Invariant Relationships for Heterogeneous Reaction Systems - - PowerPoint PPT Presentation

▶
invariant relationships for heterogeneous
SMART_READER_LITE
LIVE PREVIEW

Invariant Relationships for Heterogeneous Reaction Systems - - PowerPoint PPT Presentation

Jan 13, 2024 41 likes •258 views

Invariants - Chemical Reaction Systems Introduction Motivation Definitions Heterogeneous Invariant Relationships for Heterogeneous Reaction Systems Chemical Reaction Systems in Open Reactors System Description Transformation to Vessel

slide-1
SLIDE 1

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Invariant Relationships for Heterogeneous Chemical Reaction Systems in Open Reactors Sriniketh Srinivasan, Julien Billeter and Dominique Bonvin Laboratoire d’Automatique EPFL, Lausanne, Switzerland

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 1 / 20

slide-2
SLIDE 2

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Outline

1

Introduction Motivation Definitions

2

Heterogeneous Reaction Systems System Description

3

Transformation to Vessel Extents Linear Transformation Invariant Relationships

4

Application Data Reconciliation

5

Conclusion

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 2 / 20

slide-3
SLIDE 3

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Motivation

Consider the following homogeneous reaction system - Hydrodealkylation process C7H8 + H2 → C6H6 + CH4 2 C6H6 → C12H10 + H2 Reaction system is operated in a batch reactor

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 3 / 20

slide-4
SLIDE 4

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Motivation

The material balance equations can be written as: ˙ nC7H8 = −V r1 nC7H8(0) = nC7H8,0 ˙ nH2 = −V r1 + V r2 nH2(0) = nH2,0 ˙ nC6H6 = V r1 − 2V r2 nC6H6(0) = nC6H6,0 ˙ nCH4 = V r1 nCH4(0) = nCH4,0 ˙ nC12H10 = V r2 nC12H10(0) = nC12H10,0

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 4 / 20

slide-5
SLIDE 5

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Motivation

In a batch reactor, we have: ˙ nC7H8(t) + ˙ nCH4(t) = −Vr1 + Vr1 = 0 nC7H8(t) + nCH4(t) remains constant: nC7H8(t) + nCH4(t) = nC7H8,0 + nCH4,0 Similarly, we can get other invariant relationships:

2nH2(t) + nC6H6(t) − nC7H8(t) = 2nH2,0 + nC6H6,0 − nC7H8,0 nC7H8(t) + nC12H10(t) − nH2(t) = nC7H8,0 + nC12H10,0 − nH2,0

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 5 / 20

slide-6
SLIDE 6

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Motivation

Invariant relationships are straightforward for an homogeneous reaction system operated in batch mode What are the invariant relationships for heterogeneous reaction systems with mass transfer between phases? What are the invariant relationships when the reactor has inlet and outlet streams? This contribution gives a systematic procedure for deriving the invariant relationships

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 6 / 20

slide-7
SLIDE 7

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Definitions

Reaction & Flow Variants: Variables that vary with time due to the effects of chemical reactions and physical flows Flow Variants (but Reaction Invariants): Variables that vary with time due to other rate processes (mass transfer, inlets/outlet) but are independent of chemical reactions Reaction & Flow Invariants: Variables that do not vary with time and stay constant during the course of the reaction

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 7 / 20

slide-8
SLIDE 8

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Heterogeneous Reaction System: Mass Balance Equations

Consider a multiphase reaction system For phase F containing Sf species, Rf reactions, pm mass transfers, pf inlet streams and one outlet stream, the mole balance equation can be written as: Mole balances for Sf species

˙ nf (t) = NT

f Vf (t) rf (t)+Wm,f ζ(t)+ Win,f uin,f (t) − uout,f (t) mf (t) nf (t), nf (0) = nf 0 (Sf × 1) (Sf × Rf ) (Rf × 1) (Sf × pm) (Sf × pf ) Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 8 / 20

slide-9
SLIDE 9

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Decoupling to Vessel Extents

Bhatt et al. introduced a linear transformation to convert the number of moles to vessel extents Recently, the transformation of Bhatt et al. has been explicited as a decoupling system inversion The transformation generates:

  • an extent (variant) for each of the rate processes
  • a number of invariants
  • N. Bhatt, M. Amrhein and D. Bonvin, Incremental Identification of Reaction and Mass - Transfer Kinetics

Using the Concept of Extents, Industrial & Engineering Chemistry Research, 50(23), 12960 - 12974 (2011) Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 9 / 20

slide-10
SLIDE 10

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Decoupling to Vessel Extents

Assumption: rank([NT

f

Wm,f Win,f nf 0]) = Rf + pm + pf + 1 Linear transformation Tf from nf (t) to extents:       xr,f (t) xm,f (t) xin,f (t) xic,f (t) xiv,f (t)       = Tf nf (t)

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 10 / 20

slide-11
SLIDE 11

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Decoupling to Vessel Extents

Then the system reduces to:

˙ xr,f (t) = rv,f (t) − uout,f (t) mf (t) xr,f (t) xr,f (0) = 0Rf ˙ xm,f (t) = ζ(t) − uout,f (t) mf (t) xm,f (t) xm,f (0) = 0pm ˙ xin,f (t) = uin,f (t) − uout,f (t) mf (t) xin,f (t) xin,f (0) = 0pf ˙ xic,f (t) = −uout,f (t) mf (t) xic,f (t) xic,f (0) = 1 xiv,f (t) = 0qf .

