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Optimal selection of reference components and measurements in reaction systems Bala Shyamala Balaji, Nirav P Bhatt, and Sridharakumar Narasimhan Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02993 • Publication Date (Web): 09 Oct 2018 Downloaded from http://pubs.acs.org on October 14, 2018

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Industrial & Engineering Chemistry Research

Optimal selection of reference components and measurements in reaction systems Bala Shyamala Balaji, Nirav Bhatt,∗ and Sridharakumar Narasimhan∗ Department of Chemical Engineering, Indian Institute of Technology-Madras, Chennai 600036, India E-mail: [email protected]; [email protected] Phone: +91 44 2257 4177 Abstract The number of moles of components in reaction systems can be decomposed into reaction variants (states that change with the progress of reactions) and reaction invariants (states that do not change with progress of reactions). The concept of reaction variants/invariants is used in modeling, control and design applications. For computation of reaction variants, it has been shown that a subset of components needs to be measured and labelled as reference components. This idea ensures that measurement of a subset of components is sufficient in order to estimate the unmeasured components using reference components and known information of flowrates, inlet compositions, volume and initial conditions. There are several feasible subsets of components which can be measured for computation of reaction variants and estimation of unmeasured components. In practice, measurements are corrupted with noise. Further, there is a cost associated with measurement of each component due to the nature of instrument or protocol. In this work, we address the problem of selection of reference components or measurements. First, structural conditions for selecting minimum number of reference components are revisited for homogeneous and heterogeneous reaction systems. It

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is shown that the component mole numbers lie in a lower dimensional affine subspace under assumption of known inlet flowrates and compositions, initial conditions and volumes. Then, we formulate optimization problems for optimal selection of reference components that maximize the quality of the estimates with constraints on cardinality (number of components to be measured) and cost associated with measurement. These formulations are demonstrated via homogeneous reactions and heterogeneous reactions for different reactor configurations and cost associated with sensors.

1

Introduction

Models of reaction systems are important for design, optimization and model based control in chemical engineering and systems biology. During analysis of reaction systems, we deal with two kinds of state variables, (i) reaction variant states (often known as the extents of reaction) which vary with the progress of reactions, and (ii) reaction invariant states which do not vary with the progress of reactions. These concepts have been exploited to study modelling, control, and design related applications in literature 1 . For example, the concept of reaction invariants has been used to describe the composition space for reactive distillation 2 (for more applications, see references in Gadewar et al. 3 ) using noisy measurements of concentrations. Alternatively, the concept of reaction variants has been used to identify reaction kinetics from concentration measurements 1,4 . It is known that the measurements of a subset of the components would suffice to compute the reaction variants 3,5–7 . This subset is referred here as reference components which can be generalised to other systems as well. The problem of selection of reference components is a very important study in chemical engineering and systems biology. 3,8 For instance, for studying vapor-liquid equilibrium with multiple reactions, reference components have been used to reduce the problem dimension 5 and hence, simplify the analysis. Further, the authors observed that the lower dimensional representation using reference components helps in visualisation and interpretation. The idea of reference components has been used to obtain 2


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simplified expression for Gibbs free energy, chemical potential and other thermodynamic functions. 9,10 It has been shown that the thermodynamic potentials can be expressed as a set of new composition variables. Gadewar et al. 11 have demonstrated use of the concept of reference components in validating experimental data, material balance and process design. Further, Omtveit et al. 12 used reference components to construct “attainable region” for large reaction systems. On the other hand, the concept of reference components have been widely used in process control related applications. 13–17 Waller and Makila 13 applied the concept of reference components for model-order reduction. Hammarström 14 showed that reaction and control variants lie in a lower dimension space than the state–space model for reaction systems and used the analysis to select components to be measured for a control related application. Further, the concept of reference components has been used in construction of asymptotic and interval observers for state and interval estimation. 15,17,18 For batch reaction systems, it has been shown that the number of reference components required for computing reaction variants is equal to the number of independent reactions. 8 For example, in a batch reaction system consisting of S components and R independent reactions, the number of reference components required is R. In addition, the central condition for selecting R reference components is that the stoichiometric matrix corresponding to the R reference components should be of full rank. However, there are several feasible sets satisfying this condition 3 . Further, when the measurements are noisy, certain choice of reference components may lead to poor results for the given applications. Hence, selecting a suitable subset of reference components is essential for the end applications. For other reactor configurations (e.g., with inlets and outlets or heterogeneous systems), reference components should satisfy appropriate structural conditions. In particular, it is shown that the number of moles lie in a lower dimensional affine subspace. There are several choices of measurement combinations that satisfy these conditions. One metric to rank these combinations is to use a quantitative measure based on the covariance matrix of the estimates of the measured and

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reconstructed concentrations or mole numbers. Given the affine relationship between the mole numbers, it is possible to express the covariance matrix of the estimates in terms of the linear model, sensor locations and the respective instrument variances. The objective of this work is to develop a method for selecting reference components in reaction systems by minimizing a scalar norm of the covariance matrix subject to constraints such as cost, cardinality etc. Here, the problem of selection of reference components can be posed as a combinatorial problem which is shown to be a mixed integer cone program and hence can be solved to obtain globally optimal solutions. The methodology is demonstrated by considering various reaction systems. Dynamic models of homogeneous and heterogeneous reaction systems in the extents domains are given in Section 2. Definitions and conditions for selection of reference components and optimaization formulations for optimal selection of reference components are presented in Section 3. Section 4 demonstrates optimal selection of reference components for two different types of reaction systems. This is followed by conclusions with recommendations for future work in Section 5.

