Variant and invariant states for chemical reaction systems

Accepted Manuscript Title: Variant and Invariant States for Chemical Reaction Systems Author: D. Rodrigues S. Srinivasan J. Billeter D. Bonvin PII: DOI: Reference:

S0098-1354(14)00304-4 http://dx.doi.org/doi:10.1016/j.compchemeng.2014.10.009 CACE 5064

To appear in:

Computers and Chemical Engineering

Received date: Revised date: Accepted date:

8-7-2014 14-10-2014 21-10-2014

Please cite this article as: D. Rodrigues, S. Srinivasan, J. Billeter, D. Bonvin, Variant and Invariant States for Chemical Reaction Systems, Computational Biology and Chemistry (2014), http://dx.doi.org/10.1016/j.compchemeng.2014.10.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Variant and Invariant States for Chemical Reaction Systems

Laboratoire d’Automatique

October 28, 2014

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Abstract

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CH-1015 Lausanne, Switzerland

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Ecole Polytechnique F´ed´erale de Lausanne

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D. Rodrigues, S. Srinivasan, J. Billeter and D. Bonvin∗

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Models of chemical reaction systems can be quite complex as they typically include information

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regarding the reactions, the inlet and outlet flows, the transfer of species between phases and the

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transfer of heat. This paper builds on the concept of reaction variants/invariants and proposes a

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linear transformation that allows viewing a complex nonlinear chemical reaction system via decoupled

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dynamic variables, each one associated with a particular phenomenon such as a single chemical

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reaction, a specific mass transfer or heat transfer. Three aspects are discussed, namely, (i) the

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decoupling of reactions and transport phenomena in open non-isothermal both homogeneous and

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heterogeneous reactors, (ii) the decoupling of spatially distributed reaction systems such as tubular

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reactors, and (iii) the potential use of the decoupling transformation for the analysis of complex

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reaction systems, in particular in the absence of a kinetic model.

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Keywords: Chemical reaction systems, state decoupling, reaction variants, reaction invariants, re-

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action extents, state reconstruction, kinetic identification.

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The (bio)chemical industry utilizes reaction processes to convert raw materials into desired products that

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include polymers, organic chemicals, vitamins, vaccines and drugs. If these processes involve chemical

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reactions, they also deal with (i) material exchange via inlet/outlet flows, mass transfers, convection,

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diffusion, and (ii) energy exchange via heating and cooling. Hence, modeling these phenomena is essential

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for improved process understanding, design and operation.

Introduction

∗ Corresponding

author, Phone: +41 21 693 3843, Fax: +41 21 693 2574, Email: [email protected]

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Models of chemical reaction processes are typically first-principles models that describe the state

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evolution (the mass, the concentrations, the temperature) by means of balance equations of differential

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nature (continuity equation, molar balances, heat balances) and constitutive equations of algebraic nature

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(e.g. equilibrium relationships, rate expressions). These models usually include information regarding

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the underlying reactions (e.g. stoichiometries, reaction kinetics, heats of reaction), the transfers of mass

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within and between phases, and the operating mode of the reactor (e.g. initial conditions, external

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exchange terms, operating constraints). A reliable description of reaction kinetics and transport phe-

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nomena represents the main challenge in building first-principles models for chemical reaction systems.

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In practice, such a description is constructed from experimental data collected both in the laboratory

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and during production 1 .

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The presence of all these phenomena, and in particular their interactions, complicates the analysis

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and operation of chemical reactors. The analysis would be much simpler if one could somehow separate

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the effect of the various phenomena and investigate each phenomenon individually. Ideally, one would

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like to have true variants, whereby each variant depends only on one phenomenon, and invariants that

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are identically zero and can be discarded. Note that some of the state variables are often redundant,

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as there are typically more states (balance equations) than there are independent source of variability

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(reactions, exchange terms). Hence, one would like to have a systematic way of discarding the redundant

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state variables, thereby reducing the dimensionality of the model.

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Asbjørnsen and co-workers 2–4 introduced the concepts of reaction variants and reaction invariants and

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used them for reactor modeling and control. However, the reaction variants proposed in the literature

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encompass more than the reaction contributions since they are also affected by the inlet and outlet

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flows. Hence, Friedly 5,6 proposed to compute the extents of “equivalent batch reactions”, associating the

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remainder to transport processes. For open homogeneous reaction systems, Srinivasan et al. 7 developed

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a nonlinear transformation of the numbers of moles to reaction variants, flow variants, and reaction and

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flow invariants, thereby separating the effects of reactions and flows. Later, the same authors 8 refined

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that transformation to make it linear (at the price of losing the one-to-one property) and therefore more

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easily interpretable and applicable. They also showed that, for a reactor with an outlet flow, the concept

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of vessel extent is most useful, as it represents the amount of material associated with a given process

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(reaction, exchange) that is still in the vessel. Bhatt et al. 9 extended that concept to heterogeneous

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G–L reaction systems for the case of no reaction and no accumulation in the film, the result being

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decoupled vessel extents of reaction, mass transfer, inlet and outlet, as well as true invariants that are

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identically equal to zero. An extension regarding the incorporation of calorimetric measurements into

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the extent-based identification framework has been proposed recently by Srinivasan et al. 10 .

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Various implications of reaction variants/invariants have been studied in the literature. For example,

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Srinivasan et al. 7 discussed the implications of reaction and flow variants/invariants for control-related

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tasks such as model reduction, state accessibility, state reconstruction and feedback linearizability. On

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the one hand, control laws using reaction variants have been proposed for continuous stirred-tank reac-

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tors 11–14 . The concept of extent of reaction is very useful to describe the dynamic behavior of a chemical

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reaction since a reaction rate is simply the derivative of the corresponding extent of reaction. Bonvin

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and Rippin 15 used batch extents of reaction to identify stoichiometric models without the knowledge

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of reaction kinetics. Reaction extents have been used extensively for the kinetic identification of both

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homogeneous and G–L reaction systems using either concentration 16 or spectroscopic 17 measurements.

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On the other hand, the fact that reaction invariants are independent of reaction progress has also been

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exploited for process analysis, design and control. For example, reaction invariants have been used to

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study the state controllability and observability of continuous stirred-tank reactors 4,18 . Reaction invari-

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ants have also been used to automate the task of formulating mole balance equations for the non-reacting

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part (such as mixing and splitting operations) of complex processes, thereby helping determine the num-

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ber of degrees of freedom for process synthesis 19 . Furthermore, Waller and M¨ akil¨ a 12 demonstrated the

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use of reaction invariants to control pH, assuming that the equilibrium reactions are very fast. Gr¨ uner

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et al. 20 showed that, through the use of reaction invariants, the dynamic behavior of reaction-separation

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processes with fast (equilibrium) reactions resembles the dynamic behavior of corresponding non-reactive

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systems in a reduced set of transformed variables. Aggarwal et al. 21 considered multi-phase reactors op-

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erating at thermodynamic equilibrium and were able to use the concept of reaction invariants, which

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they labeled invariant inventories, to reduce the order of the dynamic model and use it for control.

