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Process Systems Engineering

An Iterative Real-time Optimization Scheme for the Optimal Operation of Chemical Processes under Uncertainty. Proof of Concept in a Miniplant Reinaldo Hernandez, Jens Martin Dreimann, Andreas Vorholt, Arno Behr, and Sebastian Engell Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00615 • Publication Date (Web): 05 Jun 2018 Downloaded from http://pubs.acs.org on June 5, 2018

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Industrial & Engineering Chemistry Research

An Iterative Real-time Optimization Scheme for the Optimal Operation of Chemical Processes under Uncertainty. Proof of Concept in a Miniplant Reinaldo Hernandez,∗,† Jens Dreimann,‡ Andreas Vorholt,¶ Arno Behr,‡ and Sebastian Engell† †Group of Process Control and Operations, Faculty of Biochemical and Chemical Engineering, TU Dortmund, Emil-Figge Strasse 70, D-44227 Dortmund, Germany ‡Laboratory of Chemical Process Development, Faculty of Biochemical and Chemical Engineering, TU Dortmund, Emil-Figge Strasse 66, D-44227 Dortmund, Germany ¶Max-Planck Institute for Chemical Energy Conversion, Stiftstrasse 34-36, 45740 Mülheim, Germany E-mail: [email protected] Phone: +49 (0)231-755-5135. Fax: +49 (0)231-755-5129

Abstract Real-time Optimization (RTO) has gained growing attention during the last few years as a useful approach to boost process performance while safety and environmental constraints are satisfied. Despite the increasing acceptance of RTO in traditional industries such as petrochemical and refineries, its application to novel chemical processes remains limited. This can be partially explained by the fact that only inaccurate

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models are available and the performance of the traditional RTO scheme suffers in the presence of plant-model mismatch. During the last few years, the so-called modifieradaptation schemes for real-time optimization have been gaining popularity as an efficient tool to handle plant-model mismatch. So far, there are only few published works regarding experimental implementations. In this contribution, a reliable RTO scheme which is able to deal with model uncertainty and measurement noise is applied to a novel transition metal complex catalyzed process that is performed in a continuously operated miniplant. The experimental results show that the proposed scheme is able to drive the process to an improved operation despite significant plant-model mismatch demonstrating the applicability of the method to real processes.

1

Introduction

Currently, the process industry faces multiple challenges including increasing global competition, tight product quality standards, tight environmental regulations and the quest for the processing of renewable raw materials instead of feedstock from crude oil. During the last 30 years Real-time Optimization (RTO) has become an attractive approach to improve the efficiency of plant operation with an increasing acceptance in refineries and petrochemical processes. 1 The key idea in RTO is to make use of a rigorous stationary nonlinear model of the process to compute the operating conditions that optimize a predefined performance index e.g. minimization of the operating costs, maximization of product yields, minimization of waste. Unfortunately, in the presence of plant-model mismatch and process disturbances, the traditional RTO scheme can lead to a suboptimal operation or to constraint violations. While it is true that by the use of a highly accurate model that difficulty could be overcome, the effort needed to develop a very accurate model is an impediment to the application of model-based approaches. In the traditional (two-step) approach, the issue of plant-model mismatch is usually addressed by online adaptation of some model parameters. However, this 2


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approach will only resolve the problem if the model is structurally correct and the parameters are identifiable. Under structural plant-model mismatch, it has been demonstrated that the two-step approach can lead to suboptimal operation 2,3 . Dedicated online experiments for the estimation of model parameters can involve significant costs in large scale processes, but not lead to a better operation because of the structural mismatch. 4 A different approach to address the issue of plant-model mismatch is the addition of gradient and bias correction terms to the objective function and to the constraints of the nominal optimization problem in the so-called Modifier Adaptation approach. 3,5–7 Under the assumption of model adequacy, it has been shown that the algorithm generates a sequence of inputs that converges to a KKT point of the plant. 8 The main challenge of this method is the accurate estimation of the plant gradients. In the work of Roberts, 9 the approximation of the derivatives by finite differences was proposed, limiting the applicability of the method to low-dimensional problems with negligible noise levels. More sophisticated algorithms have been proposed during the last years, including dual Modifier Adaptation (Dual-MA), 10 Nested Modifier Adaptation, 11 and more recently the use of ideas from the derivative freeoptimization (DFO) framework in the Modifier Adaptation with Quadratic Approximation (MAWQA) algorithm. 12 Despite the significant progress during the last few years on improvements and extensions of the framework, there is still a lack of literature regarding the experimental validation of the method, with only few published works concerning relatively simple processes. 8,13,14 This work reports the implementation of the efficient MAWQA Real-time Optimization scheme 12 to a continuously operated novel chemical process in a miniplant. Specifically, a transition metal complex catalyzed hydroformylation process is considered. The relevance of this work lies not only in the validation of the methodology but also in the considered case study itself. Transition metal complex catalysis (also known as organometallic catalysis) has become one of the most active research fields in industrial chemistry. The high selectivity and activity at mild conditions make them suitable in pharmaceutical and fine chemistry for

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numerous reactions, including hydroformylation, carbonylation, oxydation, hydrogenation, metathesis and hydrocyanation. 15 This is illustrated in the fact that during the last 15 years, three Nobel prizes in chemistry (2001, 2005 and 2010) have been awarded to researchers working in this field. Furthermore, it has been argued that organometallic catalysis is one of the fundamental blocks in green chemistry, due to the minimization of waste and its potential application to the processing of renewable raw materials. 16 Despite the aforementioned promising results, commercial applications of transition metal complex catalysis in comparison to their heterogeneous counterparts are still very limited. The high price of the required metals and ligands restrains its implementation to processes with a highly efficient catalyst recovery concept. 17 The potential benefits of applying RTO based on Modifier Adaptation with Quadratic Approximation to the optimal operation of multiphase homogenously catalyzed processes have been already explored in previous works by the authors, 18,19 but in these works only simulation studies were presented. In this work, a robust RTO scheme which is able to deal with model uncertainties is proposed and validated experimentally in a continuously operated miniplant for the rhodium catalyzed hydroformylation of 1-dodecene. This contribution addresses the challenges of optimizing a chemical process in the early stages of development, where only an inaccurate models is available. This paper is structured as follows: A brief review of the methods used in this work is presented in section 2. In section 3, the case study is introduced along with a brief description of the experimental set-up. Then, the process model is presented in section 4. The RTO problem for the miniplant is formulated in section 5. In section 6, the materials and methods are presented. Simulations and experimental results are presented in section 7. In the final section 8, the paper is concluded and future directions of work are proposed.

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2

Real-time Optimization (RTO) under uncertainty

In this section, a short overview of Real-time Optimization is presented. Specifically, the combination of process models and plant measurements in the so-called Modifier Adaptation algorithm is discussed in detail, which is the core of the proposed methodology. For sake of completeness a brief discussion about steady state identification and data reconciliation is also presented.

2.1

Problem statement

The objective behind Real-time Optimization (RTO) is to steer the plant to an equilibrium point (steady state) where a performance index of the process is optimized (e.g. yields, costs, profits). This index P can be computed or estimated based on the process inputs us ∈ Rnu and outputs ys ∈ Rny : P = J (ys (xs ), us ) .

(1)

As xs depends on us , RTO can be stated as finding the optimal set of inputs us ∗ and the associated states xs ∗ according to: [us ∗ , xs ∗ ] = arg min

us ,xs

P = J (ys , us )

s.t:. 0 = f (xs , us , θ)

(2a) (2b)

ys = h (xs , us , θ)

(2c)

g (xs , us ) ≤ 0

(2d)

us lb ≤ us ≤ us ub .

(2e)

Here θ ∈ Rnp is the vector of model parameters and the vectors us lb ∈ Rnu and us ub ∈ Rnu represents the input lower and upper bounds on the decision variables. The map f : Rnx × Rnu ×Rnp → Rnx is the state equation of the system; h : Rnx ×Rnu ×Rnp → Rny is the output equation and g : Rnx × Rnu → Rnr defines safety, process and enviromental restrictions, e.g. 5



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quality specifications, maximum reaction temperature, maximum emision level. Under the assumption that there is a unique steady state, the dependency on the states xs can be omitted from the previous expression and the problem can be written in a compact form by grouping together the constraints in the vector function G : Rnu × Rnθ → RnG :

min φ (us , θ)

(3a)

s.t:. G (us , θ) ≤ 0.