2

Models of Reaction Systems

In this section, we describe two forms of models for reaction systems: (i) number of moles as state variables, and (ii) extents as state variables.

2.1

Model in mole number domain

Consider a general fluid–fluid reaction system with two phases: G (gas) and L (liquid) phases. A schematic diagram of the system is shown in Fig. 1. 6 The G phase involves Sg components, pg inlets, and an outlet, while the L phase involves Sl components, pl inlets, Rl reactions and an outlet. For the sake of simplicity, it is assumed that the Rl reactions take place in the L phase only. All the results in this work are also applicable when reactions take place 4


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in both phases. Both phases are connected by the pm mass–transfer rates. The following assumptions are made: (1) The gas and liquid phases are homogeneous, (2) the total volume of the reactor is constant, (3) the reactions occur only in the liquid bulk, and (4) there is no accumulation in the boundary layer. Then, the material balance equations can be written for both phases as follows: uout,g (t) ng (t), ng (0) = ng0 mg (t) uout,l (t) nl (t), nl (0) = nl0 n˙ l = NT Vl rl (t) + Win,l uin,l (t) + Wm,lζ (t) − ml (t)

n˙ g = Win,g uin,g (t) − Wm,gζ (t) −

(1) (2)

where ni (t) is the Si -dimensional vector of the number of moles with i ∈ {g, l}. N is Gas outlet

pg gas inlets

uout,g

Win,g , uin,g Gas phase ng , mg

G to L pl liquid inlets Win,l , uin,l

L to G Mass transfer

Liquid phase nl , ml

Liquid outlet uout,l

Figure 1. Schematic diagram of a fluid-fluid reaction system with bulk phases, G and L. The two bulk phases are connected with mass transfer steps between them.

the (Rl × Sl )-dimensional stoichiometric matrix; Vi (t) is the volume of the ith phase; rl = [r1 (nl , Vl ), r2 (nl , Vl ), . . . , rRl (nl , Vl )]T is the Rl –dimensional vector of reaction rates; Win,i = ˜ M−1 w,i Win,i is the (Si × pi )-dimensional inlet composition matrix with Mw,i being the Si ˜ in,i = [w1 , w2 , . . . , wpi ] where wk dimensional diagonal matrix of molecular weight; W in,i in,i in,i in,i 5



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is the weight fraction distribution of k th inlet in the ith phase; uin,i is the pi -dimensional vector ˜ of the inlet flowrate into the ith phase; Wm,i = M−1 w,i Em,i is the Si × pm dimensional mass m ˜ m,i = [˜ ˜pm,i ˜km,i being the Si -dimensional transfer matrix to the ith phase, E e1m,i , . . . , e ] with e

vector with the element corresponding to the kth transferring components equal to unity and the other elements equal to zero; ζ = [ζ1 (nl , ng , V, Vl ), ζ2 (nl , ng , V, Vl ), . . . , ζpm (nl , ng , V, Vl )]T is the pm -dimensional vector of the mass–transfer rates, V is the reactor volume, and uout,i is the outlet flowrate of the ith phase. mi is the mass of the ith phase and is related to ni as follows: mi (t) = 1TSi Mw,i ni (t)

(3)

where 1 is a column vector having one as elements of appropriate size. Eq. (3) shows that the mass for each phase at time t can be computed from the number of moles of a particular phase at time t. Hence, there is no need to write a total mass balance equation for each phase.

2.2

Model in extents domain

The concept of the extent of reaction is widely used in the literature for modelling of batch reactions. 1 The extent of each reaction is essentially the contribution of a particular reaction and decoupled from the other reactions. For a closed reaction system with R independent reactions and S species, the change in the extent of ith reaction, ξi (t), is defined as: ξ˙i (t) = V (t) ri (t),

ξi (0) = 0.

(4)

where ri is the rate of ith reaction. Recently, the concept of the batch extents has been extended to open reaction systems, and the concept of the extents of reaction has been introduced by accounting the effect of the outlet term. 19 In open reaction systems, such as, continuous stirred tank reactor (CSTR) or multiphase reactors, the outlet steam removes some amount of all the species present in the reactor. Hence, the change in the extent of ith 6


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reaction in Eq. (4) has to be modified to account for the outlet term. Then, the extent of reaction xr,i by accounting for the the outlet term is defined as:

x˙ r,i (t) = V (t) ri (t) −

uout (t) xr,i (t), m(t)

xr,i (0) = 0.

(5)

For reactors without an outlet stream, note that the extent of the reaction given in Eq. (4) reduces to Eq. (5) for closed reaction systems. Similarly, the concept of the extents of inlet and outlet flowrates and mass–transfer has been introduced for open homogeneous and heterogeneous reactions. 6 In this section, these definitions of the extents of reaction, inlet flow, outlet flow, and mass–transfer are used to write model equations in the extents domain. Then, for a G–L reaction system, the model equations in terms of the extents as state variables for the G and L phases can be written as follows: 7 G Phase uout,g xm,g , xm,g (0) = 0, mg uout,g xin,g , xin,g (0) = 0, x˙ in,g = uin,g − mg uout,g λ˙ g = − λg , λg (0) = 1, mg x˙ m,g = ζ −

(6)

