Variant and Invariant States for Reaction Systems S. Srinivasan, J. - - PowerPoint PPT Presentation

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Variant and Invariant States for Reaction Systems S. Srinivasan, J. Billeter and D. Bonvin Laboratoire dAutomatique EPFL, Lausanne, Switzerland TFMST 2013, Lyon Laboratoire dAutomatique EPFL Variant and Invariant States 1 / 13

SLIDE 1

Variant and Invariant States for Reaction Systems

S. Srinivasan, J. Billeter and D. Bonvin

Laboratoire d’Automatique EPFL, Lausanne, Switzerland

TFMST 2013, Lyon

Laboratoire d’Automatique – EPFL Variant and Invariant States 1 / 13

SLIDE 2

Outline

Concept of Variants and Invariants Concept of Vessel Extents

Each extent linked to the corresponding rate process Presence of outlet(s) → vessel extents Vessel extents of reaction, mass transfer, heat transfer

Applications

Model reduction Static state reconstruction Incremental kinetic identification

Conclusions

Laboratoire d’Automatique – EPFL Variant and Invariant States 2 / 13

SLIDE 3

Homogeneous reaction systems

Balance equations

Homogeneous reaction system consisting of S species, R independent reactions, p inlet streams, and 1 outlet stream Mole balances for S species

˙ n(t) = NT rv(t) + Win uin(t) − ω(t) n(t), n(0) = n0

(S) (S × R) (R) (S × p) (p)

rv(t) = V (t) r(t) considered as endogenous signal ω(t) = uout (t)

m(t)

Global macroscopic view Generally valid regardless of temperature, catalyst, solvent, etc.

Win, uin n N rv n, uout

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Reaction variants and reaction invariants in the literature1

Linear transformation using N: yr(t) yiv(t)

NT+ PT

n(t)

with N P = 0R×(S−R) Reaction variants yr and reaction invariants yiv describe the reactor state: ˙ yr(t) = rv(t) + N

T+ Win uin(t) − ω(t) yr(t)

yr(0) = N

T+n0

˙ yiv(t) = P

T Win uin(t) − ω(t) yiv(t)

yiv(0) = P

Tn0

yr are reaction and flow variants yiv are reaction invariants but flow variants yr are pure reaction variants and yiv are true invariants only for batch reactors (with uin = 0, ω = 0) Can we compute pure reaction variants and true invariants for open reactors?

1Asbjornsen et al. (1970), Chem. Eng. Sci., 25:1627-1639.

Laboratoire d’Automatique – EPFL Variant and Invariant States 4 / 13

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Vessel extents and true invariants

Assumption: rank ([NT Win n0]) = R + p + 1. Linear transformation: x(t) := 2 6 6 4 xr(t) xin(t) xic(t) xiv(t) 3 7 7 5 = 2 6 6 4 R F cT Q 3 7 7 5 n(t) = T n(t) Vessel extents of reaction xr and flow xin, discounting factor xic, and invariants xiv: ˙ xr(t) = RN

| {z }

rv(t) + RWin | {z } uin(t) − ω(t) xr(t) xr(0) = 0R ˙ xin(t) = FN

|{z} rv(t) + FWin | {z }

uin(t) − ω(t) xin(t) xin(0) = 0p ˙ xic(t) = c

TN T

| {z } rv(t) + c

TWin

| {z } uin(t) − ω(t) xic(t) xic(0) = 1 ˙ xiv(t) = QN

| {z } rv(t) + QWin | {z } uin(t) − ω(t) xiv(t) xiv(0) = 0q

q = S − R − p − 1

Laboratoire d’Automatique – EPFL Variant and Invariant States 5 / 13

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Four subspaces

invariant subspace effet of outlet

n IC subspace

reaction subspace inlet subspace

P orthogonal to NT, Win and n0 x(t) = T n(t) T =

NT Win n0 P

−1

˙ xr,i(t) = rv,i(t) −ω(t) xr,i(t) xr,i(0) = 0 ˙ xin,j(t) = uin,j(t) −ω(t) xin,j(t) xin,j(0) = 0 ˙ xic(t) = −ω(t) xic(t) xic(0) = 1 xiv = PT n(t) = 0q n(t) = NT xr(t) + Win xin(t) + n0 xic(t)

NTR Win F n0cT PQ

R p 1 q

S-dimensional space, R + p + 1 variants amount that is still in the reactor

1 Bhatt et al. (2010), I&EC Research, 49:7704-7717 Laboratoire d’Automatique – EPFL Variant and Invariant States 6 / 13

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Homogeneous CSTR – Experimental data

Ethanolysis reaction with seven species (S = 7), three reactions (R = 3), two inlets (p = 2) and one outlet Stoichiometric matrix (N) and inlet-composition matrix (Win): N = h −1 −1

1 1 0 0 0 −1 −1 1 1 0 0 −1 −1 0 1 1

i Win = h win,A

0 0 0 0 0 win,B 0 0 0 0 0

iT

5 10 15 20 25 30 35 40 45 50 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 A B C D E F G

win,A, uin,A win,B, uin,B

A,B,C, D,E,F, G

N

n, uout Measured numbers of moles n(t) [kmol] Time [h]

Reaction extents ?