(3b)

us

In general, the nominal model is only an approximation to the true plant. Here, the notation (.)p is introduced to denote the true (unknown) plant map for the objective function and for the constraints. If the true plant map were perfectly known, the optimal inputs could be calculated according to:

min φp (us )

(4a)

s.t:. Gp (us ) ≤ 0.

(4b)

us

Therefore, the goal of a robust RTO scheme can be stated as finding the value of the inputs us ∗ that solves the optimization problem (4), under the limitation that only the nominal optimization problem (3) and plant measurements ys are available.

2.2

Addressing model errors in RTO

In order to handle uncertainty in optimization, different approaches have been proposed in the literature which can be classified into two main groups: robust approaches and adaptive approaches. In robust approaches, conservatism is introduced in order to guarantee feasibility and/or optimality for the entire range of expected variations. Under the assumption of parametric

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mismatch, the nominal optimization problem can be extended with additional constraints that ensure feasible operation for any realization of the uncertainty. On the other hand, adaptive approaches attempt to fit the nominal process model or the optimization problem to the actual plant behavior by using the available plant measurements in such a way that the convergence to the plant optimum is ensured under uncertainties that result from plant-model mismatch and/or process disturbances. 3 Unlike robust approaches, adaptive strategies do not need prior assumptions regarding uncertainty bounds and therefore provide less conservative operating points that are adjusted iteratively. As a consequence, these approaches can react to disturbances and process changes. Within adaptive schemes different approaches are available: 3 • Model-parameter adaptation (also called two-step RTO): The plant measurements are used to adjust the model parameters. Then, the optimization is performed using the updated model. This has been the traditional approach to RTO. However, there are several limitations of this methodology. The selection of the model parameters to be regressed is not trivial as pointed out by Forbes et al.; 20 the parameters should be identifiable despite the limited amount of information (in terms of number of measurements and excitation of the signals) which is commonly available in continuously operated processes. On the one hand, the use of a large number of parameters can potentially improve the model quality but their precise identification is limited by the excitation that is given to the plant. On the other hand, a low number of adapted parameters can produce an inaccurate model for which the optimal operating point differs from the actual plant optimum 20 . It has been proposed to integrate experimental design for the estimation of the model parameters into the RTO system, 4 but a high cost is associated with any experiment in a real process. What is even more important is that the approach is in general not able to cope with structural plant-model mismatch which is always present. • Direct input adaptation: Here, the RTO problem is transformed into a control prob7



lem and the optimization problem is not solved explicitly but indirectly online, which is the main advantage of this approach. Within this category a variety of methods can be found in the literature including Self-optimizing control (SOC), 21,22 Necessary Conditions of Optimality (NCO)-Tracking, 23 and Extremum-seeking Control (ESC). 24 The applicability of these methods is limited to specific cases and their performance depends on the particular problem. SOC relies on a perfect knowledge of the process model and assumes that the disturbances are known. NCO-tracking is only valid under the assumption that the set of active constraints does not change with the uncertainty. ESC requires significant perturbations to the process for the estimation of the plant gradient due to the assumption that no process model is available. • Modifier adaptation: Instead of updating the model parameters, bias and gradient correction terms (also called modifiers) are added to the objective function and to the constraints of the nominal optimization problem. The modifiers are updated based upon plant measurements. This approach has been gaining importance during the last years as an efficient combination of the process model with the plant measurements. 5,6 This is the method used in this work and it will be discussed in more detail below.

2.2.1

Modifier Adaptation (MA)

The idea of introducing bias terms (known as modifiers) to the objective function of the optimization problem dates back to the work of Roberts as a modification to the two-step approach 9 . The method was proven successful to achieve the optimal operating point of the real process in spite of structural mismatch in the model. This seminal work gave rise to the family of ISOPE (Integrated System Optimization and Parameter Estimation) algorithms. The method was later reformulated in a new scheme where parameter estimation is not required in each iteration. 7 In the work of Gao and Engell 6 the algorithm was generalized to the constrained optimization case and applied to the set-point optimization of a chromatographic process. The term modifier adaptation was coined by Bonvin and coworkers, 3,8 8


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who in addition presented a proof of convergence. For a complete review of the method, its extensions, and applications, the reader is referred to Marchetti et al.. 5 Starting from the nominal operating point u0 (that is found by using the process model), the basic idea of the algorithm is to generate a sequence of inputs u1 , u2 , u3 .....uk , uk+1 by iteratively adapting the nominal problem (3) and iteratively solving the modified problem (5):

us k+1 = arg min φ (us ) + kφ + ψφk

T

us

k s.t:. G (us ) + kG + ψG

us − us k

T

us − us k ≤ 0

(5a) (5b)

k where the terms kφ , kG , ψφk , ψG are the modifiers which are the correction terms of zeroth

and first order (gradients) of the objective function and of the constraints. Note that the dependency of the objective function and of the constraints to the parameters θ has been omitted, as the model is assumed to be fixed and the convergence of the algorithm does not depend on the correctness of the model parameters; furthermore, a parameter estimation step is not needed. 7 The modifiers are defined as:

kφ = φp us k − φ us k kG = Gp us k − G us k ψφk = ∇φp us k − ∇φ us k k ψG = ∇Gp us k − ∇G us k

(6a) (6b) (6c) (6d)

It can be easily demonstrated that if the iterative solution of problem (5) converges to a point us ∞ which is a KKT point of the modified problem, us ∞ is also a KKT point of the plant. 8 The Modifier-Adaptation scheme is summarized in the algorithm presented in table

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1. Table 1: Simplified Modifier Adaptation Algorithm Algorithm Modifier-Adaptation Algorithm Step 1: Given the initial point u0 , set k = 0 Step 2: Calculate the modifiers according to (6) Step 3: Compute the next input by solving (5) Step 4: Set k ← k + 1, return to Step 2

The most challenging aspect of MA is the estimation of the plant derivatives. The seminal work of Roberts 9 suggested to introduce perturbations around the current operating point with the aim of approximating the derivatives by finite differences. However, this leads to additional set point changes which may not be welcomed by the plant operators and the choice of the size of the perturbations is difficult in the presence of measurement noise. 6 A recently proposed approach makes use of quadratic approximation as it is used in the derivative-free optimization framework. The basic idea behind Modifier Adaptation with Quadratic Approximation (MAWQA) is the estimation of the process derivatives by fitting a quadratic model to the data that was obtained at previously visited operating points. 12 For instance, for the objective function, the quadratic approximation φ˜p is defined as: φ˜p (us , π) =

nu X nu X

ai,j ui uj +

i=1 j=1

nu X

bi ui + c,

(7)

i=1

with the parameter set π = {a1,1 , · · · , anu ,nu , b1 , · · · , bnu , c} obtained from solving the leastsquares problem: min π

nr X

2 φp us (ri ) − φ˜p us (ri ) , π ,

(8)

i=1

where us (ri ) is an element of the regression set U (k) composed of past set-points which are selected to guarantee well-poisedness of the problem. A minimum of (n + 1)(n + 2)/2 points must be collected before φ˜p can be computed. The values of the constraint functions are approximated in a similar fashion. Instead of introducing additional perturbations to the process, MAWQA computes the plant derivatives from the derivatives of the quadratic model. 10


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The new set-point obtained from the solution of (5) is additionally restricted by an elliptical trust region which is defined by the covariance matrix of the regression set. The algorithm is presented in detail in Gao et al.. 12

2.3

Addressing measurements errors

As was stated before, RTO makes use of plant measurements which are usually contaminated with random noise and may also exhibit systematic errors. The application to a real process requires the use of additional components for the automatic identification of a steady state and data reconciliation. In this section, a brief review of these methods is presented.