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This paper addresses the computation of variant and invariant states for reaction systems. It presents

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both existing approaches and novel techniques on a unified basis, which eases comparison. One will see

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that, not only reaction-variant states can be separated from reaction-invariant states, but a much finer

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separation can be achieved. The objective of this paper is therefore to sketch new avenues that could

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possibly lead to improved analysis, estimation, control and optimization of reaction systems.

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The paper is organized as follows. Section 2 presents a novel way of computing the vessel extents of

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reaction and flow for open non-isothermal homogeneous reactors. The approach is extended to models

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that include a heat balance in Section 3 and to fluid-fluid reaction systems in Section 4, while Section

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5 generalizes the transformation to distributed tubular reactors. The applicability of the decoupling

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transformation is discussed in Section 6, while Section 7 concludes the paper.

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2

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This section presents the computation of the extents of reaction and flow for an homogeneous reaction

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system with several inlets and one outlet. Although the computed extents are exactly the same as those

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in Amrhein et al. 8 , the computational approach is different and provides considerable insight in the

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transformation. This insight will help extend the transformation to more complex reaction systems in

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Sections 3-5.

Homogeneous Reaction Systems

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2.1

Mole Balance Equations

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Let us consider a general open non-isothermal homogeneous reactor. The mole balance equations for a

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reaction system involving S species, R reactions, p inlet streams, and one outlet stream can be written

93

as follows:

n(0) = n0 ,

with rv (t) := V (t) r(t)

(1b)

uout (t) , m(t)

cr

ω(t) :=

(1a)

ip t

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

(1c)

where n is the S-dimensional vector of numbers of moles, r the R-dimensional reaction rate vector, uin

95

the p-dimensional inlet mass flowrate vector, uout the outlet mass flowrate, V and m the volume and the

96

ˇ mass of the reaction mixture. N is the R × S stoichiometric matrix, Win = M−1 w Win the S × p inlet-

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p 1 ˇ in = [w ˇ in ˇ in composition matrix, Mw the S-dimensional diagonal matrix of molecular weights, W ···w ]

98

j ˇ in with w being the S-dimensional vector of weight fractions of the jth inlet flow, and n0 the S-dimensional

99

vector of initial numbers of moles. Note that ω(t) corresponds to the inverse of the reactor residence

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time.

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The mole balance equations (1a) hold independently of the operating conditions since the reaction

102

rates are simply modeled as the unknown time signals rv (t). The operating conditions such as the

104

105

106

concentrations c(t) and the temperature T (t) affect the reaction rates through the relations rv (t) =
V (t) r c(t), T (t) , but these dependencies are not needed at the level of equation (1a). If needed, the

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concentrations can be computed as c(t) = n(t)/V (t), while the temperature can be described by a heat balance as shown in Section 3. Note that the signals rv (t) represent endogenous inputs.

107

The flowrates uin (t) and uout (t) are considered as independent (input) variables in equation (1a). The

108

way these variables are adjusted depends on the particular experimental situation; for example, some

109

elements of uin can be adjusted to control the temperature in a semi-batch reactor, or uout is a function

110

of the inlet flows in a constant-mass reactor. The continuity equation (or total mass balance) is given

111

by:

m(t) ˙ = 1Tp uin (t) − uout (t),

m(0) = m0 ,

(2)

112

where 1p is the p-dimensional vector filled with ones and m0 the initial mass. Note that the mass m(t)

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can also be computed from the numbers of moles n(t) as m(t) = 1TS Mw n(t),

114

(3)

which indicates that equations (1a) and (2) are in fact linearly dependent. Hence, the continuity equation

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is not needed per se, but it is often useful to express the mass as a function of the flows rather than the numbers of moles. The volume V (t) can be inferred from the mass and knowledge of the density ρ as
V (t) = m(t)/ρ c(t), T (t) . The analysis that follows will use intensively the following four integer numbers:

• S: the total number of species (reacting or not) in the reactor,

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• R: the total number of independent reactions,

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• p: the total number of independent inlet streams,

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• q: the number of invariant (redundant) states (will be introduced later).

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2.2

Reaction Variants/Invariants in the Literature

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The concept of reaction variants/invariants as introduced by Asbjørnsen and co-workers 2–4 is briefly

125

reviewed next. The idea is to use the stoichiometric matrix to construct a linear transformation of the

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states n to the reaction-variant states yr and the reaction-invariant states yiv . This transformation T

127

involves the stoichiometric matrix N and its null space of dimension q = S − R described by the S × q

128

matrix P, that is N P = 0R×q : yr (t)







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

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R

 −1 T = NT P .

  = T n(t) :=   n(t) yiv (t) Q

+

130

131

132

"  IR N P =

d

It follows from N P = 0R×q and T T

" #

=

R  Q

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−1

T

0

0 Iq

#

(4)

+

that R = NT and Q = P+ , where

NT and P+ represent the Moore-Penrose pseudo-inverse of NT and P, respectively.∗ The resulting dynamical system contains the R state variables yr that depend on the reactions and the q state variables yiv that do not:

+

+

y˙ r (t) = rv (t) + NT Win uin (t) − ω(t) yr (t)

yr (0) = NT n0

y˙ iv (t) = P+ Win uin (t) − ω(t) yiv (t)

yiv (0) = P+ n0 .

(5)

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More specifically, one sees that the reaction variants are decoupled with respect to the reaction rates,

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that is, yr,i (t) depends on rv,i (t) but not on the other reaction rates: +

y˙ r,i (t) = rv,i (t) + (NT Win )i uin (t) − ω(t) yr,i (t)

135

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+

yr,i (0) = (NT )i n0

i = 1, · · · , R ,

(6)

where (.)i represents the ith row of the matrix (.). However, note that ∗ Note

that both PT and P+ span the q-dimensional null space of the stoichiometric matrix. Note also that the stoichiometric matrix N satisfies the relationship M NT = 0A×R , where M is the A×S atomic matrix indicating the number of each atom type in the various species 15,22 . The maximum number of independent reactions is given by Rmax = S − rank(M) ≥ R. 23 It follows that q = S − R ≥ S − Rmax = rank(M).

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• yr are reaction and flow variants, and not solely reaction variants,

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• yiv are reaction invariants but flow variants, hence not truly invariants,

139

• yr are pure reaction variants and yiv are true invariants only for batch reactors , that is, for uin (t) = 0p and ω(t) = 0, for which one can write:

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+

y˙ r (t) = rv (t)

yr (0) = NT n0

(7)

ip t

yiv (t) = P+ n0 ,

• yr and yiv are abstract mathematical quantities with no direct physical meaning.

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Hence, the question arises whether it is possible to compute true reaction variants and true invariants

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for open reactors, thereby removing the effect of the inlet and outlet flows. The next section will show

144

that this is possible with the concept of vessel extents.

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2.3

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For the sake of deriving the transformation, we will express the initial conditions as a unit impulse at

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time 0.† As a result, the balance equation (1a) can be written with zero initial conditions:‡

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Vessel Extents and True Invariants

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

n(0) = 0S .