Laboratoire d’Automatique – EPFL Variant and Invariant States 7 / 13

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Homogeneous CSTR – Computation of extents

5 10 15 20 25 30 35 40 45 50 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 A B C D E F G

5 10 15 20 25 30 35 40 45 50 10 20 30 40 50 60 70 5 10 15 20 25 30 35 40 45 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 35 40 45 50 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Numbers of moles

N Win n0

n [kmol] T = 2 6 6 4 R F cT Q 3 7 7 5 Extents of reaction

xr [kmol] xr,1 xr,2 xr,3

Extents of inlet

xin [kg] xin,1 xin,2

Effect of outlet on IC

Time [h] Time [h] Time [h] Time [h] xic 1 Bhatt et al. (2010), I&EC Research, 49:7704-7717 Laboratoire d’Automatique – EPFL Variant and Invariant States 8 / 13

SLIDE 9

Extension to fluid-fluid reaction systems

invariant subspace effet of outlet

n IC subspace

reaction subspace inlet subspace mass-transfer subspace NT R Win F Wm M n0 cT P Q R p pm 1 q S-dimensional space R + pm + p + 1 variants

For one of the phases

˙ xr(t) = rv(t) − ω(t) xr(t) xr(0) = 0R ˙ xm(t) = ζ(t) − ω(t) xm(t) xm(0) = 0pm ˙ xin(t) = uin(t) − ω(t) xin(t) xin(0) = 0p ˙ xic(t) = −ω(t) xic(t) xic(0) = 1 xiv = PT n(t) = 0q n(t) = NT xr(t) + Wm xm(t) + Win xin(t) + n0 xic(t)

1 Bhatt et al. (2010), I&EC Research, 49:7704-7717 Laboratoire d’Automatique – EPFL Variant and Invariant States 9 / 13

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Extension to reaction systems with heat balance equation

x(t) := T

n(t)

m(t) cp T(t)

dimension S + 1

“Decoupled” system ˙ xr(t) = rv(t) − ω(t) xr(t) xr(0) = 0R ˙ xex(t) = qex(t) − ω(t) xex(t) xex(0) = 0 ˙ xin(t) = uin(t) − ω(t) xin(t) xin(0) = 0p ˙ xic(t) = −ω(t) xic(t) xic(0) = 1 xiv = 0q Application: estimation of qex(t) or identification of heat-transfer coefficients independently of any kinetic information

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Model reduction

Dimensionality

d := R + p + 1, min(S, d) differential equations However, transformation assumes knowledge of n0, i.e., S initial conditions

Elimination of fast modes via singular perturbation

The reactions (and not the associated numbers of moles) exhibit fast or slow dynamic behavior → transformed decoupled model is well suited for input estimation: ˙ xr,i(t) = rv,i(t) − ω(t) xr,i(t) xr,i(0) = 0

Dynamical system rv,i (t) xr,i(t)

Laboratoire d’Automatique – EPFL Variant and Invariant States 11 / 13

SLIDE 12

Incremental kinetic identification via rates or extents

Computation of rates and extents

Rates

rv,i(t) =

T†

i ˙

nRV

a (t)

(at least R measured species) with ˙ nRV

a (t) = ˙

na(t) − Win,a uin(t) + ω(t) na(t) → differentiation of sparse and noisy signal na(t)

Vessel extents

xr,i(t) =

T†

i n

vRV

a (t)

(at least R measured species) with nvRV

a (t) := na(t) − Win,a xin(t) − n0,a xic(t)

xr,i(t) = Ri na(t) (at least R + p + 1 measured species) → neither integration nor differentiation of the sparse and noisy signal na(t)

Laboratoire d’Automatique – EPFL Variant and Invariant States 12 / 13

SLIDE 13

Conclusions

Transformation of numbers of moles to “decoupled” vessel extents

Transformation uses structural information N, Win, Wm and knowledge of n0 Effect of outlets is accounted for → concept of vessel extent Rates considered as time signals, e.g. rv(t) and not rv(c, T)