L Phase

x˙ r = Vl rl −

uout,l xr , xr (0) = 0, ml

uout,l xm,l , xm,l (0) = 0, ml uout,l xin,l , xin,l (0) = 0, x˙ in,l = uin,l − ml uout,l λ˙ l = − λl , λl (0) = 1, ml x˙ m,l = ζ −

(7)

where xr is the Rl -dimensional vector of the extents of reaction in the L phase, xm,i is the 7



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pm -dimensional vector of the extent of mass transfer in the ith phase, i ∈ {g, l}, xin,i is the pi -dimensional vector of the extent of inlet flowrate in the ith phase, and λi is the scalar describing the discounting of the initial condition in the ith phase, respectively. The number of moles in the G and L phases can be related to the extents by the following relationships:

ng (t) = −Wm,g xm,g (t) + Win,g xin,g (t) + ng0 λg (t) nl (t) = NT xr (t) + Wm,l xm,l (t) + Win,l xin,l (t) + nl0 λl (t)

(8)

Hence, Eqs. (6), (7), and (8) comprise the model of reaction systems with the extents as variables. Note that there are 2 pm extents of mass–transfer, xm,g and xm,l , in the model equations. For special reactors, such as batch and semi–batch reactors, the model equations in mole number and extents domain are given in Table S1 in Section 1 of supporting information.

3

Optimal selection of reference components

In this section, the reference components definition and structural conditions for minimal number of reference components are revisited first. Optimization problems for selecting reference components are formulated subsequently.

3.1

Reference components: Definition and conditions

In data-driven applications such as kinetic modelling, state estimation and state reconstruction, the quality of measurements has an important bearing on final results. In reaction systems, the inlet and outlet flowrates are often manipulated variables or can be measured without much cost at relatively high frequency and precision as compared to concentration measurements. It is often difficult or costly to measure concentrations or equivalently mole numbers of all components. Using known relationships between the concentrations (or mole

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numbers), it is possible to infer the concentrations or mole numbers of the unmeasured species from measurements of a subset of components. Definition 1 (Reference Components) A subset of components which is required to be measured in order to reconstruct or estimate or compute concentrations of the unmeasured components are called reference components. The definition of reference components in this work is closely related to the definitions available in Gadewar et al. 3 and Ung and Doherty. 5 There exist several sets of feasible reference components. Then, the following questions related to reference components for data–driven applications can be posed (a) What are the conditions for selection of reference components? (b) What is the minimal number of reference components? (c) How to select reference components in practical applications? Questions (a) and (b) are revisited from the point of view of state reconstruction and estimation in this section. The answers to Questions (a) and (b) provide conditions for selection of minimal reference components. Motivations for the relevance of Question (c) are provided in this section while optimal selection of reference components is discussed in Section 3.3. The structural conditions to select minimal reference components for homogeneous and heterogeneous reaction systems are as follows. For a homogeneous reaction system, let denote Sa the number of measured components. The condition for the measured components to be a valid set of reference components is rank (Na ) = R where Na is the stoichiometric matrix corresponding to the measured components. For a heterogeneous reaction system, denote (Sl,a +Sg,a ) to be the measured components in the L and G phases such that the same components can be measured in both phases, respectively. Further, the flow rates uin,i and uout,i are also measured without any error, and N, Win,i , Wm,i , and ni0 , where i = {g, l} are known without error. Then, the conditions for 9



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the measured components (Sl,a +Sg,a ) to be a valid set of reference components are as follows: (C1) Sl,a + Sg,a ≥ R + pm , and (C2) rank (Wmg,a ) = pmg , (C3) rank([NTa , Wml,a ]) = R + pml , where Wmg,a is the Sg,a × pmg –dimensional sub–matrix of Wm,g corresponding to the measured components in the G phase, and the Wml,a is the Sl,a × pml -dimensional sub–matrix of Wm,l corresponding to the measured components in the L phase. pmg and pml are the number of transferring components in the G and L phases, respectively. The details on structural conditions to select reference components, related theorems and reconstruction of unavailable measurements are provided in Section 2 of supporting information.

3.2

Selection of reference components

It is known that a subset of the components is sufficient to calculate or reconstruct the other unmeasured components. The conditions in Section 3.1 are necessary conditions that need to be satisfied in order to reconstruct the concentrations of the unmeasured components from the measurements of the reference components. E.g., for a homogeneous batch reactor, the number of moles of unmeasured components nu (t) can be uniquely calculated as follows nu (t) = NTu NTa + (na (t) − n0,a ) + n0,u

(9)

provided rank of NTa is R. Mathematically, all choices of reference components that satisfy the rank condition are equivalent. However, numerically, it is important to choose reference components such the matrix NTa is well conditioned. Further, if the measurements are corrupted by error, the variance of the estimates of na (t) and nu (t) will depend on the variance of the measurements. Different choices of reference components will result in different variances of estimates of na (t) and nu (t). When more than the minimum number of components are measured, the measurements can be adjusted or reconciled to obtain consistent estimates while improving the precision. 20 Further, in heterogeneous reaction systems, costs of measurement in the G and L phases can be different and can also differ from component

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to component. Hence, it is important to choose the reference components such that quality of estimates is maximized subject to constraints on individual and total cost of sensors etc.

