4,000 s) the DAE system, comprising 1067 equations, takes about 95 s of CPU time on a workstation computer.

• PROBLEM: the detailed first principles model requires a CPU time that is prohibitive for model based control purposes.

• SOLUTION

 A high efficiency numerical model in terms of CPU time is therefore required;

 Such a model should be able to describe the nonlinearities and the articulate profiles of the network of reactors;

 Artificial Neural Networks, ANN, may be the answer.

(34)

System identification

with ANN

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ANN architecture

Input variables Range

Inlet gas temperature, T _in [K] 423÷453 Inlet gas velocity, v _in [m/s] 0.0189÷0.0231

Switch time, t _c [s] 10÷50

Output variables

Mean methanol molar fraction, x _CH3OH

Outlet gas temperature, T _GAS [K]

(36)

Levels # of nodes

Activation function

1 Input 40 Sigmoid

2 Intermediate 1 15 Sigmoid

3 Intermediate 2 15 Sigmoid

4 Output 1 Linear

# of weights and

biases 840 + 31 = 871

Learning factor, a 0.716

Momentum, b 0.366

Linear activation

constant, m ^0.275

ANN architecture

(37)

Random input patterns

455 450 445 440 435 430 425

T [K]

in 425

Inlet gas temperature

Inlet gas velocity

0.023 0.022

0.021 0.020 0.019

v

in

[m/ s]

50 45 40 35 30

t

c

[s]

25 20 15 10

Switch time

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Input patterns

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Initialization of weights + biases

Forward

Backward ERMS

)

i

(n

 )

(n

o

_i

i  N ?

Random selection Load of the input

patterns

( ) OR ?

av

n  n N

MAX

 ^ ^

)

av

(n



)

i

(n x

)

i

(n y n

NO YES

NO

)

i

(n w

)

i

(n b

LM equation





  







  



av ? new

av 

 ^

new n ( )

 av

ERMS

Levemberg Marquard learning algorithm

SECOND ORDER ALGORITHM

YES

(39)

Algorithms comparison

Iterations

ERMS (l og)

1

0.1 First order Pattern Mode

10 Second order LM

10

^-2

10

^-3

10

^-4

10

^-5

10

^-6

10

^-7

10

^-8

100 500

1 0.92 s

22.32 s 16.37 s

0.86 s

50.86 s

85.37 s

230.03 s

778.42 s

3172.50 s 4581.86 s

127.80 s

110.25 s

First order Batch Mode

(40)

The overlearning problem

Time [s]

CH

3

OH mola r fr act ion

0.041 0.040 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.032

Detailed model II order

3000 4000 5000 6000 7000 9000

2000

50 40 30 20 10

t

c

[s]

8000

I order

Disturbance on the switch time (from 30 to 10 s)

(41)

Iterations

ERMS

1

0.1 Learning

10 Cross Validation

10

^-2

10

^-3

10

^-4

10

^-5

10

^-6

10

^-7

10

^-8

100 500

1

13

^th

iterat. 31

^th

iteration

The overlearning problem

(42)

0.041

0.040 0.039

0.038 0.037

0.036 0.035 0.034

0.033 0.032

Detailed model ANN

100 200 300 400 500 600

0 Pattern number CH

3

OH mola r fr act ion Re la tiv e erro r

0.45%

0.40%

0.35%

0.30%

0.25%

0.20%

0.10%

0.15%

0.00%

0.05%

700 800 900 1000

ANN cross-validation

(43)

ANN disturbance response

Time [s]

CH

3

OH mola r fr act ion

0.041 0.040 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.032

Detailed model ANN

3000 4000 5000 6000 7000 9000

2000

455450 445440 435430 425420 415

T

in

[K]

8000

Disturbance on the inlet temperature 438  443  433 [K]

(44)

Time [s]

CH

3

OH mola r fr act ion

0.041 0.040 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.032

Detailed model ANN

3000 4000 5000 6000 7000 9000

2000

50 40 30 20 10

t

c

[s]

8000

Disturbance on the switch time 30  45 [s]

ANN disturbance response

(45)

Time [s]

CH

3

OH mola r fr act ion

0.041 0.040 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.032

Detailed model ANN

3000 4000 5000 6000 7000 9000

2000

0.023 0.022 0.021 0.020 0.019

V

in

[m/ s]

8000

ANN disturbance response

Disturbance on the inlet velocity 0.021  0.0194 [m/s]

(46)

ANN CPU times

MISO MIMO

ANN output variables CH

₃

OH T

_G

CH

₃

OH+T

_GAS

# of output nodes 1 1 2

# of weights and biases 840+31=871 840+31=871 855+32=887 Jacobian matrix dimensions 4000 x 871 4000 x 871 8000 x 887

CPU time for evaluating J

^T

Dynamics of forced and