(8)

Equation (8) clearly explicits the fact that the initial conditions can be considered as a rate process

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(the process of instantaneously loading the reactor) that can be put on the same footing as the other

150

rate processes. The right-hand side of equation (8) has four contributions that indicate the effects of

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the reactions, the inlets, the initial conditions, and the outlet, respectively. Note that the first three

152

contributions have a particular structure, namely they are expressed as the product of a constant term

153

and a rate signal. This particular structure will be exploited for decoupling next.

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2.3.1

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We look for a full-rank linear transformation that transforms n(t) into the three decoupled parts xr (t),

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xin (t) and xic (t) and an orthogonal remaining part xiv (t). The first three parts correspond to the

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reactions, the inlets and the initial conditions, while the remaining part is invariant as will be seen

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Decoupling transformation

below. The linear transformation T reads: † For

˙ ˙ example, n(t) = 0S , n(0) = n0 can be written equivalently as n(t) = n0 δ(t), n(0) = 0S . way to have zero initial conditions is to work in terms of deviation variables as shown in Appendix A.

‡ Another

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



  R         xin (t)  F   = T n(t) :=   n(t),    T  xic (t)  i      xiv (t) Q xr (t)

(9)

and brings the dynamic model (8) to the following decoupled form:

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T u (t) + R n0 δ(t) − ω(t) xr (t) x˙ r (t) = RN | {z } rv (t) + RW | {z } | {z in} in

0R×p

ip t

IR

xr (0) = 0R

0R

T x˙ in (t) = FN u (t) + F n0 δ(t) − ω(t) xin (t) | {z } rv (t) + FW |{z} | {z in} in

xin (0) = 0p

0p

Ip

cr

0p×R

x˙ ic (t) = |i {z N } rv (t) + i Win uin (t) + iT n0 δ(t) − ω(t) xic (t) | {z } | {z } T

0T R

1

0T p

x˙ iv (t) = QNT rv (t) + QWin uin (t) + Q n0 δ(t) − ω(t) xiv (t) | {z } | {z } | {z } 0q×p

0q

xiv (0) = 0q ,

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0q×R

(10)

xic (0) = 0

T

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T

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where R, F and Q are matrices of dimension R×S, p×S, and q×S, respectively, and i is a S-dimensional

161

vector, with q = S − R − p − 1 being the number of invariant quantities. Upon choosing T as

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 −1 T = NT Win n0 P ,

(11)

T  where the matrix P describes the q-dimensional null space of the matrix NT Win n0 ,§ the transfor-

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mation gives the conditions shown under the braces in equation (10), namely:

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Ac ce pt e

   R I    R    F  T  0   N Win n0 P =   T  i  0    Q 0

0

0

Ip

0

0

1

0

0

0



  0 .  0  Iq

(12)

  It follows from P being orthogonal to NT Win n0 and Q P = Iq that Q = P+ . Furthermore,

165

166

NT R + Win F + n0 iT + P P+ = IS , where NT R represents the R-dimensional reaction subspace, Win F

167

the p-dimensional inlet subspace, n0 iT the one-dimensional subspace describing the contribution of the

168

initial conditions, and P P+ the q-dimensional invariant subspace (Figure 1). All subspaces add up to the

169

S-dimensional species space RS . Note that the invariant subspace is orthogonal to the other subspaces

170

by construction, while the other subspaces are typically not orthogonal to each other. §

ˆ T ˆ ˜T ˜ N Win n0 P = 0(S−q)×q or, equivalently, PT NT Win n0 = 0q×(S−q) . In this case, the matrix P spans the space that is orthogonal to the reactions, the inlets and the initial conditions.

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PP+

n0 iT

initial condition subspace

invariant subspace

Win F

reaction subspace

p

R

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inlet subspace

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NT R

ip t

q =S−R−p−1

1

Vessel extents and invariants   If rank [NT Win n0 ] = R + p + 1, the linear transformation (11) brings the dynamic model (8) to:¶

M

2.3.2

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x˙ r,i (t) = rv,i (t) − ω(t) xr,i (t)

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Figure 1: Decomposition of the S-dimensional subspace of numbers of moles into an R-dimensional reaction subspace, a p-dimensional inlet subspace, a one-dimensional subspace describing the contribution of the initial conditions and a q-dimensional invariant subspace.

x˙ in,j (t) = uin,j (t) − ω(t) xin,j (t) x˙ ic (t) = −ω(t) xic (t)

xr,i (0) = 0

i = 1, · · · , R

(13a)

xin,j (0) = 0

j = 1, · · · , p

(13b)

xic (0) = 1

xiv (t) = 0q ,

(13c) (13d)

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where xr,i (t) is the extent of the ith reaction at time t expressed in kmol, xin,j (t) the extent of the jth

173

inlet flow at time t expressed in kg, xic (t) the dimensionless extent of initial conditions that jumps from

174

0 to 1 and then indicates the fraction of the initial conditions that is still in the reactor at time t, and

175

xiv (t) the vector of invariants at time t. Note that each extent is affected by its corresponding rate

176

process (rv,i (t), uin,j (t), δ(t)) and, in the presence of an outlet (ω(t) > 0), also by the inlet and outlet

177

flows. Since each extent is discounted by the amount that has left the reactor and thus represents the

178

amount of material associated with the corresponding rate that is still in the vessel, these extents are

179

called “vessel extents”. The numbers of moles n(t) can be reconstructed from the various extents by ¶ The

differential equation x˙ ic (t) = δ(t)− ω(t) xic (t), xic (0) = 0, can be written as equation (13c), which is easier to simulate.

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  pre-multipliying (9) by T −1 = NT Win n0 P and considering the fact that xiv (t) = 0q : (14)

n(t) = NT xr (t) + Win xin (t) + n0 xic (t).

181

Note that the mass m(t) can also be reconstructed from xin (t) and xic (t) as follows: (15)

182

ip t

m(t) = 1Tp xin (t) + m0 xic (t). Remarks

• The invariants xiv (t) are identically equal to zero and can be discarded from the model. The

184

invariant relationships P+ n(t) = 0q , which represent q constraints prevailing among the variables

185

n(t), can be rather useful in practice to reduce the number of degrees of freedom in a given problem.

186

• The extents xin (t) and xic (t) can be computed from uin (t) and uout (t) in equations (13b) and (13c)

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and the continuity equation (2): x˙ in (t) = uin (t) − ω(t) xin (t)

xin (0) = 0p

x˙ ic (t) = −ω(t) xic (t)

xic (0) = 1

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187

m(t) ˙ = 1Tp uin (t) − uout (t),

(16)

m(0) = m0 .

• The extent of reaction xr,i (t) depends only upon the corresponding reaction rate rv,i (t) and the

189

inlet and outlet flows, but not on the other rate processes. It follows that rv,i (t) can be computed

190

directly from xr,i (t), its time derivative and ω(t), that is, without having to know the other extents.