The number of invariants is: qf = Sf − Rf − pm − pf − 1 The number of moles can be reconstructed as:

nf (t) = N

T

f xr,f (t) ± Wm,f xm,f (t) + Win,f xin,f (t) + nf 0 xic,f (t)

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 11 / 20

slide-12
SLIDE 12

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Invariant Relationships

The transformation matrix is computed as

Tf = [N

T

f Wm,f Win,f nf 0 Pf ]−1

where

P

T

f [N

T

f Wm,f Win,f nf 0]

T = 0qf ×(Rf +pm+pf +1)

The invariant relationships is

P

T

f nf (t) = 0qf

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 12 / 20

slide-13
SLIDE 13

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Invariant Relationships - Flow rate Measurements

If flowrate measurements are available, then

˙ xin,f (t) = uin,f (t) − uout,f (t) mf (t) xin,f (t) xin,f (0) = 0pf ˙ xic,f (t) = −uout,f (t) mf (t) xic,f (t) xic,f (0) = 1

Compute the vector of reduced numbers of moles:

nvRMV

f

(t) = nf (t) − Win,f xin,f (t) − nf 0 xic,f (t) = N

T

f xr,f (t) + Wm,f xm,f (t).

Assumption: rank ([NT

f Wm,f ]) = Rf + pm

  xr,f (t) xm,f (t) xiv,f (t)   = T vRMV

f

nvRMV

f

(t)

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 13 / 20

slide-14
SLIDE 14

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Invariant Relationships

The transformation matrix is given by

Tf = [N

T

f Wm,f PvRMV f

]−1

where

(PvRMV

f

)

T [N T

f Wm,f ] = 0qf ×(Rf +pm)

The invariant relationships is

(PvRMV

f

)

T nvRMV

f

(t) = 0qf

Note that the number of invariants is : qf = Sf − Rf − pm

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 14 / 20

slide-15
SLIDE 15

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Application - Data Reconciliation

The invariant relationships can be used directly to improve the accuracy of the measurements Additional positivity constraints can also be imposed minimize

ˆ nf (t)

(˜ nf (t) − ˆ nf (t))T Σ−1

n

(˜ nf (t) − ˆ nf (t)) subject to PT

f ˆ

nf (t) = 0q ˆ nf (t) ≥ 0S

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 15 / 20

slide-16
SLIDE 16

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Example - Data Reconciliation

Consider the following gas-liquid reaction system BA(l) + Cl2(g) → MBA(l) + HCl(l) BA(l) + 2Cl2(g) → DBA(l) + 2HCl(l) Chlorine is fed in the gas phase and transfers to the liquid The liquid phase has an outlet stream with a measured flowrate BA is present initially in the liquid phase

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 16 / 20

slide-17
SLIDE 17

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Example - Data Reconciliation

Compute nvRMV Compute: (PvRMV

l

)T [NT

l Wm,l]T = 02×3

Nl = −1 −1 1 1 −1 −2 1 2

  • Wm,l =
  • 1
  • (PvRMV

l

)T = 2 1 1 1 1 1

  • Laboratoire d’Automatique – EPFL

Invariants - Chemical Reaction Systems 19th November, 2014 17 / 20

slide-18
SLIDE 18

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Example - Data Reconciliation

The measured numbers of moles are corrupted with 10% gaussian white noise

20 40 60 80 100 −2 2 4 6 8 10 12 Time (min) Number of Moles (mol)

(a) Measured Concentrations

20 40 60 80 100 −2 2 4 6 8 10 12 Time (min) Number of moles (mol)

(b) Reconciled Concentrations

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 17 / 20

slide-19
SLIDE 19

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Example - Data Reconciliation

Sum of squares between the simulated (true) and the measured/reconciled numbers of moles in the liquid phase Species Residual sum of squares Measure- ments With Rec-

  • nciliation

With Rec-

  • nciliation

+ positivity BA 30.5 6.7270 4.4418 Cl2 0.1804 0.1804 0.1799 MBA 12.5 6.7910 4.5385 DBA 15.6 6.7008 4.3820 HCl 0.019 0.0189 0.0184

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 18 / 20

slide-20
SLIDE 20

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

Conclusion

Systematic procedure for deriving the invariant relationships for heterogeneous chemical reaction systems in open reactors Successful application to data reconciliation Thank you for your attention

Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 19 / 20

slide-21
SLIDE 21

Invariants - Chemical Reaction Systems Introduction

Motivation Definitions

Heterogeneous Reaction Systems

System Description

Transformation to Vessel Extents

Linear Transformation Invariant Relationships

Application

Data Reconciliation

Conclusion

References

  • M. Amrhein, N. Bhatt , B.Srinivasan and D. Bonvin, Extents of Reaction and Flow for Homogeneous

Reaction Systems with Inlet and Outlet Streams, AIChE Journal, 56(11), 2873 - 2866 (2010)

  • N. Bhatt, M. Amrhein and D. Bonvin, Incremental Identification of Reaction and Mass - Transfer

Kinetics Using the Concept of Extents, Industrial & Engineering Chemistry Research, 50(23), 12960

  • 12974 (2011)
  • S. Srinivasan, J. Billeter and D. Bonvin, Extent-Based Incremental Identification of reaction systems

using concentration and calorimetric measurements, Chem. Eng. Journal, 207-208, 785-793 (2012)

  • J. Billeter, S. Srinivasan and D. Bonvin, Extent-based Kinetic Identification using Spectroscopic

Measurements and Multivariate Calibration, Analytica Chimica Acta, 767, 21-34 (2013)

  • S. Srinivasan, J. Billeter and D. Bonvin, Variant and invariant states for reaction systems, 1st IFAC

Workshop on Thermodynamic Foundations of Mathematical Systems Theory, Lyon, 2013. Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 20 / 20