3.3

Optimization formulation for selection of reference components

The formulation of the problem of optimal components selection in practice (Question (c) in Section 3.1) is addressed in this subsection. First, we develop affine representations for the components mole numbers showing that they lie in a lower dimensional subspace. This is followed by deriving an expression for the covariance of estimates of the mole numbers which is used to formulate an optimization problem for selection of reference components. The objective is to maximize the quality of the estimates subject to constraints on cost and or the number of components to be measured. To show an affine representation of number of moles, we exploit the relationship between the number of moles and the extents. Consider reactions occurring in a homogeneous batch reactor, the number of moles n(t) satisfies the following relationship

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

(10)

Hence, if n0 is known, n(t) are constrained to lie in an affine subspace and can be compactly expressed as

n = Ax + b,

(11)

where A = NT , x = xr and b = n0 . Similarly, it can be shown that the mole numbers in different reaction systems lie in an affine subspace. Table I consolidates the values of A and b for various reactor configurations (refer Section 3 in the supporting information for detail derivations.). Depending on the reactor configuration, b contains the effect of initial conditions, inlet flows or both. It is assumed that the initial conditions, flow rates and volumes are known precisely, and hence, b is known without error. 11



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TABLE I. Affine representations for n x Batch process Semi batch process

Only inlet Both inlet and outlet (reduced form)

A Homogeneous system NT NT Heterogeneous system

xr xr

xr xm xr xmg

0 NT

0 −Wm,g NT W m,l −Wm,g ,β = βWm,l

Vl qout,g . Vg qout,l

b n0 Win xin (t) + n0

xin,g Win,g 0 + n0 Win,l xin,l 0 xin,g Win,g 0 xin,l 0 Win,l

Now, we consider a general reaction system in the affine form, with the measurements of mole numbers corrupted with noise as follows:

n ∈ Rn , A ∈ Rn×m , x ∈ Rm , b ∈ Rn , v ∈ Rn

n = Ax + b + v;

(12)

where n is an n-dimensional vector of mole numbers, x is an m-dimensional state vector with the extents of reaction and/or mass transfer as states. In Eq. (12), the measurements (n) and the state variables (x) are related linearly through a measurement model matrix A, an offset term b which is known precisely and additive Gaussian white noise vector v with 2 vi ∼ N (0, σv2 ) with σv,i refers to the respective instrument variances. It is to be noted that

x is the set of internal variables (e.g., extents of reaction, mass transfer etc. as the case may be) and typically n ≥ m. The measurements n clearly do not lie in the affine subspace due to measurement error and hence are not consistent. This can be treated as a reconciliation problem where the objective is to reconcile measured n so as to obtain a consistent estimate ˆ . A weighted least–squares formulation can be used to estimate x. The formulation can be n given as ˆ = arg min (n − Ax − b)T Q(n − Ax − b) x x

(13)

2 where Q is the n-dimensional diagonal matrix with 1/σv,i where i = 1, . . . , n as diagonal

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entries. Using the formulation in Eq. (13), the estimates of x can be obtained as follows:

ˆ = (AT QA)−1 AT Q(n − b) x

(14)

Consistent and improved estimates of n can be obtained as

ˆ = Aˆ n x + b = A(AT QA)−1 AT Q(n − b) + b

(15)

ˆ are also normally distributed with the With the Gaussian noise assumption, the estimates n following mean and covariance matrix: 21,22

E[ˆ n] = n Σn = A(AT QA)−1 AT .

(16)

From Eq. (16), it can be observed that the estimates are unbiased and the quality of the estimates as measured by their covariance depends on the sensor variances and choice of measurements. It is possible to use this treatment to pose an appropriate sensor placement 2 problem by redefining Q matrix as a decision variable as follows: Q = diag(qi /σv,i ), ∀ i =

1, . . . , n, where qi is a binary variable indicating whether a variable is measured (qi = 1) or not (qi = 0). Then, using covariance matrix with redefined Q, Σn , an optimal components selection problem can be formulated where the objective is to maximize the quality of the estimates subject to constraints. This can be achieved by minimizing some scalar norm of the covariance matrix, e.g., A–optimal (min T r(Σn )), D–optimal (min det(Σn )), E–optimal (max min(eig(Σn ))) or T–optimal (max T r(Σ−1 n )). Here, we chose the A-optimal criterion, i.e., to minimize the trace of Σn subject to a constraint on the number of measurements.

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The formulation is given as below min T r(Σn (qi )) qi

Problem P1

s.t.

n X

(17) qi = R

i=1

where R is the required number of components to be selected and n is the total number of components. A lower value of the objective function implies that the estimate quality is better and vice versa. It should be noted that in order to ensure that the error–covariance matrix is invertible, it is essential that the components selected form a valid set of reference components. The diagonal matrix Q will vary according to the variances of instruments specified and result in selection of appropriate components from the scaled matrix. There are occasions when cost of sensors plays a major role. It could be possible to measure some components with ease and difficult or more expensive to measure some other components. There could also be difficulty in measuring some component in the liquid state as against measuring it in the gas phase and vice versa. All this can be accounted for in the following optimization formulation where cost can be incorporated as a constraint to the problem: min T r(Σn (qi )) qi

Problem P2

n X

(18) c i qi ≤ c

∗

i=1

where ci is the cost associated with the ith sensor and c∗ is the maximum cost allowed. The advantages and disadvantages of these formulations are demonstrated in Section 4. A combined formulation can be obtained from the formulations P1 and P2 in Eq. (17) and Eq. (18) as:

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min γ1 qi

Problem P3

N X

ci qi + γ2 T r(Σn (qi )) (19)

i=1

n X

c i qi ≤ c ∗

i=1

where γ1 and γ2 are the weights to each of the objectives. The formulations P1 , P2 and P3 in Eqs. (17), (18) and (19) are directly applicable for determining the optimal set of reference components in reaction systems depending on the constraints imposed. Depending on the reactor configuration, x, A, and b are chosen appropriately from Table I. The Integer NonLinear Program (INLP) problems P1 , P2 , and P3 can be reformulated as Mixed Integer Cone Programs (MICP), and hence be solved to obtain globally optimal solutions 23 . E.g., consider the problem P3 in Eq. (19) which can be reformulated as:

min γ1 qi ,Y

n X i=1

n X

ci qi + γ2 T r(Y)

i=1





A  Y  c i qi ≤ c ∗ ,  0 T T A A Q(qi )A

(20)

Some remarks on the formulations P1 , P2 , and P3 are in order: 1. Invertibility of AT QA ensures that the conditions on the number of selected components and rank conditions (appropriately defined in Section 3.1; for details, refer Section 2 of SI) are satisfied. However, if R is lower than the minimal number as described in Section 3.1 or the rank conditions are not satisfied, the covariance matrix Σn is singular or equivalently, the unmeasured mole numbers cannot be reconstructed uniquely. The selected components using the formulations described in this section conform to all the required conditions mentioned above and are maximally informative. 2. Measurement of lesser number of components reduces the cost. This can be controlled by the parameter R in the problem P1 (Eq. (17)). Cost of each sensor can be explicitly added to the constraints using the formulations P2 and P3 (Eqs. (18) and (19)). 15



3. Selection of appropriate reference components ensures that the unmeasured components can be estimated from the measurements of the reference components and b which includes flow rates, known information about unmeasured components (initial conditions). E.g. for homogeneous reaction systems, in Theorem 1 in Section 2 of SI, the selected components, b and known information are used to compute the extents of reaction first and then, the extents of reaction and b are used to compute the unmeasured components. 4. All species are given equal importance in the above formulations. However, a weighted selection problem can be formulated by minimizing T r(WΣn ) where W is a weighting matrix. E.g, the entries in W can be chosen according to the ability to measure or importance of measuring respective components. 5. Flow rates and volumes are relatively easier to measure with higher precision as compared to concentrations or mole numbers. Hence, it is reasonable to assume that all flow rates (if applicable) and volumes are known without error. 6. The formulations are developed in terms of mole numbers. If concentrations are the measured quantities, mole numbers in the ith phase can be expressed in terms of concentrations as ni = ci Vi , where Vi is the volume of the ith phase. Hence, variances of mole number measurements can be linearly related to the variances of the concentration sensors. 7. When all measurements have identical variances (σ 2 ), the formulation P1 in Eq. (17) indirectly ensures numerical observability. Consider reactions occurring in a homogeneous batch reactor with A = NT . Without loss of generality, let NT = [Na , Nu ]T , where Na and Nu correspond to the measured and unmeasured components. We have motivated the need for NTa to be well conditioned (refer Eq. (9)). T r(Σn ) = P T r(AT A(AT QA)−1 ) ≤ σ 2 T r(AT A) s12 , where si are singular values of NTa . Hence, i

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minimizing T r(Σn ) indirectly ensures that the singular values of NTa are relatively large and NTa is well conditioned. 8. We have exploited only spatial redundancy (affine relationships between the mole numbers) for selection of reference components. Temporal relationships require reaction and mass transfer models which are not used. Hence, the analysis in this section is particularly suitable for measurement selection when carrying out experiments for identification of reaction and mass transfer rate parameters. The ideas presented in this section are specifically brought out in the examples provided in Section 4.

4

Case studies

The formulations in Section 3 for optimal selection of reference components are illustrated via different reactor configurations for both homogeneous and heterogeneous systems. The optimization problems have been solved using a desktop computer with an i5 processor and 4 GB RAM. The problems have been reformulated as MICPs (as explained in Section 3) and in particular Linear Matrix Inequality (LMI) problems with binary variables in the YALMIP 24

toolbox and solved using SeDuMi solver to obtain globally optimal solutions. Alternatively,

algorithms such as Outer Approximation (OA), 25 Extended Supporting Hyperplane (ESH) 26 or their variants can be used for larger sized and complex problems.

4.1

Homogeneous reaction systems

In this section, an example of ethanolysis of phthalyl chloride in a semi-batch reactor is considered. Another example of silicon chemical vapor deposition in a batch reactor is thoroughly explained in Section 4 of the supporting information. We consider ethanolysis of phthalyl chloride in a semi-batch reactor 27 involving three independent reactions (R = 3) between seven components (S = 7) which are described as 17



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follows:

R1 : A + B → C + D R2 : C + B → E + D (21)

R3 : D + B ↔ F + G

where phthalyl chloride monoethyl ester (C) is the desired product, phthalyl chloride (A) and ethanol (B) are reactants, phthalic diethylester (E) and hydrochloric acid (D) are byproducts. Further, ethanol and hydrochloric acid react to form ethyl chloride (F) and water (G). The stoichiometric matrix is 

1 0 0 0  −1 −1 1  A=  0 −1 −1 1 1 0 0  0 −1 0 −1 0 1 1

T     

(22)

with n = [nA , nB , nC , nD , nE , nF , nG ]T . Component A is charged initially in the reactor while component B is fed continuously. The mass flowrate of B is assumed to be known without any errors for all time. Further details regarding the reaction system are provided in Bhatt. 27 Using conditions in Section 3.1, measurement of three components would suffice in order to estimate the concentrations of the rest of the components. Under the assumption of equal variances for all components, the solutions represent the optimal selection obtained by solving the optimization problem P1 for the various values of R. It is observed that the trace of the covariance matrix T r(Σn ) = 4.75 when the minimal number of components (B, C, D) are selected. As the value of R increases to 4, 5 and 6, the components selected are (B, C, D, F),(A, B, C, D, F) and (A, B, C, D, E, G), respectively. The trace was found to decrease from 4.75 to 3.33 as redundancy is introduced. The objective function values, time and iterations taken for the solution to be achieved can be referred from Table S3 in 18