191

However, this apparent decoupling is somewhat misleading: indeed, since rv,i (t) is an endogenous

192

signal (and not an exogenous input), it also depends on what happens in the reactor, that is, it feels

193

the effect of most other processes. Hence, the prevailing decoupling is limited to the relationships

194

between rate and extent signals, as given in (13a)-(13c).

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2.4

Special Cases

196

We consider next the transformed systems for three special cases, namely, batch reactors, semi-batch

197

reactors and CSTRs.

198

2.4.1

199

In a batch reactor with p = 0 and no outlet, the linear transformation (11) reduces to:

Batch reactors

 −1 T = NT n0 P ,

(17)

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with q = S − R − 1, and the transformed system reads: x˙ r,i (t) = rv,i (t)

xr,i (0) = 0

i = 1, · · · , R

(18a)

xic (t) = 1

(18b)

xiv (t) = 0q .

(18c)

Note that since there is no outlet, xic = 1 is also invariant. Hence, there are R reaction variants and

201

S − R invariants.The numbers of moles n(t) can be expressed as:

ip t

200

(19)

2.4.2

Semi-batch reactors

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cr

n(t) = NT xr (t) + n0 .

A semi-batch reactor has p inlets but no outlet. With the linear transformation (11), the transformed

an

system becomes: x˙ r,i (t) = rv,i (t) x˙ in,j (t) = uin,j (t)

i = 1, · · · , R

(20a)

xin,j (0) = 0

j = 1, · · · , p

(20b)

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xic (t) = 1

xr,i (0) = 0

xiv (t) = 0q ,

(20c) (20d)

with q = S − R − p − 1. There are R reaction variants, p inlet variants and S − R − p invariants. The

204

numbers of moles n(t) can be expressed as:

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203

n(t) = NT xr (t) + Win xin (t) + n0 .

(21)

205

2.4.3

CSTR

206

In a CSTR, uout (t) is computed from equation (2) and m(t) = V0 ρ(t), with V0 the constant volume, as

207

follows:

uout (t) = 1Tp uin (t) − V0 ρ(t) ˙ .

208

209

(22)

• If the density varies, the transformed system (13a)-(13d) cannot be simplified and thus holds with q = S − R − p − 1.

• If the density is constant, uout (t) = 1Tp uin (t) and thus ω(t) = computed algebraically from the states xin (t) as xic (t) = 1 −

1T p uin (t) . m0 T 1p xin (t) , m0

In this case, xic (t) can be with m0 = V0 ρ. This can

be shown by differentiating the last expression and writing x˙ in (t) and x˙ ic (t) using equations (13b)

10

Page 10 of 28

and (13c). The transformed system becomes: x˙ r,i (t) = rv,i (t) − ω(t) xr,i (t)

xr,i (0) = 0

i = 1, · · · , R

(23a)

x˙ in,j (t) = uin,j (t) − ω(t) xin,j (t)

xin,j (0) = 0

j = 1, · · · , p

(23b)

xic (t) = 1 −

1Tp xin (t) m0

(23c) (23d)

ip t

xiv (t) = 0q .

Since xic (t) is linearly dependent on xin (t), the system is of order R + p, q = S − R − p − 1, and

211

there are q + 1 = S − R − p invariants . The numbers of moles n(t) can be expressed as:

cr

210

(24)

us

 n0 1Tp  n(t) = NT xr (t) + Win − xin (t) + n0 . m0

3

Homogeneous Reaction Systems with Heat Balance

213

Let us consider an open non-isothermal homogeneous reactor that involves heat exchange via a heat-

214

ing/cooling jacket.

215

3.1

216

The model includes the mole balance equations (1a) and a heat balance around the reactor 24 :

M

an

212

Ac ce pt e

d

Mole and Heat Balance Equations

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

ˇ T uin (t) − ω(t) Q(t) ˙ Q(t) = (−∆ H)T rv (t) + qex (t) + T in

n(0) = n0

(25)

Q(0) = Q0 ,

217

where Q = m cp T is the heat, with T the reactor temperature and cp the heat capacity of the reaction

218

ˇ in the p-dimensional vector of mixture, qex is the heat flow from the jacket to the reaction mixture, T

219

specific heat of the inlet streams with Tˇin,j = cp,in,j Tin,j and Tin,j the temperature of the jth inlet, and

220

221

222

∆ H the R-dimensional vector of reaction enthalpies. Obviously, the heat flow term qex (t) depends on
the reactor temperature, for example qex (t) = U A Tj (t) − T (t) with the heat transfer coefficient U A and the jacket temperature Tj (t), but this dependency is not needed at the level of equation (25). This

223

model assumes that most physical properties are not temperature dependent, which is well justified in a

224

nearly isothermal reactor. Furthermore, for simplicity, let us assume that the inlet specific heat vector

225

ˇ in is constant. T

11

Page 11 of 28

226

227

228

3.2

Vessel Extents and Invariants

For the sake of deriving the transformation, we will express the initial conditions as a unit impulse "at time # 0. The model can be written in compact form using the (S + 1)-dimensional state vector z(t) =

n(t)

:

Q(t)

˙ z(t) = A rv (t) + b qex (t) + C uin (t) + z0 δ(t) − ω(t) z(t) where A =

NT

(−∆ H)T

#

,b=

"

0S 1

#

,C=

"

Win ˇT T in

#

and z0 =

"

n0

Q0

#

.

230

231

The right-hand side of equation (26) has five contributions that indicate the effects of the reactions, the heat exchange, the inlets, the initial conditions, and the outlet, respectively.

233

3.2.1

cr

232

Decoupling transformation

us

234

(26)

ip t

229

"

z(0) = 0S+1 ,

 −1 The linear transformation T = A b C z0 P transforms z(t) into five parts, namely, xr (t), xex (t),

xin (t) and xic (t) that are associated with the reactions, the heat exchange, the inlets and the initial

236

conditions, and xiv (t) that is orthogonal to the other extents and invariant as will be seen below: 

xr (t)



an

235



R



(27)

d

M

      T  h  xex (t)          xin (t)  = T z(t) :=  F  z(t).       T  i   xic (t)      xiv (t) P+

238

239

T  The matrix P describes the q-dimensional null space of the matrix A b C z0 , with q = S − R − p − 1 . 3.2.2

Ac ce pt e

237

Vessel extents and invariants   If rank A b C z0 = R + p + 2, the linear transformation (27) brings the dynamic model (26) to: x˙ r (t) = rv (t) − ω(t) xr (t)

xr (0) = 0R

x˙ ex (t) = qex (t) − ω(t) xex (t)

xex (0) = 0

x˙ in (t) = uin (t) − ω(t) xin (t)

xin (0) = 0p

x˙ ic (t) = −ω(t) xic (t)

xic (0) = 1

(28)

xiv (t) = 0q , 240

where xex is the extent of heat exchange expressed in kJ. Compared to the transformed model (13a)-

241

(13d), the model (28) has the one-dimensional state xex (t) to describe the evolution of the heat exchange.