2.3.1

Steady State Identification (SSI)

The first step for data processing in any RTO scheme is steady-state identification (SSI). A false identification of the steady state can lead to a poor performance of data reconciliation. While a late identification of steady state leads to a poor performance of the RTO scheme, a false identification of stationarity might result in the use of transient measurements. Different approaches have been proposed in the literature, the reader is referred to Roux et al. 25 for a brief review. In Polynomial Interpolation Test (PIT) a window of data is selected and a polynomial of a predefined degree is fitted (similar to a Savitzky-Golay filter 26 ) and then the slope of the polynomial at the middle of the window is assessed for stationarity. Reverse Arragements Test (RAT) and Rank von Neumann tests are used to detect if a sequence of data is derived from independent observations or if a significant trend underlies the data. 27 In a recent work, it has been proposed to use wavelet multiscale decomposition of the temporal signals where the high frequency noise is easily filtered-out and the low frequency signal is used to identify stationarity. 28 All the before mentioned methods make use of a significant amount of data in order to detect the steady state and are particularly suitable in the case of very noisy data . A simpler algorithm for SSI is the F -test, 29 which makes use of a statistical approach by 11



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comparing the variance obtained by the raw data with the variance obtained from filtered data. This is evaluated at each sampling point which leads to a fast detection of stationarity. Due to its simplicity and relatively good accuracy in the estimation of steady states, it has been used in data reconciliation of industrial processes. 30,31 As will be discussed in section 7, due to the characteristics of the signals in the miniplant, this method has been selected and integrated into the RTO scheme that is applied here. The basic idea behind the method is to compute the ratio R of two estimates of the variance of the noisy data. The null hypothesis is that the data points in the signal are independent observations from random variables i.e. the process is at steady state and the variation is only given by random measurement noise. The algorithm can be summarized in the following steps: • Step 1: An exponential moving average filter is applied to the raw data y i according to:

y i,f = λ1 y i + (1 − λ1 ) y i−1,f

(9)

where y i,f is the filtered signal and 0 ≤ λ1 ≤ 1 is a filter constant. • Step 2: The filtered squared deviation (v i,f )2 from the previous filtered value is calculated according to:

(v i,f )2 = λ2 y i − y i−1,f

2

+ (1 − λ2 ) v i−1,f

2

(10)

with 0 ≤ λ2 ≤ 1 as a filter constant for the squared deviation. • Step 3: The filtered squared difference (di,f )2 of successive data is calculated by:

(di,f )2 = λ3 y i − y i−1

12

2

+ (1 − λ3 ) di−1,f


2

(11)

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with a third filter parameter 0 ≤ λ3 ≤ 1 • Step 4: The R statistic is defined by:

R=

(2 − λ1 ) v f,i (df,i )2

2 .

(12)

Finally the value of R is compared with a critical value Rcrit . If R > Rcrit then the process is considered not to be at steady state at the corresponding level of significance. The performance of the algorithm depends on the values of the filter parameters λ1 , λ2 and λ3 . Ideally, they have to be chosen in order to have few Type I errors (incorrect rejection of a true null hypothesis) and Type II errors (incorrectly retaining a false null hypothesis) as well as the delay in the detection within acceptable limits. The smaller the value of λ1 , the stronger is the filtering and less Type II errors occur but at the expense of a delayed steady state detection. A large value of λ1 will fail to detect the steady state by associating the measurement noise with a transient behaviour of the process. While it was proposed to choose both values as equal in the original paper of Cao and Rhinehart, 29 Baht et al. 30 reported benefits by selecting λ2 and λ3 . 30 An algorithm for estimation of the optimal value of the fiilter parameters was proposed by Bhat and Saraf 30 based upon an analysis of historical process data. The procedure can be summarized as follows: 1. Step 1: Select a small value of λ3 . 2. Step 2: Select a small value of λ2 ≥ λ3 . 3. Step 3: Start with a low value of λ1 and calculate Rcrit , Type II errors, and the delay in the detection of the steady state with these values. 4. Step 4: Increment the value of λ1 by 0.02 and return to Step 3 until the predefined range of Type II errors is reached.

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5. Step 5: Increment λ2 by 0.05 and return to Step 3 until an early enough detection of the steady state is obtained. 6. Step 6: Increment the value of λ3 by 0.01 and return to Step 2 until a satisfactory result is obtained.

2.3.2

Data Reconciliation (DR)

Data Reconciliation (DR) is understood as the process of correcting measurements in such a way that the reconciled data satisfies a predefined process model and constraints. The method has been succesfully applied to different processes including heat exchanger networks, chemical reactors, gas processing plants, see e.g.Mobed et al., 32 Rafiee and Behrouzshad. 33 For the steady state case, DR can be formulated as follows: Given the set of steady state measurements ys after applying the inputs us to the plant, DR attempts to estimate the vector of states x ˆ and reconciled variables y ˆ which satisfy the process model and constraints while the likelihood of making the observations is maximized. 34 DR can be formulated in a more general form as:

max p (ys − y ˆs )

(13a)

s.t:. 0 = m (ˆ xs , us )

(13b)

x ˆs ,ˆ ys

y ˆs = n (ˆ xs , us ) ,

(13c)

where the maps m, n must be consistent but not necessarily the same as the model used in the optimization and p (y − y ˆ) is the probability density function of the error. It is common to take the negative logarithm and to represent the problem as:

min

x ˆs ,ˆ ys

ρ (ys − y ˆs )

(14a)

s.t:. Model Equations and Constraints m, n,

14


(14b)

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where ρ (ys − y ˆs ) = − log (p (ys − y ˆs )). Under the assumption that the error follows a Gaussian distribution, the conventional weighted least square formulation provides unbiassed estimates. In that case, the optimization problem results as:

min

x ˆs ,ˆ ys

(ys − y ˆs )T W −1 (ys − y ˆs )

s.t:. Model Equations and Constraints m, n,

(15a) (15b)

where W is the covariance matrix of the measurements. If the mesurements are independent, each measurement is weighted with the inverse of the variance, which means that more reliable measurements are corrected less and vice versa. The assumption of a Gaussian distribution is one of the main limitations of this method. In the presence of outliers and gross errors, the quality of the estimation is affected. An approach that attempts to increase the robustness in the presence of outliers is the use of M-estimators, where the objective function of the problem is modified in such a way that a less weight is given to measurements with large errors. 35 Different functions have been proposed such as e.g. the ν−contaminated normal, Huber, Cauchy, Loretzian, Tukey, Hampel, Andrews and Welsh function.

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Process Description: Hydroformylation of 1-dodecene in a Thermomorphic Solvent System

3.1

Hydroformylation of higher olefins

Hydroformylation, also known as the oxo process, is the most important process for the production of aldehydes from alkenes by addition of carbon monoxide and hydrogen and is the best known application of homogeneous transition metal catalysis. Figure 1 shows the case of the hydroformylation of propene to butanal, the most important industrial example of this reaction. The estimated world capacity of hydroformylation processes accounts for

15



107 t/a. 36 The produced aldehydes are manly used as intermediates for the production of plasticizers and fine chemicals.

Figure 1: Hydroformylation of propene to butanal Since the discovery of the reaction by Otto Roelen, the history of hydroformylation is filled with milestones in the evolution of homogeneous catalysis, not only in research but also in industry. The first oxo-process was developed by BASF and made use of the cobalt-complex tetracarbonyl hydride HCo(CO)4 at relatively severe reaction conditions of temperature (120−180 ◦ C) and pressure (270−300 bar). It represented the first commercial application of transition metal catalysis. With the introduction of the significantly more active rhodium catalysts, it was possible to reduce the severity and to achieve high selectivity at a lower temperature (85−95 ◦ C) and pressure (15−18 bar). 15 A breakthrough was the development of the Ruhrchemie/Rhône−Poulenc (RCH/RP) oxo process for the hydroformylation of propene in 1984. In this process, the water soluble catalyst (a complex of rhodium with a sulfonated phosphine ligand) remains dissolved in water and can be easily recovered and separated from the organic products without additional thermal separation stages that can compromise the catalyst stability. 37 The success of aqueous multiphase catalysis in hydroformylation of propene and butene has been transferred to other chemical reactions at the industrial level: selective hydrogenation of unsaturated alcohols, hydrodimerization and C − C bond formation. 38 Despite recent efforts in the use of cheaper metals as e.g. ruthenium or iridium, 39 the selectivity and activity of the rhodium catalyst is the reason for its dominant industrial application despite its high price and scarcity. Higher oxo products are used in the production of plasticizer alcohols in the range C8 −C11 and synthetic detergent alcohols in the range C12 −C18 . Unfortunately, higher alkenes (> C4 ) are not suitable for the RCH/RP process, therefore different alternatives have been proposed 16


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in order to overcome the before-mentioned limitations. 40

