20th European Symposium on Computer Aided Process Engineering – ESCAPE20S. Pierucci and G. Buzzi Ferraris (Editors)c© 2010 Elsevier B.V. All rights reserved.

Minimal state representation for homogeneousreaction systemsNirav Bhatt, Michael Amrhein, Dominique Bonvin

Laboratoire d’Automatique, École Polytechnique Fédérale de Lausanne,CH-1015 Lausanne, Switzerland

AbstractMinimal state representations are parsimonious models having no redundant states.For homogeneous reaction systems with S species, R independent reactions, p in-dependent inlet streams and one outlet stream, a nonlinear diffeomorphism of thenumbers of moles to reaction variants, flow variants and constant invariants is pro-posed. The conditions under which this transformed system is a minimal staterepresentation of order (R + p + 1) are presented. A simulation example illustratesthe theoretical developments.Keywords Model reduction, minimal order, homogeneous reaction systems, reac-tion variants, flow variants, invariants.

1. Introduction

Detailed first-principles models of reaction systems are important for process analy-sis, control and optimization. However, these detailed models often contain a largenumber of either redundant or negligible dynamic elements. Model-order reductionis often used to eliminate both redundant and negligible elements, thus providingdeeper insight into the reaction system [1, 2]. In the literature, approaches such aslumping, time-scale analysis and sensitivity analysis have been applied to reducethe model order [2, 3, 4].First-principles models describe the state evolution by means of conservation equa-tions of differential nature, such as material and energy balances, and constitutiveequations of algebraic nature, such as kinetic expressions. For homogeneous reac-tion systems, Srinivasan et al. [5] proposed a nonlinear transformation of the origi-nal states (the numbers of moles of S species) to R-dimensional reaction variants,(p+1)-dimensional reaction invariants but flow variants, and (S−R−p)-dimensionalreaction and flow invariants. This transformation is quite useful but suffers fromthe fact that the reaction and flow variants are merely mathematical quantities thatdescribe a space rather than individual physically meaningful reaction and flow ex-tents. Recently, Amrhein et al. [6] simplified and extended that transformationto obtain true extents of reaction and flow. The simplification results from beingable to neglect the continuity equation since the mass can be computed from thenumbers of moles and the molecular weights of the various species. The extensiondeals with the possibility to isolate the effect of the outlet flow from the reactionand inlet-flow extents.This work proposes a minimal state representation for open homogeneous reactionsystems. In contrast to reduced-order representations that are based on time-scaleand sensitivity analysis, the proposed representation is not an approximation. Iteliminates redundant states and does it without knowledge of reaction kinetics. Thetransformation leads to a description of the manifold on which the reaction systemevolves and to conditions ensuring minimal state representation. The approach is il-lustrated in simulation via the startup of a continuous stirred-tank reactor (CSTR).

1

2. Mole balance equations

Consider an homogeneous liquid-phase reaction system involving S species, R reac-tions, p inlet streams, and one outlet stream.1 The mole balance equations can bewritten as follows:

n(t) = V (t)NT r(t) + Win uin(t) −uout(t)

m(t)n(t), n(0) = n0, (1)

with

V (t) =m(t)

ρ(t), m(t) = 1T

SMwn(t), ρ(t) = ρL(n(t), T (t)),

r(t) = r(c(t), T (t)), c(t) = n(t)/V (t),

T (t) = f(T (t),n(t),uin(t), uout(t), Qext(t)), T (0) = T0,

where n is the S-dimensional vector of numbers of moles, r the R-dimensionalreaction rate vector expressed in terms of the molar concentrations c and the tem-perature T , uin the p-dimensional inlet mass flowrate, uout the outlet mass flowrate,V , m and ρ the volume, mass and density of the reaction mixture, N the R×S stoi-chiometric matrix, Win = M−1

w Win the S×p inlet-composition matrix with Mw theS-dimensional diagonal matrix of molecular weights and Win = [w1

in, · · · , wp

in] withw

jin being the S-dimensional vector of weight fractions of the jth inlet streams, Qext

the net external heat flow due to heating, cooling, mixing, n0 the S-dimensional vec-tor of initial numbers of moles, T0 the initial temperature, and 1S the S-dimensionalvector of ones. Without loss of generality, it is assumed that the R reactions andthe p inlets are independent [6].

3. Transformations to reaction variants and invariants

We summarize here the results of previous investigations dealing with the trans-formation of the number of mole vector to various types of reaction variants andinvariants. In the context of this work on minimal state representation, we areparticularly interested in being able to generate constant invariants, i.e. invariantsfor which the dynamics have been removed.

3.1. Linear transformation to reaction variants and invariants

The two-way decomposition found in the literature [7, 8] transforms the space ofnumbers of moles into mutually orthogonal reaction variant and reaction invariantspaces. The resulting variants are abstract mathematical quantities that are devoidof any physical meaning. Furthermore, the invariants are constant only for batchreaction systems, i.e. reaction systems without inlet and outlet streams.

3.2. Linear transformation to reaction variants, inlet-flow variants, and invariants

Amrhein et al. [6] proposed a linear transformation to reaction variants, inlet-flowvariants, and invariants, which is summarized next.