242

Note that the extents xr , xin and xic in equation (28) are those in equations (13a)-(13c), which confirms

243

the fact that the transformed model (13a)-(13c) can be used to describe the reactions and flows in a

12

Page 12 of 28

244

non-isothermal reactor also in the absence of a heat balance. The numbers of moles n(t) and the heat Q(t) can be reconstructed from the transformed variables

245

246

as follows: (29)

z(t) = A xr (t) + b xex (t) + C xin (t) + z0 xic (t). 247

248

A possible use of this decoupling regards the estimation of qex (t) or the identification of the heat-transfer
coefficient U A in the expression qex (t) = U A Tj (t) − T (t) , independently of any kinetic information,

from discrete measurements of z(t) and computation of xex (t).

250

4

251

This section extends the results obtained for homogeneous reaction systems in the previous sections to

252

heterogeneous fluid-fluid (F-F) reaction systems. In addition to the extents developed above, there will

253

be extents of mass transfer that describe the material transport between the two phases.

ip t

249

us

cr

Fluid-Fluid Reaction Systems

We consider here a reaction system consisting of two phases, namely, the G and L phases.k The two

255

phases are modeled separately, with the mass-transfer rates ζ connecting the two phases. The L phase

256

contains Sl species, pl inlets and one outlet, while the G phase contains Sg species, pg inlets and one

257

outlet. There are pm mass transfers taking place between the two phases. The reactions can occur in

258

both phases, with Rl reactions in phase L and Rg reactions in phase G.

259

4.1

260

The differential mole balance equations for phase F , F ∈ {G, L}, read:

Ac ce pt e

d

Mole Balance Equations

M

an

254

n˙ f (t) = NTf rv,f (t) ± Wm,f ζ (t) + Win,f uin,f (t) − ωf (t) nf (t)

261

nf (0) = nf 0 ,

(30)

with a positive sign (+) for phase L and a negative sign (-) for phase G, and where the subscript (.)f is

264

used to denote phase F, with f ∈ {g, l}. The pm mass transfers are treated as pseudo inlets with the massh i 1 m transfer rates ζ , and Wm,f = M−1 · · · epm,f w,f Em,f is the Sf × pm mass-transfer matrix, Em,f = em,f

265

equal to unity and the other elements equal to zero.

266

4.2

267

Again, we will express the initial conditions as a unit impulse at time 0, which gives the following balance

268

equations:

262

263

with ejm,f being the Sf -dimensional vector with the element corresponding to the jth transferring species

Vessel Extents and Invariants

n˙ f (t) = NTf rv,f (t) ± Wm,f ζ (t) + Win,f uin,f (t) + nf 0 δf (t) − ωf (t) nf (t) k Although

nf (0) = 0Sf .

(31)

G and L are often the gas and liquid phases, they can also refer to two distinct liquid phases.

13

Page 13 of 28

269

The right-hand side of equation (31) has five contributions that indicate the effects of the reactions, the

270

mass transfers, the inlets, the initial conditions, and the outlet, respectively.

271

4.2.1

272

273

Decoupling transformation

−1  transforms nf (t) into For phase F , the linear transformation Tf := NTf ± Wm,f Win,f nf 0 Pf

five parts, namely, xr,f (t), xm,f (t), xin,f (t) and xic,f (t) that are associated with the reactions, the mass

transfers, the inlets and the initial conditions, and xiv,f (t) that is orthogonal to the other extents and

275

invariant as will be seen below: 

xr,f (t)



(32)

us

cr

     xm,f (t)     −1   nf (t), xin,f (t) = NTf ± Wm,f Win,f nf 0 Pf      xic,f (t)    xiv,f (t)

ip t

274

278

4.2.2

M

Vessel extents and invariants   If rank [NTf ± Wm,f Win,f nf 0 ] = Rf + pm + pf + 1, the linear transformation (32) brings (31) to: x˙ r,f (t) = rv,f (t) − ωf (t) xr,f (t)

xr,f (0) = 0Rf

x˙ m,f (t) = ζ (t) − ωf (t) xm,f (t)

xm,f (0) = 0pm

x˙ in,f (t) = uin,f (t) − ωf (t) xin,f (t)

xin,f (0) = 0pf

x˙ ic,f (t) = −ωf (t) xic,f (t)

xic,f (0) = 1

Ac ce pt e

d

279

an

277

 where the matrix Pf of dimension Sf × qf describes the qf -dimensional null space of the matrix NTf ± T Wm,f Win,f nf 0 , with qf = Sf − Rf − pm − pf − 1.

276

(33)

xiv,f (t) = 0qf .

280

Compared to the transformed model (13a)-(13d), the model (33) has the pm -dimensional states

281

xm,f (t) to describe the evolution of the mass transfers. The reconstruction of the numbers of moles

282

nf (t) reads:

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

(34)

283

5

Distributed Reaction Systems

284

For lumped homogeneous reaction systems, the transformation (9) is able to isolate the contributions of

285

the reactions, inlets and initial conditions that are contained in the state vector n(t). In addition, it is

286

possible to express the invariant states as P+ n(t) = 0q , which is useful in many applications 4,12,19 . We

14

Page 14 of 28

287

develop next a similar transformation applicable to distributed reaction systems. As illustrative example,

288

we will consider a one-dimensional single-phase tubular reactor model 25 .

289

5.1

290

Consider the reaction-convection-diffusion system

Material Balance Equations

291

(35)

ip t


∂ v(z, t)c(z, t) ∂c(z, t) T = N r(z, t) + Ed φ(z, t) − , ∂t ∂z

with the initial conditions c(z, 0) = c0 (z) and the boundary conditions c(0, t) = cin (t) and lim

z→∞

∗∗

∂c(z,t) ∂z

=

0S .

293

time t in kmol/m3 , v(z, t) is the velocity of the convective flow in m/s, and φ(z, t) is the pd -dimensional

295

296

vector of diffusion rates in kmol/(m3 s). It is assumed that only pd ≤ S − R species are affected by h i diffusion as indicated by the S × pd matrix Ed = e1d · · · epdd , with ejd being the S-dimensional vector

us

294

Here, c(z, t) represents the S-dimensional vector of concentrations at the spatial coordinate z and

cr

292

with the element corresponding to the jth diffusive species equal to unity and the other elements equal

to zero. This model includes S species, R independent reactions, pd diffusive terms and one convective

298

term. The reaction and diffusion rates are modeled as the unknown signals r(z, t) and φ(z, t), which

299

hides the fact that the rates depend on z and t over the concentrations c(z, t) and the temperature

300

T (z, t).

302

M

301

an

297

The right-hand side of equation (35) has three contributions that are associated with reactions,
∂ v(z,t)c(z,t) T appear diffusion and convection. Since the corresponding terms N r(z, t), Ed φ(z, t) and ∂z

linearly, the principle of superposition is satisfied and each contribution can be computed separately. We

304

will first remove the contribution of the reactions and diffusion and compute the effect that convection

305

has on the initial and boundary conditions by solving the differential equation

Ac ce pt e

d

303


∂ v(z, t)cibc (z, t) ∂cibc (z, t) + = 0S ∂t ∂z

cibc (z, 0) = c0 (z), cibc (0, t) = cin (t),

(36)

306

where cibc (z, t) indicates the concentrations at position z and time t that are due uniquely to the non-zero

307

initial and convective boundary conditions.