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supporting information. In the second case, it is assumed that the cost of various sensors is different. Then, the formulation P2 in Eq.(18) is used. The cost of different sensors is: [1, 2, 3, 4, 5, 4, 3] units. Table II provides the components to be measured for various maximum cost imposed, components selected, computational time and iterations. TABLE II. Cost constrained components selection Sr.No. 1 2 3 4

Max cost (c∗ ) 7 8 9 10

Cost selected Time (s) Iterations 7 0.8 11 7 0.8 19 9 0.3 8 10 0.4 8

Components selected T r(Σn ) A, B, D 10 A, B, D 10 B, C, D 4.75 A, B, C, D 4.36

The summary of interpretation of the results in Table II is given below 1. The formulation ensures that the T r(Σn ) is the least possible with respect to the cost constraints. This can be observed from the simulation no. 1 and 2, where R = 3. As the maximum cost is increased to 9 units, the estimation error comes down from 10 to 4.75 in the simulation no. 3. 2. When the maximum cost is increased, it can be observed that the number of species selected also increases alongwith the quality of estimates. The objective function decreases as the number of sensors selected increases. The solutions for increasing values of cost till maximum cost can be referred from the supporting information (Table S4).

4.2

Heterogeneous reaction systems

We consider chlorination of butanoic acid, a gas-liquid reaction system. 27 In this system, chlorine transfers from the gas phase to the liquid phase and reacts with butanoic acid (A) with ethanol as solvent in the liquid phase. It has two parallel auto catalytic reactions that consume dissolved Cl2 (B). The main reaction produces α-monochlorobutanoic acid (C) and 19



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Hydrochloric acid (D). The second reaction produces α-dichlorobutanoic acid (E) and HCl (D). HCl is volatile and exists in both phases. The reactions are given as follows:

A+B →C +D A + 2B → 2D + E Chlorine and hydrochloric acid are transferring components in the system. As per the structural conditions provided in Section 3.1, the minimum number of components to be selected is four. Gas–liquid reactor configuration of continuous reactors with both inlets and outlets are studied in the following case study. Another heterogeneous reaction case study of a semibatch reactor with only gas inlet with the same system is discussed in Section 6 of supporting information. Since air is removed with the outlet, the mole balance of air needs to be considered, thus Sg = 3 with Sg = {air, B, D}. Furthermore, Sl = 5 with Sl = {B, A, C, D, E}. The volumes, inlet and outlet flow rates are measured precisely with no error. The stoichiometric matrix and extended inlet matrices for the gas and liquid phases are given by



0 0   0 0 0 −1 −1 1 1 0   N=  ; Wm,g =  0  0.0141  0 0 0 −2 −1 0 2 1 0 0.0274 





   ; Wm,l  



     =     

0.0141 0 0 0 0

0

   0    0    0.0274   0 (23)

The liquid and gas components can be combined to give: n = [nair nB,g nD,g nB,l nA,l nC,l nD,l nE,l ]T . The molecular weight matrix in the liquid phase and the gas phase

20




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are



Mw,l



0  29 0   Mw,g =  0   0 71 ,   0 0 36.45   71 0 0 0 0      0 88.12  0 0 0     = 0 122.52 0 0  0 ,     0 0 0 36.45 0    0 0 0 0 156.97

˜ m,g E

˜ m,l E



0  = 1  0  1   0   = 0   0  0



0  0   1  0   0   0    1  0

(24)

˜ m,i , i ∈ Using information of stoichiometric matrix, molecular weight matrices and the E {g, l}, the matrix A can be written as follows: 

          A=           where β =

0

0

0

0



   0 0 −0.0141 0   0 0 0 −0.0274     −1 −2 0.0141β 0     −1 −1 0 0    1 0 0 0    1 2 0 0.0274β   0 1 0 0

(25)

Vl qout,g . Vg qout,l

Assuming equal variances of all measurements, σ 2 = 1, two different sensor cost scenarios are considered: Cost 1=[1; 1.1; 1.2; 1.09; 0.99; 0.94; 0.93; 1.21 ] and Cost 2=[1; 4.2; 5.3; 4.8; 2.9; 3.3; 1.2; 6.5]. Table III provides the solution of the problem P3 (refer Eq.(19)) with equal weights to the two objectives (γ1 = 1; γ2 = 1) for various β values and the two cost scenarios. The results for Example 2 are discussed next. 21



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TABLE III. Trade off between estimation error and cost for various β values Sr. β No.