1The model is given here for a liquid-phase homogeneous reaction system. A similar model canbe written for a gas-phase homogeneous reaction system [6].

Theorem 1 (Amrhein et al. [6])Consider an homogeneous reaction system involving S species, R independent reac-tions, p independent inlets and one outlet, and let rank ([NT Win]) = R + p. Then,the transformation

n −→ z :

zr

zin

ziv

=

ST

MT

QT

n (2)

brings Eq. (1) to:

zr = V r −uout

mzr, zr(0) = STn0,

zin = uin −uout

mzin, zin(0) = MTn0,

ziv = −uout

mziv, ziv(0) = QTn0.

(3)

zr is the R-dimensional vector of reaction variants expressed in kmol, zin the p-dimensional vector of inlet-flow variants expressed in kg, and ziv the (S − R − p)-dimensional vector of the reaction and inlet-flow invariants expressed in kmol. Thematrices S, M and Q are computed using the algorithm given in Appendix D in[6]. The numbers of moles n in the reactor at time t can be computed from zr(t),zin(t) and ziv(t) as follows:

n(t) = NT zr(t) + Win zin(t) + Qziv(t). (4)

The reaction and inlet-flow variants are mathematical quantities of dimension R+p.Unfortunately, since the invariants depend on the outlet flow and thus are notconstant, the model order is larger than R + p.

3.3. Nonlinear transformation to reaction variants, flow variants, and constant in-variants

The reaction and flow invariants ziv in Theorem 1 vary with the outlet when uout > 0.In this section, ziv(t) will be transformed nonlinearly so as to result in constantinvariants also when uout > 0. By selecting the scalar variable λ(t) = ziv,i(t)such that ziv,i(0) 6= 0 and transforming the remaining flow invariants ziv,j(t), ∀j =1, . . . , S − R − p − 1, j 6= i, nonlinearly as follows:

ziv −→

[

xiv

λ

]

:

[

{xiv,j}λ

]

=

[

{ziv,j

λ}

ziv,i

]

, (5)

xiv becomes an (S − R − p − 1)-dimensional vector of constant flow invariants.The scalar λ is the only time-varying flow variant. The nonlinear transformation isdetailed in the next theorem. The reaction and inlet-flow variants zr and zin aredenoted xr and xin, respectively.

Theorem 2Consider a homogeneous reaction system involving S species, R independent reac-tions, p independent inlets and one outlet, and let rank ([NT Win]) = R + p. Then,

there exists the diffeomorphism ℑ: [ n ] ↔ [ xλ ]:

[

n]

→

[

x

λ

]

:

xr

xin

xiv

λ

=

ST n

MTn

QT

ivn/qTn

qTn

, (6)

[

x

λ

]

→[

n]

: n = NT xr + Win xin + Q

[

1xiv

]

λ, (7)

that transforms Eq. (1) to the following form:

xr = V r −uout

mxr, xr(0) = STn0,

xin = uin −uout

mxin, xin(0) = MTn0,

xiv = 0S−R−p−1, xiv(0) = QT

ivn0/q

Tn0,

λ = −uout

mλ, λ(0) = qTn0.

(8)

xr, xin and λ are the reaction variants, the inlet-flow variants and the outlet-flowvariant of dimensions R, p and 1, respectively, while xiv are the constant reactionand flow invariants of dimension (S −R− p− 1). Without loss of generality, q is anS-dimensional vector corresponding to the first column of Q, Qiv is S×(S−R−p−1)-dimensional matrix corresponding to the remaining (S − R − p − 1) columns of Q.For n0 = 0S , xiv(0) = 0S−R−p−1.

4. Minimal state representation

Since the states xiv of (8) are constant, Eq. (8) can be reduced to:

ξ =

xr

xin

λ

=

fr(ξ,xiv)0p

0

+

0R×p −xr

m

Ip −xin

m

01×p − λm

u, ξ(0) =

STn0

MTn0

qTn0

, (9)

with (R + p)-dimensional state vector ξ, the p-dimensional input vector u = [ uinuout

],p = p + 1, fr = V r, and xiv = QT

ivn0/q

Tn0. Eq. (9) can be written as:

ξ = f(ξ,xiv) +

p∑

j=1

gj uj , (10)

where f and gj are (R + p)-dimensional vector fields.

Theorem 3Consider the homogeneous reaction system given by Eq. (1) with p > 0 and define

the R× (p + 1)-dimensional matrix J=[fr∂fr

∂xin

1

m( ∂fr

∂xrxr + ∂fr

∂xinxin + ∂fr

∂λλ− fr)].

If rank ([NT Win]) = R + p and rank (J(t)) = R for some finite time interval, thenEq. (9) is a minimal state representation of Eq. (1).