308

309

310

We will then remove the effect of the initial and boundary conditions by writing the concentrations as deviations from cibc (z, t),

δc(z, t) := c(z, t) − cibc (z, t),

(37)


∂ v(z, t)δc(z, t) ∂δc(z, t) T = N r(z, t) + Ed φ(z, t) − , ∂t ∂z

(38)

with which equation (35) becomes:

∗∗ These

boundary conditions correspond to what has been called “approximation by a reactor of infinite length” 26 , since the boundary conditions are given at z = 0 and z → ∞. Note that the separation of the various effects can also be achieved when different boundary conditions are assumed, as shown in Appendix B for the case of the Danckwerts boundary conditions 27 .

15

Page 15 of 28

311

with the initial conditions δc(z, 0) = 0S and the boundary conditions δc(0, t) = 0S and lim

z→∞

∂δc(z,t) ∂z

=

312

0S .†† The right-hand side of equation (38) has three contributions, with the first two associated with

313

the R reactions and the pd diffusion terms, respectively.

314

5.2

315

This section proposes a transformation that generates R extents of reaction, pd extents of diffusion and,

316

as will be seen, q = S − R − pd invariants.

317

5.2.1

ip t

Decoupling transformation

 −1 The linear transformation T = NT Ed P transforms δc(z, t) into three parts, namely, xr (z, t) and

cr

318

Extents and Invariants

xd (z, t) that are associated with the reactions and the diffusion, and the part xiv (z, t) that is invariant

320

as will be seen below:



xr (z, t)





R



us

319

5.2.2

Extents and invariants

xr (z, 0) = 0R ,

xd (z, 0) = 0pd ,

xr (0, t) = 0R

(40a)

xd (0, t) = 0pd , lim

z→∞

Ac ce pt e


∂ v(z, t)xr (z, t) ∂xr (z, t) + = r(z, t) ∂t ∂z
∂ v(z, t)xd (z, t) ∂xd (z, t) + = φ(z, t) ∂t ∂z

M

322

T  where the matrix P describes the q-dimensional null space of the matrix NT Ed .

d

321

(39)

an

         xd (z, t)  = T δc(z, t) :=  D  δc(z, t),     P+ xiv (z, t)

xiv (z, t) = 0q ,

∂xd (z, t) = 0pd ∂z

(40b) (40c)

323

where xr (z, t) indicates the change in concentrations at position z and time t due to the reactions; xd (z, t)

324

indicates the change in concentrations at position z and time t due to diffusion; xiv (z, t) represents

325

variables that are orthogonal to these extents and therefore invariant.

326

327

The concentrations δc(z, t) can be reconstructed from the various extents by pre-multiplying (39) by   T = NT Ed P and considering the fact that xiv (z, t) = 0q : −1

δc(z, t) = NT xr (z, t) + Ed xd (z, t).

328

Finally, the concentrations c(z, t) are obtained from equations (37) and (41) as: c(z, t) = NT xr (z, t) + Ed xd (z, t) + cibc (z, t).

329

(41)

(42)

Remarks †† The

last boundary condition follows from the assumption lim

z→∞

∂cibc (z,t) ∂z

= 0S .

16

Page 16 of 28

• The concentration of the S species at a given position and time can be transformed into R extents   of reaction and pd extents of diffusion. The only requirement is rank NT Ed = R + pd , which

330

331

implies that the number of diffusing species pd can be at most S − R.

332

333

• The linear transformation is independent of z and t and does not assume that the diffusivities of

334

the species are equal. Furthermore, the transformation is valid for any profile of the convective

335

velocity and of the initial and inlet concentrations.

ip t

A possible use of this decoupling regards the identification of reaction rates independent of diffusion,

336

and the identification of diffusion rates independently of any kinetic information.

338

6

339

This section is not meant to discuss a specific case study but rather to guide the reader through potential

340

uses of the concept of extents. The decoupling transformation can be used for two different classes of

341

applications, namely, (i) to simplify the dynamic model, its analysis and possibly the design of model-

342

based estimation, control and optimization schemes, and (ii) to process measured data by either isolating

343

certain signals that are useful for kinetic identification or reconstructing certain quantities in the absence

344

of a kinetic model. These two classes of problems are briefly discussed next.

345

6.1

cr

337

Model-based applications

M

an

us

Application of the Decoupling Transformation

• Model reduction. The reaction system can be described by either the S numbers of moles n(t)

347

given by the differential equations (1a) or the (R + p + 1) extents given by the differential equations

348

(13a)-(13c). The dimensionality of the system is therefore d := R + p + 1. However, note that

350

351

352

Ac ce pt e

349

d

346

the transformed model (13a)-(13c) is not a minimal-state representation of the system (1a) since
the reaction rates rv (t) = V (t) r c(t) cannot be computed solely from the reduced states xr (t), xin (t) and xic (t) 28,29 . Indeed, the description of the concentrations c(t) = n(t)/V (t) requires the knowledge of n0 , of dimension S, to reconstruct n(t) as per equation (14).

353

• State accessibility. The transformed system (13a)-(13d) indicates that xiv is inaccessible. Hence,

354

the maximal dimension of the accessible part is R + p + 1. In other words, since the dimension of

355

q need to be zero for full state accessibility, there should be at least S − R − 1 inlet streams. Note

356

that this results, which has been reported previously in the literature 18 , follows trivially from the

357

decoupling transformation.

358

• Control. The idea is to work with the reduced model that does not include the inaccessible invariant

359

part. With the inlet flows or the heat input as manipulated variables, the resulting models are

360

typically input-affine and lends themselves well to control via feedback linearization, for which

361

certain conditions are necessary 7,30 . Application of such control approaches are available in the

362

literature 14,31,32 . 17

Page 17 of 28

363

6.2

Data-driven applications

364

• Reconstruction of extent and rate profiles. With the knowledge of N, Win and n0 , the transforma-

365

tion T allows computing the various extents at the time instant th from the discrete measurements

366

n(th ). On the other hand, as expressed by equation (13a), the extent of reaction xr,i (t) is the

367

output of a dynamic system whose input is the reaction rate rv,i (t). It follows that the rate profile

368

rv,i (th ) can be reconstructed from xr,i (th ) by system inversion 33 . • Kinetic identification. Kinetic identification is performed by comparing the rates or extents com-

370

puted from measured concentrations to the values predicted by the model. This comparison can

371

be done individually for each reaction 34–36 . This way, several rate expressions can be compared

372

to experimental data, one at the time, until the correct expression has been found and the corre-

373

sponding parameters identified. The route over extents has certain advantages, in particular in the

374

presence of noisy and scarce measurements (see Appendix C) 16,37 .

us

cr

ip t

369

• State reconstruction. The objective is to reconstruct the state n(th ) of dimension S from Sa < S

376

measured species na (th ). When all the kinetic models are available and the model is observable,

377

the unmeasured states can be reconstructed via dynamic reconstruction (state observers). Alterna-

378

tively, when there are at least as many measured species as independent reactions, that is Sa ≥ R,

379

n(th ) can be reconstructed without knowledge of kinetics 17 . This is a two-step procedure, whereby

380

the extents are first estimated using either equation (9) or the approach described in Appendix C.