3.2

Thermomorphic Multicomponent Solvent (TMS) System

In order to achieve an efficient recovery of the homogeneous catalyst a Thermomorphic Multicomponent Solvent System (TMS) exploits the temperature dependence of the miscibility gap of a multicomponent solvent. The principle is illustrated in Figures 2-3. The feed (S3) is mixed with a polar (S1) and a nonpolar solvent (S2), which are selected in order to provide a strong dependence of the miscibility gap on temperature. At a high temperature in the reactor (T1) the mixture is homogeneous and therefore any mass transfer limitations between the liquid phases are overcome, at a low temperature in the decanter (T2), phase separation takes place and the catalyst rich phase is recovered and recycled to the reactor. The product phase is sent to further separation stages for solvent recovery and product purification. The process concept has been applied to different model reactions: co-oligomerization

Figure 2: Principle behind a Thermomorphic Multicomponent Solvent (TMS) System of double unsaturated fatty substances with ethylene, isomerization - hydroformylation reaction of trans-4-octene to n-nonanal, the hydrosilylation of methyl 10-undecenoate with triethoxysilane, etc. In all the cases a high catalytic activity is observed without mass transfer limitations, together with an easy and efficient catalyst recovery. 15,40

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Figure 3: Simplified process diagram of a thermomorphic solvent system

3.3

Case Study: Hydroformylation of 1-dodecene in a TMS System

In this work, the hydroformylation of 1-dodecene to the linear product tridecenal is considered as a case study. This system has been extensively investigated as a model reaction for multiphase homogeneous catalysis. 41 It was chosen not only for its industrial relevance (hydroformylation is one of the key steps in the chemical chain), but also as a model reaction due to its similarity to other transition metal catalyzed processes such us hydrocarbonylation, hydroesterification, amination, hydroaminomethylation and hydrosilylation. The catalytic system consists of the precursor (acetylacetonato) - dicarbonylrhodium(I) Rh(acac)(CO)2 and the ligand Biphephos. The use of this bidentate ligand ensures a very high selectivity towards the linear isomer, with n/iso ratios of up to 99:1. 42 In previous works, different solvent mixtures were investigated and it was found that the system N,N dimethylformamide (DMF)/n-decane with a composition of 50/50, %m/%m yields high conversion and selectivity with the lowest catalyst leaching. 41 In the same work it was also demonstrated that a high product concentration (as a consequence of high conversion) can lead to an increase in the loss of catalyst due to its solubilizing effect. An alternative approach to solvent selection reported in the literature is based on the COSMO solvation

18


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model. Based on the predicted solubility of the Biphephos ligand and predicted partition coefficients a set of suitable solvents were identified, including DMF as the polar solvent. 43 During the hydroformylation of 1-dodecene to tridecanal, several side-reactions take place in the system leading to the formation of different side products: isomers of 1-dodecene, branched aldehydes and alkanes. A complete analysis of the reaction network was published by Markert et al. 44 and a reduced reaction network containing six possible reactions was proposed (Figure 4): • Hydroformylation of 1-dodecene to the main product tridecanal: Reaction r1 • Isomerization of 1-dodecene to different internal olefins that are lumped as the pseudospecies iso-dodecene: Reaction r2 • Hydrogenation of 1-dodecene to dodecane: Reaction r3 • Hydrogenation of the isomers of 1-dodecene to dodecane: Reaction r4 • Hydroformylation of the internal olefins to branched aldehydes that are lumped as b-aldehyde: Reaction r5 • Hydroformylation of 1-dodecene to 2-methyl-dodecanal which is also lumped together with the brached aldehydes as b-aldehyde: Reaction r6

Figure 4: Reaction Network of the hydroformylation of 1-dodecene to tridecanal (Adapted from 44 ) 19



The isomerization of 1-dodecene was identified as the main side reaction, and it plays a significant role for the selectivity towards the main product. Further hydrogenation of the aldehydes to the corresponding alcohols was not observed. 44 It is worth to mention that in industrial plants the formation of heavy ends has been detected during the hydroformylation of lower olefins. As a consequence of the high reactivity of the aldehyde product, further reactions can take place: aldol condensation, acetilization and oxidation. 45

3.4

Miniplant for hydroformylation of 1-dodecene

In order to perform long term investigations of the catalyst stability and to study the accumulation of impurities and the presence of side products, a miniplant was built and a stable continuous operation for more than 200 hours was achieved. 46,47 Figure 5 shows a simplified flow diagram of the TMS miniplant with some key instrumentation, while a picture of the miniplant is presented in Figure 6. The miniplant consists of two sections: the reaction section and the catalyst recovery section. The raw material 1-dodecene is fed together with the nonpolar solvent n-decane from the vessel B1 to the reactor B3 via the dosing pump P1. Replenishment of the polar solvent DMF and catalyst (available in the vessel B2) takes place via the dosing pump P2. No flow indicators are installed on the feed streams; the flow rate is computed based on the change of the mass in the vessels B1 and B2. Therefore, an offline calibration of the pumps P1 and P2 was performed. The jacketed reactor B3 is a 1000 ml continuous stirred tank which has an overflow pipe installed that keeps the liquid holdup constant at 330 ml. A constant temperature is kept in the reactor by circulating silicone oil coming from the heating circulator WT1 (Julabo 6-ME). Carbon monoxide and hydrogen are fed to the reactor B3 under pressure and ratio control. After leaving B3, heat is removed from the reaction mixture in the heat exchanger WT3. In the temperature controlled vessel B4 (decanter) the nonpolar and the polar (catalyst stream) phases are separated. The temperature in the jacketed decanter is kept constant by circulating glycol from the refrigerated circulator WT2 (Julabo F33-M). The catalyst phase is pumped back 20


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to the reactor via pump P3. The nonpolar phase is sent to the flash drum B5 where the solubilized syngas is released after expanding to ambient pressure. The raw product is stored in the tank B6. The product is analyzed online by gas chromatography (Agilent 7890A) by sampling the nonpolar phase in B4.

Figure 5: Simplified flow diagram: TMS Miniplant for the hydroformylation of 1-dodecene

3.4.1

Regulatory Control

The controlled variables of the miniplant are: 1. Reactor temperature. This is achieved by using a silicone oil circuit which keeps the temperature of the reaction medium within ± 0.1

◦

C of the given set point.

2. Reactor total pressure. It is controlled by manipulating the flow of carbon monoxide which is fed to the process. 3. Gas feed flow fraction. It is controlled by manipulating the flow of hydrogen which is fed to the process.

21



Figure 6: TMS Miniplant for the hydroformylation of 1-dodecene (Adapted from Zagajewski 48 )

22


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4. Decanter temperature. It is controled by using an ethylene glycol cooling circuit which keeps the temperature of the decanter within ± 0.1

◦

C of the given set point..

5. Feed flow of the substrate and solvents. It is controlled in open loop based on the offline calibration of the pumps P1 and P2. 6. Catalyst dosing. This is achieved by changing the concentration of catalyst in the DMF dosing stream. 7. Level controller. The reactor level is kept constant by an overflow weir. The level of the polar and nonpolar phases in the decanter are kept constant by manipulating the product flow and the catalyst recycle flow.

3.4.2

Measurements of the variables

1. Measurements of the mass in vessels B1 and B2, reactor temperature and pressure, gas flow rate, flow rate of the catalyst recycle stream, decanter temperature and pressure are available every second. 2. Measurements of the composition of the nonpolar phase in the decanter B4 are available every 60 min.

4

Model Description

There are only few published works related to the kinetic modelling of the hydroformylation processes. van Elk et al. 49 presents a rigorous model where the mass transfer in the liquid phase is described by the Higbie penetration model, while the mass transfer in the gas phase is described by the stagnant film model. Due to the fact that only the main reaction was considered, the results of this work are of limited applicability. This contribution considers the complete reaction network and the latest results that have been published on the macro and micro kinetics. The expressions for the correspond23



ing reaction rates rj developed by Kiedorf et al. 50 and modified by Hentschel et al. 51 are used. The kinetic model was developed based on a postulated catalytic cycle mechanism, considering the different steps for the hydrogenation, isomerization and hydroformylation reactions. It was assumed that all the reactions are catalyzed by the same active species which is in equilibrium with different inactive species. An important aspect in the reactor operation is the solubility of the syngas components H2 and CO in the reaction mixture or equivalently their molar concentration in the liquid phase. These concentrations have an influence not only on the reaction kinetics but also on the amount of active catalyst in equilibrium. In Vogelpohl et al., 52,53 the gas solubility of carbon monoxide and syngas in various solvents (including n-decane, 1-dodecanal and DMF) was measured and correlated using the Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT). In this work, the thermodynamic relations for densities and solubilities are adapted from. 54

4.1

Reactor Model

The reactor model is composed of two compartments, one for the liquid phase where the reactions take place and one for the gas phase. It is assumed that both the liquid and the gas phase are well mixed. A schematic representation of the reactor is presented in Figure 7.