Proof: The idea of the proof is to show that Eq. (10) does not contain redundantstates. In mathematical terms, this means that the (R+ p) states must be reachablein finite time, i.e. Eq. (10) has to satisfy the local accessibility rank condition [9].Let ∆k be the kth distribution (k = 0, . . . , R + p − 1) corresponding to Eq. (10):

∆k = span{[f l, gm], 0 ≤ l ≤ k, 1 ≤ m ≤ p}, (11)

where [: , :] indicates the Lie-bracket and [f l, gm] the iterated Lie-bracket of [f , f l−1gm].Let consider the distribution ∆1 =

[

f ∆1

]

with k = 1:

∆1 =[

g1 . . . gp gp f [f ,g1] . . . [f ,gp]]

=

0R×p −xr

mJ(t)

Ip −xin

m0p×p+1

01×p − λm

01×p+1

. (12)

Since λ 6= 0 and the rank of J(t) = R by assumption, ∆1 has full rank (R + p),which satisfies the local accessibility rank condition.Remarks.

1. rank (J(t)) = R is a sufficient condition.

2. If rank (J(t)) < R and thus rank(

∆1

)

< R + p, the rank of the distribution

with k > 1 should be computed to check local accessibility.

3. The derivatives ∂fr

∂xr, ∂fr

∂xin, and ∂fr

∂λare the sensitivities of the reaction rates

fr with respect to the reaction and flow variants.

The minimal state representation and the model dimension for three special reac-tors are summarized in Table 1. Since in batch reactors p = 0, the minimal modeldimension corresponds to the number of independent reactions. In semi-batch reac-tors, uout = 0, p = p, and λ is constant. In a constant-density CSTR, uout = 1T

puin

and thus the additional state λ can be expressed algebraically in terms of xin and

m0, namely λ =m0−1

T

p xin

mivwith miv = 1T

SMwQ[

1xiv

]

.

Table 1: Minimal state representation and model dimension for batch, semi-batch, and constant-density CSTR reactors.

Reactor type Model Model order

Batch xr = V r R

Semi-batch xr = V r, xin = uin R + p

Constant-density CSTR xr = V r −1T

p uin

m0

xr, R + p

xin = uin −1T

p uin

m0

xin

5. Illustrative Example

We consider the startup of an open reactor with two inlet and one outlet streams forthe ethanolysis of phthalyl chloride (A) [6]. In two successive irreversible reactions,the desired product phthalyl chloride monoethyl ester (C) and phthalic diethylester(E) are produced from ethanol (B). Both reactions produce hydrochloric acid (D).It is assumed that B also reacts with D in a reversible side reaction to produce ethylchloride (F ) and water (G). The reaction system can be described by the followingreaction scheme with seven species (S = 7) and three independent reactions (R = 3):

R1: A + Br1−→ C + D,

R2: C + Br2−→ E + D,

R3: D + Br3

⇋ F + G.

The stoichiometric matrix is N =[

−1 −1 1 1 0 0 00 −1 −1 1 1 0 00 −1 0 −1 0 1 1

]

, and the reaction rates obey

the mass-action principle:

r =[

r1 r2 r3

]

T

with r1 = κ1 cAcB, r2 = κ2 cBcC , r3 = κ3 cBcD−κ4 cF cG.

Initially, 0.95 kmol of B and 0.05 kmol of G are placed in the reactor and thusn0 = [0, 0.95, 0, 0, 0, 0, 0.05]T kmol. Species A and B are fed with the constantmass flowrate uin,A = 7.8 kg h−1 and uin,B = 5.3 kg h−1, respectively. Hence,p = 2, and the inlet-composition matrix Win=[ 0.0049 0 0 0 0 0 0

0 0.0217 0 0 0 0 0 ]T

kmol/kg. The

density of the reaction mixture is ρ = 1/∑S

i=1

wi

ρi, with wi and ρi being the weight

fraction and the density of Species i. The total mass is m = xin,A + xin,B + mivλ,where xin,A and xin,B are the flow variants corresponding to the inlets of SpeciesA and B, respectively, and miv = −106.1091. With xr = [xr,1, xr,2, xr,3]

T andxin = [xin,A, xin,B]T, the derivatives ∂fr

∂xr, ∂fr

∂xinand ∂fr

∂λread:

ρ2 ∂fr

∂xr

= mρ∂r

∂xr

− mr∂ρ

∂xr

,

ρ2 ∂fr

∂xin

= [r 1T

p + m∂r

∂xin

]ρ − mr∂ρ

∂xin

,

ρ2 ∂fr

∂λ= [mivr + m

∂r

∂λ]ρ − mr

∂ρ

∂λ. (13)

Using the Symbolic Math toolbox of MATLAB R©, it was verified that rank (J(t)) =3 = R. Also, the condition rank ([N Win]) = 5 = R + p is fulfilled. Since the twoconditions of Theorem 3 are satisfied, Eq. (9) is a minimal state representation oforder 6 for this reaction system.

6. Conclusion

Homogeneous reaction systems can be described by (R + p + 1) reaction and flowvariants instead of the S original states. Conditions have been formulated underwhich the transformed reaction system forms a minimal state representation. Thisminimal state representation has several advantages when it comes to model reduc-tion by e.g. singular perturbation, sensitivity analysis between a reaction rate andthe corresponding kinetic parameters, parameter estimation, and feedback lineariz-ability.

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