381

The unmeasured species are then reconstructed from the estimated extents. For the case Sa < R,

382

it is no longer possible to compute all extents of reaction from na (th ), in which case only a part of

383

xr (th ) is computed statically, with the remaining extents being computed using a kinetic model.

Ac ce pt e

d

M

an

375

384

7

Conclusions

385

The concept of reaction variants/invariants has been around for nearly 60 years. The reaction invariants,

386

which are typically computed from the knowledge of stoichiometry (or, almost equivalently, from the

387

atomic matrix) are not affected by the progress of the reactions. The reaction variants are often chosen

388

orthogonal to the reaction invariants. Unfortunately, a reaction variant may also be affected by other

389

phenomena such as flows and mass transfers. Similarly, although a reaction invariant is unaffected by

390

reaction, it might sense the effect of flows and mass transfers. Hence, the applicability of the concept of

391

reaction variants/invariants has been limited to specific reactor arrangements with negligible overlap of

392

the reaction and transport phenomena.

393

This paper has addressed the computation of variant and invariant quantities for open both homo-

394

geneous and heterogeneous reaction systems. The concept of reaction variants/invariants that is used

395

extensively in the context of batch processing has been extended to take into account the effects of inlet

18

Page 18 of 28

396

and outlet flows, of mass transfers between phases, and of convection and diffusion. In contrast to previ-

397

ously defined variants, each extent developed in this work describes uniquely and completely the progress

398

of the corresponding process. Furthermore, the invariants defined in this work are true invariants that

399

are identically equal to zero and can be discarded from the dynamic model. Isolation of the individual phenomena is implemented via decoupling of the balance equations through

401

linear transformation. The transformation uses structural information about the reaction system, in

402

particular the stoichiometry, the inlet composition, the initial conditions, and the knowledge of the

403

species that transfer between phases. If these parameters are constant, the transformation is globally

404

valid and straightforward to implement. Otherwise, things are more complicated and one might need to

405

rely on a piecewise transformation. For a convective and/or diffusive system, it is also necessary to know

406

the convection velocity and/or the identity of the species that diffuse.

cr

ip t

400

The significance of this work is twofold: (i) at the scientific level, the separation of reaction and

408

transport phenomena has significant implications for kinetic identification 16 , model reduction 28 , state

409

reconstruction 38,39 and leads to a better understanding of chemical reaction systems, and (ii) at the

410

application level, a systematic procedure is available for developing kinetic models in the laboratory

411

and design targeted control and optimization schemes for production 31,32,40 . Both aspects will improve

412

process operation in the long run.

an

us

407

This paper summarizes the efforts that have been done in the last decade to extend the concept of

414

reaction variants/invariants, originally defined for batch homogeneous reactors, to multi-phase reaction

415

systems with inlets and outlets. Compared to previous work done by the same research group, this paper

416

has introduced several novel elements, namely, the key concept of linear transformation, the extension

417

of the approach to distributed reaction systems as well as the treatment of the initial conditions via

418

the introduction of deviation variables. More efforts are needed to develop the theory further and,

419

for example, make it applicable to other lumped or distributed reaction systems, such as heterogeneous

420

catalytic reaction systems and reaction-absorption or reaction-distillation columns. Finally, it is expected

421

that similar decoupling transformations can be found for a wider class of processing systems as long as

422

the balance equations can be characterized by a well-defined structure as that of equation (1a).

423

Acknowledgment

424

The authors would like to acknowledge insightful discussions with Dr. Michael Amrhein and Dr. Bala

425

Srinivasan.

426

Appendix A – Reaction System in Deviation Variables

427

This appendix proposes another way to perform the decoupling for homogeneous reaction systems, which

428

parallels the method used above for distributed reaction systems. The idea is to work with numbers of

Ac ce pt e

d

M

413

19

Page 19 of 28

moles that represent deviations from the residual effect of the initial conditions, nic (t), as shown below.‡‡

430

Mole Balance Equations

431

The right-hand side of equation (1a) has three contributions that are associated with the reactions, the

432

inlets and the outlet. Since the corresponding terms NT rv (t), Win uin (t) and ω(t) n(t) appear linearly,

433

they satisfy the principle of superposition and each contribution can be computed independently of the

434

others. We will next compute the residual effect at time t of the initial conditions, that is, the part of the

435

initial conditions that has not been removed through the outlet flow, by solving the differential equation

Since each element of nic (t) has the same dynamics, nic (t) can be expressed as

437

an

nic (t) = n0 xic (t),

438

(43)

with nic (t) indicating the numbers of moles that are due uniquely to the non-zero initial conditions.

us

436

nic (0) = n0 ,

cr

n˙ ic (t) = −ω(t) nic (t),

ip t

429

with xic (t) given by

xic (0)

= 1.

(45)

M

x˙ ic (t) = −ω(t) xic (t)

(44)

439

The extent of initial conditions xic (t) is a scalar that varies between 1 and 0 and indicates the fraction

440

of the initial conditions that is still present in the reactor at time t . Upon writing the concentrations as deviations from the concentrations nic (t):

Ac ce pt e

d

441

(46)

δn(t) := n(t) − nic (t),

442

equation (1a) becomes:

˙ δ n(t) = NT rv (t) + Win uin (t) − ω(t) δn(t),

δn(0) = 0S .

(47)

443

Vessel Extents and Invariants

444

The right-hand side of equation (47) has three contributions that indicate the effects of the reactions,

445

the inlets and the outlet, respectively.

446

Decoupling transformation

447

448

 −1 The linear transformation T = NT Win P transforms δn(t) into three parts, namely xr (t) and

xin (t) that are associated with the reactions and the inlets, and xiv (t) that is orthogonal to the other ‡‡ An

alternative consists in working with numbers of moles that represent deviations from the initial conditions n0 . However, this way, the formulation and the interpretation of xic (t) is not as straightforward.

20

Page 20 of 28

449

extents and invariant as will be seen below: 

xr (t)





R



        xin (t) =  F  δn(t) = T δn(t),     P+ xiv (t)

451

452

Note that the invariant space corresponding to equation (8) is of lower dimension, q = S − R − p − 1 , as

one dimension is needed to represent the contribution of the initial conditions.

Vessel extents and invariants   If rank [NT Win ] = R + p, the linear transformation (48) brings the dynamic model (47) to:

us

cr

453

T  where the matrix P describes the q-dimensional null space of the matrix NT Win , with q = S − R − p.

ip t

450

(48)

xr,i (0) = 0

i = 1, · · · , R

(49a)

x˙ in,j (t) = uin,j (t) − ω(t) xin,j (t)

xin,j (0) = 0

j = 1, · · · , p

(49b)

an

x˙ r,i (t) = rv,i (t) − ω(t) xr,i (t)

xiv (t) = 0q .