Figure 7: Gas-Liquid Reactor B3

24


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4.1.1

Macro Model

Material balance in the liquid compartment: For the liquid components i (with i = DMF, n-decane, 1-dodecene,....), the liquid bulk phase material balance is given by: N

r X dni,liquid ˙ ˙ = Vin Ci,in − Vout Ci,out + mcat νi,l rl , dt l=1

(16)

where ni,liquid is the amount of material (i.e. mol) in the liquid phase, V˙ in and Ci,in are the volumetric flow rate and the molar concentration of the inflow to the reactor; V˙ out and C˙ i,out are the volumetric flow rate and the concentration of the outflow; νi,j are the coefficients in the stoichiometric matrix; rl the reaction rate for the l = 1, ...Nr reactions and mcat is the mass of active catalyst. For the hydrogen and the carbon monoxide, the material balance (16) is extended to include the molar flux, Jj of the j gas component: N

l X dnj,liquid ˙ = Jj aVR,liquid − Vout Cj,out + Mcat νj,l rl . dt l=1

(17)

In this equation a is the surface area of the G-L interface per unit of volume and VR,liquid is the liquid holdup. Material balance in the gas compartment: It is assumed that at reaction conditions, the gas phase contains only j = H2 , CO. The material balance is given by: dnj,gas = n˙ j,in − Jj aVR,liquid , dt

(18)

where nj,gas denotes the amount of material in the gas phase and n˙ j,in is the inflow flow rate. Due to the relatively low total pressure, it can be assumed that the ideal gas law acurately describes the behaviour of the gas phase. Therefore, the partial pressure of each

25



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gas is determined according to

pj =

nj,gas RTReactor . VR,gas

(19)

Reaction kinetics. Equations 20 present the rate equations used in this work. 50,51 In all expression Ci denotes the molar concentration of the species i=1-dodecene,... etc, in the bulk liquid.

r1 =

r2 =

r3 =

k1,0 C1−dodecene CH2 CCO 1 + K1,1 C1−dodecene + K1,2 Ctridecanal + K1,3 CH2 Cidodecene k2,0 C1−dodecene − Ke,2 1 + K2,1 C1−dodecene + K2,2 Cidodecene k3,0 C1−dodecene CH2 − C1−dodecene Ke,3 1 + K3,1 C1−dodecene + K3,2 Cdodecane + K3,3 CH2

(20a)

(20b)

(20c)

r4 = k4,0 Cidodecene CH2

(20d)

r5 = k5,0 Cidodecene CH2 CCO

(20e)

r6 = k6,0 C1−dodecene CH2 CCO

(20f)

The reaction rate constants kl depend on the reaction temperature TReactor and can be described by the Arrhenius equation 21 with pre-exponential factor kl,0 and activation energy Eal , with the reference temperature Tref :

Eal kl = kl,0 exp − R

1 TReactor

−

1 Tref

.

(21)

The reaction equilibrium constants Ke,l for the isomerization (l = 2) and the hydrogenation (l = 3) are computed according to:

Ke,l

∆GR,l = exp − , RT

26


(22)

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where ∆GR,l is the Gibbs free energy of reaction for the species involved. They can be calculated based on the following correlation

2 ∆GR,l = a0,l + a1,l TReactor + a2,l TReactor .

4.1.2

(23)

Micro Model

In order to determine the molar fluxes Jj that are needed in the bulk-phase material balances, different theories and methods are available, including the stagnant two-film model with the film thickness δ as parameter, the penetration model in which a residence time θ is associated to the fluid element and the surface renewal model. 55–57 In this work the two-film theory is used. It is assumed that a stagnant film is present near the interface, where film transport is governed essentially by molecular diffusion (depicted in Figure 8).

Figure 8: Phases in a gas-liquid reactor according to the two-film theory Under the assumption that the reaction only takes place in the liquid phase and that Fick‘s law is valid, the material balance within the films for the component j is given as: film film ∂Cj,gas ∂ 2 Cj,gas = Dj,gas ∂t ∂x2 Nl film film X ∂Cj,liquid ∂ 2 Cj,liquid = Dj,liquid + νi,l rl , ∂t ∂x2 l=1

27


(24a) (24b)


Page 28 of 67

film film where Cj,gas and Cj,liquid are the concentrations in the gas and liquid films and Dj,gas and

Dj,liquid are the diffusivities. In order to solve these partial differential equations, the following boundary conditions are assumed:

film bulk Cj,gas = Cj,gas

x = −δgas

(25a)

x = +δliquid

(25b)

film film ∂Cj,liquid ∂Cj,gas = −Dj,liquid −Dj,gas ∂x ∂x

x=0

(25c)

film film = Kj Cj,gas Cj,gas

x = 0,

(25d)

bulk film = Cj,liquid Cj,liquid

where the first and the second conditions are stated by the two-film model. The third equation is based on the assumption that there is no accumulation of material at the interface and therefore the fluxes through the gas and liquid films are the same. At the interface, chemical equilibrium is assumed with an equilibrium constant Kj . If it is assumed that the reaction is relatively slow and therefore no chemical reaction occurs in the liquid film and considering stationarity, the equations can be solved analytically, which yields the following expression for the estimation of the flux: bulk bulk − Kj Cj,liquid Cj,gas Jj = , (Kj /kj,l ) + (1/kj,g )

(26)

where the film coefficients kj,l and kj,g are defined as:

kj,l =

Dj,liquid , δliquid

kj,g =

Dj,gas . δgas

(27)

Due to the difficulties in measuring concentrations at the interface, the overall mass transfer coefficients (kj,G ) based on the difference between the bulk concentration in one phase eq and the concentration that would be in equilibrium (Cj,liquid ) with the bulk concentration in

28


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the other phase are used in this work. The flux is computed as:

eq bulk Jj = kj,G Cj,liquid − Cj,liquid ,

(28)

with the equilibrium concentration given by:

eq Cj,liquid

=

VR,gas 1 bulk p + Cj,liquid RTReactor VR,liquid j , VR,gas 1 H +1 RTReactor VR,liquid j

(29)

where Hj is the Henry coefficient Pjeq = Cjeq Hj .

(30)

The dependence of the Henry coefficient Hj on the reaction temperature TReactor is modeled by an Arrhenius expression (31) with activation energy Ej Hj = Hj,0 exp −

4.1.3

Ej

RTReactor

.

(31)

Residence Time Distribution

In order to characterize and to model a real reactor, deviations from a non-ideal behavior as a consequence of a non-perfect mixing should be identified. 58 Specifically, for the reactor in consideration, the Residence-Time Distribution (RTD) function was estimated at different flow rates. 48 This was achieved by pulse input experiments where a concentrated solution of sodium chloride was used as tracer while demineralized water was flowing through the reactor at the same stirring speed as in normal operation.The results showed that there is a good agreement between the measured distribution function and the predicted distribution function for an ideal reactor, with a deviation of around 5% in the mean residence time. This suggests that the reactor is well mixed and that there are no bypassing or dead zones. 48

29



4.1.4

Page 30 of 67

Catalyst Concentration

Different catalytic species are simultaneously present in the reaction medium including the Rh-di-carbonyl (I), the Rh-dimer (II) and the catalyst in resting state (III) which is the active species in the catalytic mechanism (Figure 9). Under the assumption that the precursor Rh(acac)(CO)2 is completely converted into coordination complexes, the following relation between their concentrations can be stated:

CRh,precursor = Ccat,I + 2Ccat,II + Ccat,III ,

(32)

Furthermore, if it is assumed and that there is a fast catalyst pre-equilibrium, the mass action law can be used with order nCO for the carbon monoxide:

K1 =

Ccat,I CH2 , 2 Ccat,II

K2 =

Ccat,I nCO Ccat,III CCO

(33)

Then, the catalyst in resting state is given by: 59

Ccat,III

nCO CH (K2 CCO + 1) =− 2 + 2n 4K1 K22 CCOCO

s

nCO CH2 (K2 CCO + 1) 2n 4K1 K22 CCOCO

2 +

CRh CH2 2nCO 2K1 K22 CCO

(34)

which provides a relationship netween the concentration of catalyst in resting state as a function of the concentration of disolved gases and the total concentration of rhodium CRh . A simpler expresion has been proposed in 51 and will be used in this work:

Ccat,III =

CRh,precursor 1 + Kcat,1 CCO + Kcat,2 CCCO H

(35)

2

4.2

Decanter Model

The next section of the miniplant is the catalyst recovery section. As was stated before, after reducing the temperature of the outflow stream from the reactor, phase separation 30


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Figure 9: Equilibrium between the different catalyst species takes place and the polar phase which is rich in catalyst is separated from the product phase and sent back to the reactor. The economic feasibility of the complete process depends on the efficient recovery of the catalyst. The decanter model consists of three compartments which correspond to the two liquid phases and to the gas phase (Figure 10). It can be assumed that the phases are well mixed and therefore a lumped model is sufficient to describe the dynamics of the system.

Figure 10: Schematic of the compartments in the liquid-liquid decanter B4 Material balance in the liquid compartments: The material balance for the nonpolar and polar liquid phases is given by: dCi,nonpolar = n˙ i,in,nonpolar − V˙ nonpolar C˙ i,out,nonpolar + Ji aVdecanter dt dCi,polar Vpolar = n˙ i,in,polar − V˙ polar C˙ i,out,polar − Ji aVdecanter , dt

Vnonpolar

31


(36a) (36b)


Page 32 of 67

where Ci,nonpolar , Ci,polar are the molar concentrations in the nonpolar and polar phases, Vnonpolar , Vpolar are the liquid hold-ups (in volume), n˙ i,in,nonpolar , and n˙ i,in,polar are the molar flows coming from the reactor and Ji is the molar flux over the interface. The total interfacial area is given by aVdecanter , with a the surface specific area. Due to the long residence time in the vessel, it can be assumed that the phases are in equilibrium. Different LLE models are available in the literature for this system. In Merchan and Wozny 60 three different approaches were evaluated: the predictive UNIFAC-Dortmund model, a combination of equation of state (Soave-Redlich-Kwong) with the activity coefficient model NRTL (non-random two liquid model) and the heterosegmented PCP-SAFT (hs-PCPSAFT). It was reported that the second approach shows the best correlative capabilities (with the exception of volumetric properties) with a lower computational effort than the PCP-SAFT EoS, while the UNIFAC Dortmund does not describe the experimental data in a satisfactory way. In Steimel, 61 a simplified model of the LLE in the decanter was presented based on simple expressions of the equilibrium constants Ki as a function of the decanter temperature Tdecanter for a fixed composition of the TMS (DMF/n-decane 50:50): Ki = exp A1,i +

A2,i Tdecanter

+ A3,i Tdecanter ,

(37)

where the constants A1,i ,A2,i and A3,i were obtained by regression of experimental data 62 . If it is assumed that the separation after the reactor is instantaneous, the molar flow rates of the inflow streams to the nonpolar and polar phases are calculated by:

n˙ i,in,nonpolar = ζi n˙ i,in n˙ i,in,polar = (1 − ζi ) n˙ i,in ,

(38a) (38b)

where n˙ i,in is the molar flow rate of the inflow stream to the decanter and ζi is the split factor defined as: ζi =

Ki . 1 + Ki 32


(39)

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This is a similar to the approach used by Müller 63 for the modelling of a hydroformylation process in a micellar solvent system. Material balance in the gas compartment: For the gas components, the material balance is given by: dnj,gas = n˙ j,in − n˙ j,off-gas , dt

(40)

where n˙ j,in is the molar flow of gas in the feed stream.

4.3

Reaction Performance Indicators

The main performance indicators: Conversion (X), product yield (Y ) and regioselectivity (S) are calculated based on the mass flow (m ˙ substrate ) and on the purity (w1−dodecene,substrate ) of the feed stream and the mass flow (m ˙ product ) and composition of the product stream (wi,product ): w1-dodecene,product m ˙ product w1-dodecene,substrate m ˙ substrate wtridecanal,product m ˙ product M1−dodecene Y = w1-dodecene,substrate m ˙ substrate Mtridecanal wtridecanal,product m ˙ product S= . (wtridecanal,product + wb-aldehyde,product )m ˙ product

X =1−

4.4

(41) (42) (43)

Model Implementation

In steady state, all the previously formulated differential equations corresponding to the different material balances are transformed into algebraic equations by setting x˙ = 0, where x is the state vector. Two kinetic models were considered and their prediction capabilities were compared (see section 7.1): 1. Model I: Model structure given by equations with the kinetic parameters reported by Kiedorf et al.; 50 2. Model II: The same general model structure as Model I but with the kinetic parameters 33



Page 34 of 67

reported by Hentschel et al.. 51 The main kinetic and thermodynamic model parametes are given in the Supporting Information. The models were implemented in Matlab R2015b.

5

RTO: Problem Formulation and implementation

In this section, the RTO problem for the hydroformylation miniplant is formulated and a brief description of the implementation of the scheme is presented,

5.1

RTO Problem Formulation

The RTO problem is formulated as the minimization of the operating costs (raw material, catalyst and energy in e/h) per unit of tridecanal produced (Ftridecanal in kmol/h) subject to the model equations (grouped in the map H(u)) and the input constraints, as expressed by:

min φ = u

Cost1−dodecene + CostCatalyst + Costenergy Ftridecanal

s.t. H(u) = 0

(44a) (44b)

ulb ≤ u ≤ uub .

(44c)

where Cost1−dodecene is the cost of the raw material, CostCatalyst is the cost of the continuous addition of catalyst to the process and Costenergy is the cost associated to cooling and heating of the process streams. Preliminary studies showed that CostCatalyst >> Costenergy , therefore, an accurate estimation of the heating a cooling requirements is not relevant for the computation of the operating costs. The objective function is expressed in e/kmol of tridecanal. As degrees of freedom (u), the following variables were considered: • Reactor temperature set-point TReactor . 34


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• Reactor pressure set-point PReactor . • Molar fraction of carbon monoxide in feed gas set-point yCO . • Catalyst dosing rate n˙ catalyst,in .

5.2

Proposed Scheme

Figure 11 depicts a block diagram of the RTO scheme that was implemented in the miniplant. It is similar to the typical structure used in commercial RTO systems based on the twostep approach. 31 The main difference is the iterative optimization via Modifier Adaptation in order to update the decision variables i.e. the set points that are sent to the lower control layer. By means of this scheme, the robustness of the process to model uncertainties is improved. A steady-state detector module and a data reconciliation with M-estimator module were implemented to provide robustness to measurements errors. The RTO layer can be triggered when a steady-state has been detected based on the raw data from the GC and the data from the process control system of the miniplant (temperature, flows, pressure). The raw data is processed by the Data Reconciliation Module and the reconciled data is used to estimate the objective function and the constraints that are used by the iterative optimization algorithm. Finally the optimization module generates the set points for the regulatory control layer.

5.3

System Implementation

Figure 12 illustrates the architecture of the system. The miniplant is connected to the main work station that runs a LabVIEW application, where the different process variables are displayed in real-time. A sample of the product in the decanter is taken automatically by the GC analyzer. The signal is sent to a second work station where the raw data is processed to estimate the actual composition. This is realized by using a Java Script (which is able to read the .pdf report of 35



Figure 11: Block Diagram of the RTO scheme

Figure 12: System Architecture of the RTO implementation

36


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the GC) or by using a Visual Basic Script (which is able to read the .xls report of the GC), both alternatives were implemented. Furthermore, a TCP/IP protocol suite was chosen to achieve communication between the workstations. The LabVIEW application in the main workstation acts as a server and opens the ports and keeps listening until the client (a Java application on the sub workstation) requests to establish a connection and the client and the server exchange data. The composition of the product stream and the operating conditions that are recorded by the LabVIEW application is used by the MATLAB algorithm in order to perform the optimization and to compute the optimal operating points. Finally, the operator validates the output from the optimizer i.e. the set-points that were computed by the RTO scheme, and manually introduces the respective values. A sampling time of 60 min is adequate for the process at hand which settles slowly to a new equilibrium due to the slow kinetics and the presence of the recycle. The process model consists of 422 nonlinear equations. In average it takes less than 1 min to solve each R optimization in MATLAB 2013a which runs in a Intel CoreTM i7-4770 Processor 3.40 GHz.