Max Cost cost selected

1 2 3 4 5

0.1 0.5 1 5 10

4.5 4.5 4.5 4.5 4.5

4.16 4.16 4.06 4.06 4.06

6 7 8 9 10

0.1 14 0.5 14 1 14 5 14 10 14

14 14 13.5 13.5 13.5

Time (s) Iterations Components selected Cost 1 0.5 5 Dg , B l , C l , D l 0.5 7 Dg , B l , C l , D l 1.3 24 B g , B l , C l , Dl 0.2 3 B g , B l , C l , Dl 0.3 3 B g , B l , C l , Dl Cost 2 1.6 27 B g , Dg , C l , Dl 0.8 13 B g , Dg , C l , Dl 1.2 19 B g , B l , C l , Dl 0.5 9 B g , B l , C l , Dl 0.5 9 B g , B l , C l , Dl

T r(Σn )

Net objective

5.02 5.68 8.5 5.1 5.02

9.18 9.84 12.56 9.16 9.08

106.01 10.37 8.5 5.10 5.02

120.01 24.37 22 18.60 18.52

1. Air does not appear in the optimal selection. This is to be expected since air has no role in the measurement matrix A and does not affect the quality of estimates. 2. The value of β affects the quality of estimates for a fixed upper bound on maximum cost as can be seen from the Table III. 3. It happens that the quality of estimates is same for the costs 1 and 2 for simulations 3-5 and 8-10 even though β is varied. It is only because the selected sensors are the same. 4. Alternatively, for β = 0.1 and 0.5 the estimation error is different in both the cost scenarios. It could be attributed to cost of species in the two cases which results in selection of less costly components, but which results in poor quality estimates. The solution also depends on the upper bound on the cost. It is observed that if the upper bound on cost is increased from 14 units to 15 units, the quality of estimates improve drastically. E.g., when β = 0.1 and the maximum cost available is increased marginally to 15, the selected sensors are Dg , Bl , Cl , Dl and the quality of estimates improves significantly with T r(Σn ) = 5.02. Similar situation is observed when β = 0.5 and available cost =15, the selected sensors are Dg , Bl , Cl , Dl and T r(Σn ) = 5.68. 22


Page 23 of 30

It should be noted that β influences the estimate quality and hence the sensor selection and the total objective. To illustrate the effect of β on estimate quality alone, we consider the formulation P1 in (17) with only estimation error minimization as an objective (or equivalently all sensor costs are the same). Figure 2 consolidates the effects of β in the component selection problem for β values ranging from 0.1 to 10 and R = 4, 5, 6. It can 8.5 R=4 R=5 R=6

8 7.5 7

Tr(Σ y)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6.5 6 5.5 5 4.5 4 0

1

2

3

4

5

6

7

8

9

10

β

Figure 2. Influence of β on estimation error be seen that quality of estimates changes with β with significant variation when R = 4. However, β can be tuned to improve the quality of estimates as observed from Figure 2. It is most important to select minimal number of components with regard to cost. Hence, this would serve as a good factor for designing experiments where only a limited number of measurements is possible. A table providing the effects of β on estimation error has been provided for different values of β in supporting information for reference (Table S8, Section 7).

23



5

Conclusions

In practice, it is difficult to measure all the components participating in reaction systems. The quality of components measured plays an important role in data–driven applications such as kinetic modelling, state and parameter estimations etc. In this work, optimal selection of reference components in homogeneous and heterogeneous reaction systems has been addressed based on measurement models. First, it has been shown that component mole numbers lie in a lower dimensional affine subspace under assumptions of perfect knowledge of operational variables such as flowrates, compositions and initial conditions when the extents are used as the state variables. Further, it is shown that the quality of the estimates of component mole numbers depends on their covariances. Then, three integer optimization problems (P1 , P2 and P3 ) have been formulated to select reference components for maximation of quality of estimates under different constraints. In Problem P1 , the trace of covariance matrix of estimates of component mole numbers is minimized with the number of components to be selected as constraint. Since the cost of sensors plays a major role in selection of reference components, Problem P2 minimizes the trace of co-variance matrix subject to the maximum total cost allowed on selecting reference components. Then, Problem P3 combines both formulations for selection of reference components. Utility of the proposed optimization formulations for selection of reference components has been demonstrated for homogeneous and heterogeneous reaction systems with different reactor configurations. It has been shown that the cost of sensors, maximum allowed cost, and quality of measurements play role in selecting reference components. Further, in heterogeneous reaction systems with outlets, it is observed that the operating condition also plays an important role. As mentioned earlier, optimal selection of reference components and measurements plays important role in data-driven applications. In future, it is proposed to study the impact of optimal selection of reference components in kinetic modelling, and state and parameter estimation. The proposed formulations only require measurement models and do not require 24


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any information of reaction kinetics and/or mass–transfer rates. Hence, it can be used for a priori analysis for designing experiments for parameter estimation. Also, it is proposed to study the error propagation from measurements to state or parameter estimation and optimally select reference components.

Author Information Corresponding Authors: [email protected], [email protected]

Acknowledgement The financial support to NB from Department of Science & Technology, India through INSPIRE Faculty Fellowship is acknowledged.