455

The numbers of moles δn(t) can be reconstructed from the various extents by pre-multipliying (48)   by T −1 = NT Win P and considering the fact that xiv (t) = 0q :

M

454

Ac ce pt e

Finally, the numbers of moles n(t) is obtained from (50) with (46) and (44) as: n(t) = NT xr (t) + Win xin (t) + n0 xic (t).

457

(51)

Similarly, the mass m(t) can be reconstructed from xin (t) and xic (t) as follows: m(t) = 1Tp xin (t) + m0 xic (t).

458

(50)

d

δn(t) = NT xr (t) + Win xin (t).

456

(49c)

(52)

21

Page 21 of 28

459

Appendix B – Danckwerts Boundary Conditions This appendix shows that the transformation in Section 5 still holds when the reaction-convectiondiffusion system (35) is written with the following Danckwerts boundary conditions 27 : c(0, t) = cin (t) −

1 Ed J(0, t), v(0, t)

(53a) (53b)

ip t

J(l, t) = 0pd ,

460

for a tubular reactor ranging from z = 0 to z = l, where J(z, t) is the pd -dimensional vector of diffusion

461

fluxes in kmol/(m2 s).

1 Ed J(0, t) The partial differential equation (38) has now the boundary conditions δc(0, t) = − v(0,t)

463

and J(l, t) = 0pd . The diffusion flux J(z, t) is unknown, but it can be computed from the diffusion rates

464

as follows:

us

cr

462

∂J(z, t) = −φ(z, t) ∂z  NT

Ed



(54)

= R + pd , the linear transformation (39) brings (38) with the corresponding

an

If rank

J(l, t) = 0pd .

Danckwerts boundary conditions to:

xr (z, 0) = 0R , xr (0, t) = 0R

(55a)

1 J(0, t) xd (z, 0) = 0pd , xd (0, t) = − v(0,t)

(55b)

M


∂xr (z, t) ∂ v(z, t)xr (z, t) + = r(z, t) ∂t ∂z
∂xd (z, t) ∂ v(z, t)xd (z, t) + = φ(z, t) ∂t ∂z

d

xiv (z, t) = 0q ,

(55c)

which corresponds to (40a)-(40c) with different boundary conditions. The boundary condition J(0, t) is

466

computed from J(l, t) = 0pd using the ordinary differential equation (54).

Ac ce pt e

465

467

Note that the Danckwerts boundary conditions assume that there is a discontinuity in the value of

468

the diffusion fluxes at z = 0, 41 which is clearly illustrated by the non-zero values of the extents xd (0, t).

469

A detailed discussion of the differences between the Danckwerts boundary conditions and alternative

470

formulations for tubular reactors can be found elsewhere 42 .

471

Appendix C – On Isolating Reaction Contributions

472

We present two different ways of eliminating the effect of the inlet and outlet flows in the measured

473

numbers of moles, and we will compare these approaches with that based on the linear transformation.

474

In particular, we are interested in reconstructing the rate rv,i (t), the batch extent ξi (t) and the vessel

475

extent xr,i (t).

22

Page 22 of 28

476

Numbers of Moles in Reaction-Variant Form

477

Re-writting equation (8) as: ˙ n(t) − Win uin (t) − n0 δ(t) + ω(t) n(t) = NT rv (t),

Z

Z

t

uin (τ ) dτ − n0 +

0

t

ω(τ ) n(τ ) dτ = NT 0

rv (τ ) dτ. 0

t

Z

t

uin (τ ) dτ +

and the batch extents of reaction, ξ (t) :=

t

ω(τ ) n(τ ) dτ,

0

rv (τ ) dτ,

an

0

equation (57) gives:

Z

cr

Z

0

481

t

Upon defining the numbers of moles in reaction-variant (RV ) form, nRV (t) := n(t) − n0 − Win

480

Z

nRV (t) = NT ξ (t), or in differential form,

M

482

n˙ RV (t) = NT rv (t),

nRV (0) = 0S .

(57)

ip t

n(t) − Win

479

(56)

and integrating gives:

us

478

n(0) = 0S ,

(58)

(59)

(60)

(61)

Numbers of Moles in Vessel Reaction-Variant Form

484

If the inlet and outlet flowrates uin (t) and uout (t) are known, one can compute xin (t) and xic (t) according

485

to (16) and then use (14) to compute the contribution of the reactions, labeled the numbers of moles in

486

vessel reaction-variant (vRV ) form, as follows:

488

Ac ce pt e

487

d

483

nvRV (t) := n(t) − Win xin (t) − n0 xic (t),

(62)

nvRV (t) = NT xr (t),

(63)

which leads to:

or in differential form,

n˙ vRV (t) = NT rv (t) − ω(t) nvRV (t),

nvRV (0) = 0S .

(64)

489

Expression (63) is very useful for analysis purposes as will be seen later. Note that nvRV (t) 6= nRV (t) in

490

the presence of an outlet.

23

Page 23 of 28

491

Comparison with Linear Transformation

492

A key feature of the linear transformation T is its ability to isolate and remove the effect of the inlet and

493

outlet streams from the measured numbers of moles. Next, we will compare the experimental implications

494

of computing rv,i (t) from n˙ RV (t), ξi (t) from nRV (t) and xr,i (t) via the linear transformation from either

495

n(t) or nvRV (t). It is assumed here that the measured numbers of moles n(t) are noisy and only available

496

unfrequently at the time instants th , h = 1, 2, · · · .

498

1. Computation of rv,i (th ) from n˙ RV (th ). This is done by inversion of equation (61), with n˙ RV (th ) computed from equation (58) as:

(65)

One sees that this operation requires the differentiation of the noisy and scarce signal n(th ).

us

499

cr

˙ h ) − Win uin (th ) + ω(th ) n(th ). n˙ RV (th ) = n(t

ip t

497

2. Computation of ξi (th ) from nRV (th ). This is done by inversion of equation (60), with nRV (th )

501

computed from equation (58). One sees that this operation requires the integration of the noisy

502

and scarce signal n(th ).

an

500

3. Computation of xr,i (th ) via linear transformation from n(th ). This is done via equation (9) as

504

xr,i (th ) = Ri n(th ), which does not require differentiation nor integration of noisy signals. The

505

transformation requires at least R + p + 1 measured species. 4. Computation of xr,i (th ) via linear transformation from nvRV (th ). T+

)i n

d

506

M

503

This is done as xr,i (th ) =

(th ), which does not require differentiation nor integration of noisy signals either.

507

(N

508

The transformation requires only R measured species.

Ac ce pt e

vRV

509

This shows that the path over the linear transformation, in particular Path (4) using nvRV (th ), has a

510

clear experimental advantage.

511

References

512

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27

Page 27 of 28

ip t cr us an

reaction subspace

Ac c

invariant subspace S

1

ep te

d

M

initial condition subspace

R

inlet subspace p

Page 28 of 28