6

Materials and methods

The chemicals used for the experiment in this work are listed in table 2. All the chemicals were used without further purification. Table 2: Chemicals used in this work Substance 1-Dodecene DMF n-Decane Biphephos Rh(acac)(CO)2 CO H2

Supplier VWR Carl Roth Merk Molisa Umicore Messer Industriegase Messer Industriegase

Purity [%] 95 99 94 97 39.9 Rh > 99.99 > 99.99

A brief description of the experimental set-up was already given in section 3.4. Before starting the experiment, the reactor was filled with the DMF+catalyst solution and pres37



surized with the gas mixture for catalyst activation. Reactor and decanter temperature set-points were adjusted to the nominal values. Once the reaction conditions have been achieved, the pumps P1 and P2 are switched on to ensure continuous dosing of the substrate and of the DMF+catalyst solution. When the level switch in the decanter is active, the recycle pump P3 is switched on. The composition of the product phase is analysed by gas chromatography in an Agilent Gas Chromatography HP6890A with a capillary column ( HP5, 30m × 0.32mm × 0.25µm) and a flame ionization detector.

7

Results

In this section, the main results regarding the performance of the proposed RTO scheme are presented. It starts with the assessment of the model quality by comparing the predictions with experimental data and the validation of the algorithms for steady state identification and data reconciliation. The complete RTO scheme is validated first by simulations and then experimentally in the continuously operated miniplant.

7.1

Model predictions vs. experimental results

To evaluate the performance of the models, data regarding the composition of the raw product stream (i.e. composition of the nonpolar phase in B4) was compared to the prediction of the models for two long-term experiments in the miniplant that were previously reported: 1. Experiment A: Reported by Zagajewski et al. 46 with the operating conditions presented in table 3. 2. Experiment B: Reported by Dreimann et al. 17 with the same operating conditions as experiment A, but with half the catalyst concentration (0.025 % mol vs. 0.05% mol, based on the amount of substrate).

38


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Table 3: Operating conditions of the miniplant for experiments A and B reported by Zagajewski et al. 46 and Dreimann et al. 17 Operating Parameters Feed 1-dodecene Feed n-decane Flows Feed DMF Catalyst Precursor Ligand Pressure . Temperature 1-dodecene/precursor Reactor Ligand/precursor CO/H2 Stirrer Speed Decane/DMF/Substrate Decanter Temperature

Units [g/h] [g/h] [g/h] [mg/h] [mg/h] [bar] [◦ C ] [mol/mol] [mol/mol] [mol/mol] [rpm] [% mass] [◦ C]

Value 12.2 32.1 4.3 0.25 31.00 20.0 90.0 4000/1 5/1 1:1 750 42/42/16 5.0

Figures 13 and 14 show the raw product composition (composition of the nonpolar phase in B4) and the product yield for both experiments together with the predictions given by the models. There is a relatively good agreement between the predictions of both models and the experimental data, particularly in terms of the yield of tridecanal. Some differences between the predictions and the experiments arise when the yield of dodecane and of b-aldehyde are compared. This can be explained by the high uncertainty in the rate of isomerization that has been reported in the literature. Figure 15 and 16 display the parity plot for both experiments. As can be seen here, Model II provides better predictions compared to Model I in terms of raw product composition and yield of the different products. This can be verified by calculating the root-mean-square error (RMSE) rP RMSE =

(yî − yi )2 , N

(45)

where yˆ1 is the estimation of the process model and yi is the measured value. When different variables are compared, scaling of the data was performed. Regarding the product composition, the RMSE for Model I (4.89 %) was higher than for Model II (2.47 %). The difference becomes even higher when the yields of the products are 39



Figure 13: Comparison of experimental with predicted values for the miniplant run reported by Zagajewski et al. 46 (Experiment A). The operating conditions are listed in table 3 (a) Composition of the raw product stream (nonpolar phase in B5) (b) Product yield.

40


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Figure 14: Comparison of experimental with predicted values for the miniplant run reported by Dreimann et al. 17 (Experiment B). The operating conditions are listed in table 3 (a) Composition of the raw product stream (nonpolar phase in B5) (b) Product yield.

41



compared. In this case, the RMSE of Model I is 59.36 %, which can be explained by the large overestimation of the amount of b-aldehyde, while for Model II the RMSE is 15.18 %. As a result, Model II will be used in the proposed RTO scheme.

Figure 15: Parity plot of the raw product composition for the experiments presented in table 3

Figure 16: Parity plot of the yields for the experiments presented in table 3

7.2

Validation of the Steady State Identification Module

In this work, due to the low frequency sampling of the Gas Chromatography, SSI should be performed with the smallest possible number of points. Among the measured variables, the mass fractions of the components in the raw product stream were considered as key to the 42


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performance of the process. Therefore, the stationarity test was applied to these measurements and the plant is considered at steady state only if these variables are stationary. The tuning of the filter parameters was performed by using historical data of the process and according to the algorithm suggested by Bhat and Saraf. 30 Figure 17 illustrates the performance of the algorithm for λ1 = 0.7, λ2 = 0.95, λ3 = 0.63 and Rcrit = 2.0. As can be seen, the algorithm is able to distinguish between windows were the system is in transient state (state=0) and windows were the system is in steady state (state=1). Despite the simplicity of the approach it works well due to the low level of noise in the signal. As it has been reported in the literature, under low noise the Modified F -test can provide a similar performance as other more complicated algorithms. 25

0.25 Tridecanal in Product [% mass]

0.2 0.15

Filtered data GC measurements

0.1 0.05 0 0

500

1000

1500

2000

2500

1500

2000

2500

Time [min]

1 State

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


0 0

500

1000 Time [min]

Figure 17: Validation of the steady state identification algorithm. The system is in steady state when State = 1, otherwise the system is in transient state (State = 0)

43



7.3

Data Reconciliation (DR)

It was observed that at stationary conditions the material balance is not completely satisfied. This can be explained by the presence of measurement noise and gross errors. Malfunction or inaccurate calibration of the sensors might lead to wrong estimates of the objective function in the RTO scheme which can lead to suboptimal operation. In this case study, the following sources of errors in the material balance have been identified: 1. Superposition of the peaks for the isomers of dodecene, the feed 1-dodecene and dodecane. As a result, the calculated areas for the estimation of the mass fractions are associated with some uncertainty. 2. Lack of flow meters in the feed streams of substrate and DMF. As was mentioned in section 3.4, the feed pumps P1 and P2 were calibrated offline by adjusting the electrical current supply. In normal operation, the performance is quite reliable. However, if there is a gas pocket in the intake line, the actual flow delivered will be different from the expected according to the calibration. This is particularly important in the case of the pump P2 where as a result of the low flow rate of 3.5 − 4.5 ml/h, any error is magnified. Figure 18 shows the global material balance envelope for the miniplant. First, the mass flows of the different components in the product stream were calculated based on the assumption that the total mass flow of the product is equal to the sum of the mass flows of the liquid inflow streams. i.e. m ˙3 ≈ m ˙1+m ˙ 2 . Then, the mass flows of the individual components were computed by m ˙ i,3 = wi,3 m ˙ 3 , where wi,3 is the mass fraction obtained from the GC analysis. Table 4 shows the results of the material balance by using that aproach. As can be observed, there is a significant mass imbalance for the nonpolar (decane) and polar (DMF) solvents, equivalent to 4.23% and 22.88% respectively. The total mass imbalance is estimated as −1.59%. The basis of data reconciliation are the balance equations. For the polar and nonpolar 44


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Figure 18: Material balance envelope used in the data reconciliation model

Table 4: Material balance for the TMS miniplant, without data reconciliation Component DMF decane 1-dodecene iso-dodecene dodecane b-aldehyde tridecanal Total Mass Flow