Supporting Information Available • Section 1: Models for special reactor configurations • Section 2: Structural conditions for selection of reference components • Section 3: Affine representation of components mole numbers • Section 4: Case study for homogeneous batch reactor: Silicon chemical vapor deposition process (Table S2) • Section 5: Case study for homogeneous reaction systems: Additional results for semi batch reaction of ethanolysis of phthalyl chloride (Tables S3 and S4) • Section 6: Case study: Heterogeneous reactor with only gas inlet and no outlet (Tables S5, S6 and S7)

25



• Section 7: Effect of β on estimation error for heterogeneous reaction system with inlets and outlet (Table S8)

References (1) Rodrigues, D.; Srinivasan, S.; Billeter, J.; Bonvin, D. Variant and invariant states for chemical reaction systems. Comput. Chem. Eng. 2015, 73, 23–33. (2) Slaughter, D. W.; Doherty, M. F. Calculation of solid-liquid equilibrium and crystallization paths for melt crystallization processes. Chem. Eng. Sci. 1995, 50, 1679 – 1694. (3) Gadewar, S. B.; Schembecker, G.; Doherty, M. F. Selection of reference components in reaction invariants. Chem. Eng. Sci. 2005, 60, 7168 – 7171. (4) Bhatt, N.; Kerimoglu, N.; Amrhein, M.; Marquardt, W.; Bonvin, D. Incremental identification of reaction systems - A comparison between rate-based and extent-based approaches. Chem. Eng. Sci. 2012, 83, 24–38. (5) Ung, S.; Doherty, M. F. Synthesis of reactive distillation systems with multiple equilibrium chemical reactions. Ind. Eng. Chem. Res. 1995, 34, 2555–2565. (6) Bhatt, N.; Amrhein, M.; Bonvin, D. Extents of Reaction, Mass Transfer and Flow for Gas- Liquid Reaction Systems. Ind. Eng. Chem. Res. 2010, 49, 7704–7717. (7) Bhatt, N.; Amrhein, M.; Srinivasan, B.; Müllhaupt, P.; Bonvin, D. Minimal state representation for open fluid-fluid reaction systems. Am. Control Conf., 2012. 2012; pp 3496–3502. (8) Flockerzi, D.; Bohmann, A.; Kienle, A. On the existence and computation of reaction invariants. Chem. Eng. Sci. 2007, 62, 4811 – 4816.

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(9) Ung, S.; Doherty, M. F. Theory of phase equilibria in multireaction systems. Chem. Eng. Sci. 1995, 50, 3201–3216. (10) Grüner, S.; Mangold, M.; Kienle, A. Dynamics of reaction separation processes in the limit of chemical equilibrium. AIChE J. 2006, 52, 1010–1026. (11) Gadewar, S. B.; Doherty, M. F.; Malone, M. F. A systematic method for reaction invariants and mole balances for complex chemistries. Comput. Chem. Eng. 2001, 25, 1199–1217. (12) Omtveit, T.; Tanskanen, J.; Lien, K. M. Graphical targeting procedures for reactor systems. Comput. Chem. Eng. 1994, 18, S113–S118. (13) Waller, K. V.; Makila, P. M. Chemical reaction invariants and variants and their use in reactor modeling, simulation, and control. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 1–11. (14) Hammarström, L. G. Control of chemical reactors in the subspace of reaction and control variants. Chem. Eng. Sci. 1979, 34, 891–899. (15) Srinivasan, B.; Amrhein, M.; Bonvin, D. Reaction and flow variants/invariants in chemical reaction systems with inlet and outlet streams. AIChE J. 1998, 44, 1858–1867. (16) Fjeld, M.; Asbjørnsen, O.; Åström, K. J. Reaction invariants and their importance in the analysis of eigenvectors, state observability and controllability of the continuous stirred tank reactor. Chem. Eng. Sci. 1974, 29, 1917–1926. (17) Dochain, D. State and parameter estimation in chemical and biochemical processes: a tutorial. J. Process Control 2003, 13, 801–818. (18) Rapaport, A.; Dochain, D. Interval observers for biochemical processes with uncertain kinetics and inputs. Math. Biosci. 2005, 193, 235–253.

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(19) Amrhein, M.; Bhatt, N. P.; Srinivasan, B.; Bonvin, D. Extents of reaction and flow for homogeneous reaction systems with inlet and outlet streams. AIChE J. 2010, 56, 2873–2886. (20) Srinivasan, S.; Billeter, J.; Narasimhan, S.; Bonvin, D. Data reconciliation for chemical reaction systems using vessel extents and shape constraints. Comput. Chem. Eng. 2017, 101, 44–58. (21) Chmielewski, D. J.; Palmer, T.; Manousiouthakis, V. On the theory of optimal sensor placement. AIChE J. 2002, 48, 1001–1012. (22) Narasimhan, S.; Jordache., C. Data reconciliation & Gross error detection: an intelligent use of process data. Gulf Publishing Co., Houston, TX, USA 2000, 6789–6797. (23) M.Nabil,; Narasimhan., S. Sensor Network Design for Optimal Process Operation based on data Reconciliation. Ind. Eng. Chem. Res. 2012, 51(19), 6789–6797. (24) Lofberg, J. YALMIP: A toolbox for modeling and optimization in MATLAB. Proceedings of the CACSD Conference, Taipei,Taiwan 2004, (25) Fletcher, R.; Leyffer, S. Solving Mixed Integer Nonlinear Programs by Outer Approximation. Mathematical Programming 1996, 66, 327–349. (26) Kronqvist, J.; Lundell, A.; Westerlund, T. The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming. Journal of Global Optimization 2016, 64, 249–272. (27) Bhatt, N. P. Extents of Reaction and Mass Transfer in the Analysis of Chemical Reaction Systems. Ph.D. thesis, EPFL, Lausanne, 2011.

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For Table of Contents (TOC) Only

Noise

Win, uin

(v)

Mole numbers

Extents of reaction and mass transfer

n1

x1 n

Reference components

N, r

xr

Measurement model

n2

n= Ax + b + v

nn

n, uout

n1

Reference components selection

Improved estimates of x and n by fusing model and measurements

29


nm


254x114mm (300 x 300 DPI)


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Optimal Selection of Reference Components and ... - ACS Publications

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