Mass Flow [g/h] Inlet Outlet 4.25 3.27 32.08 30.72 12.22 1.74 0 2.72 0 0.05 0 0.38 0 8.86 48.55 47.75

45

Molar Flow [mol/h] Inlet Outlet −2 5.81 × 10 4.48 × 10−2 −1 2.25 × 10 2.16 × 10−1 7.25 × 10−2 1.04 × 10−2 0 1.62 × 10−2 0 3.25 × 10−4 0 1.92 × 10−3 0 4.47 × 10−2 -



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solvent, the mass balances are given by:

m ˙ DM F,1 = m ˙ DM F,3

(46a)

m ˙ n−decane,1 = m ˙ n−decane,3

(46b)

where m ˙ i,j denotes the mass flow of the component i = DMF, n-decane in the stream j (Figure 18). For the components that are involved in the chemical network, individual component balances can be stated only under the assumption of perfect knowledge of the reaction kinetics. As was stated before, the objective of this work is to design a RTO scheme which is able to reach the plant optimum despite the presence of those uncertainty. Therefore, instead of setting up individual component balances, the following relation was formulated which is independent of the kinetic model:

n˙ 1−dodecene,1 =n˙ 1−dodecene,3 + n˙ Iso−dodecene,3 + n˙ tridecanal,3 + n˙ b−aldehyde,3 + n˙ dodecane,3 ,

(47)

where n˙ i,j is the molar flow of the component i in the stream j. This equation basically states that the molar inflow of 1-dodecene to the miniplant should be equal to the sum of the molar flows of the different species involved in the reaction network at the outlet. This equality is related to the concept of reaction invariants and can be generalized to more complex reaction networks and stoichiometries. 64

46


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The DR problem for the miniplant was then formulated as:

max

X

ρ(ξi )

(48a)

s.t:. m ˙ DM F,1 = m ˙ DM F,3

(48b)

y ˆ

m ˙ n−decane,1 = m ˙ n−decane,3

(48c)

n˙ 1−dodecene,1 = n˙ 1−dodecene,3 + n˙ Iso−dodecene,3 ... + n˙ tridecanal,3 + n˙ b−aldehyde,3 + n˙ dodecane,3

(48d)

ni,3 M W (i) wi,3 = P ni,3 M W (i)

(48e)

for all i,

where ξi = yi − yî in the estimation error of the i measurement. The decision variables in the objective function are the mass flows of the pumps P1 and P2 and the mass fractions in the product streams. For a robust estimation in case of outliers, the traditional least square objective function is replaced by a Welsch estimator defined by: c2 ρ(ξi , cw ) = w 2

1 − exp −

ξi cw

2 !! (49)

with tunning parameter cw . A value of cw = 2.9846 is used to obtain 95% asymptotic efficiency on the standard distribution. 65 Table 5 shows the material balance for the same set of measurements as presented before in table 4 but after applying the proposed robust data reconciliation approach. As can be observed, the material balance is satisfied. The most significant adjustment is the flow of DMF that can be associated with the presence of gross errors in the indicator. For the species participating in the reaction network, with the exception of dodecane, there were no changes of the values.

47



Table 5: Material balance for the TMS miniplant, after data reconciliation Component DMF decane 1-dodecene iso-dodecene dodecane b-aldehyde tridecanal Total gas Total Mass Flow

7.4

Mass Flow [g/h] Inlet Outlet 2.55 2.55 32.08 32.08 12.22 1.73 0 2.70 0 0 0 0.37 0 9.09 1.39 1.39 48.24 48.24

Molar Flow [mol/h] Inlet Outlet −2 3.49 × 10 3.49 × 10−2 2.25 × 10−1 2.25 × 10−1 7.25 × 10−2 1.02 × 10−2 0 1.60 × 10−2 0 0 0 1.92 × 10−3 0 4.43 × 10−2 9.26 × 10−2 9.26 × 10−2 -

Iterative Optimization via Modifier Adaptation

For the validation of the iterative optimization scheme, first, simulation studies were carried out. A sensitivity analysis was performed on the model parameters and it was found that the gas solubility and the equilibrium constants for the catalyst species have the largest influence on the cost function. In order to illustrate the algorithm, parameteric plant-model mismatch was created by decreasing the Henry coefficients Hj,0 by 50% (parametric uncertainty). Furthermore, it is asumed that the model that is used by the optimization ignores the formation of dimer catalyst specie i.e. Kcat,2 = 0 (structural uncertainty). Table 6 lists the operating intervals of the optimization variables and compares the simulated plant optimum with the model optimum. As expected, the mismatch leads to a suboptimal operation (807 vs 742 Euro/mol) if the wrong model is used without further adaptation.

For comparison purposes the traditional two-stage approach was applied in a simulation study. First, a parameter estimation step is performed in order to adapt the model parameters. Then, the new operating point is computed based on the updated model. These steps are repeated until a relative difference of less than 0.1% in the parameter values between two consecutive iterations is reached. A constraint on the change of the input variable of 10% 48


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Table 6: Operating variables and simulated plant optimum. Simulations studies

Reactor temperature [◦ C] CO fraction [-] Catalyst dosage [ppm] Gas pressure [bar] Cost [Euro/kmol]

Operating interval 85∼105 0.0∼0.99 0.25∼2.0 10∼30

Initial Optimum Set-point Simulated Plant 95.0 85.92 0.5 0.56 1.1 0.47 20.0 30.0 919.36 742.25

Optimum Model 90.5 0.67 0.13 30.0 807.87

of the operating range is imposed, i.e. ∆u ≤ 0.1(uub − ulb ). Table 7 shows the values of the parameters for the simulated plant, the process model and the values after convergence of the two-step approach. Table 7: Parameters values for the simulated plant, the nominal model and the values estimated by the two-step approach Parameter Unit HCO,0 (MPa · m3 )/kmol HH2 ,0 (MPa · m3 )/kmol Kcat,2 -

Simulated Plant Model Two-step Result 35500 17775 17907 910 455 1966 1.01 -

Figure 19 shows the trajectory of the optimization variables as a function of the iteration index. The initial operating point was taken from Zagajewski et al.. 46 In the first iteration, the model-based optimal operating point is applied to the process. As can be seen, the algorithm does not converge to the optimal operating point of the simulated plant. There are significant differences regarding the reactor temperature and the catalyst dosing that can be explained by the fact that the model does not consider the catalyst pre-equilibrium in detail. As a result of the mismatch a suboptimal operation is obtained. Figure 20 shows the trajectory of the objective function. A significant improvement of the performance is achieved after only two iterations. However, after convergence a difference in the cost function is observed in comparison to the simulated plant optimum. This is the best result that can be obtained by using the two-step approach. If a different set of parameters is used in the estimation step, a larger deviation from the plant optimum is observed. Simulation results of the modifier-adaptation approach using quadratic approximation 12 49



TR [°C]

95

Iterative Optimization by two−step approach

90

RTO Set−point Simulated Plant Optimum

Initial Operating Point Model−based Optimum

85

0

2

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8

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14

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22

RTO Iteration 0.7

[−] CO

y


Model−based Optimum

0.65 0.6 0.55

Initial Operating Point

0.5 0.45

0

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22

RTO Iteration 2.5

Catalyst Dosing [ppm]

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

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2



1.5 1


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22

RTO Iteration

Figure 19: Results of simulation studies of iterative optimization using the traditional twostep approach: Trajectory of the optimization variables as a function of the number of RTO-iterations

50


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920

Iterative Optimization by two−step approach

900

880


860

Cost (Euro/kmol)

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


Model−based Optimum 840

820

800

780

760

740 0

2

4

6

8

10

12

14

16

18

20

22

Index

Figure 20: Results of simulation studies of iterative optimization using the traditional twostep approach: Trajectory of the plant operating cost as a function of the number of RTOiterations are shown in Fig. 21, 22. The first two operating points are the same as before. In Figure 21 the trajectories of the optimization variables with respect to the number of RTO iterations (including perturbations to the process) are displayed. After reaching a steady state, in the iteration 3 the iterative optimization algorithm via Modifier Adaptation is started. In the first iterations, finite-difference approximation is used to compute the gradients. Afterwards, enough sampled points (here (nu + 1)(nu + 2)/2 = 10) are available for the quadratic approximation. As can be observed, convergence to the optimal inputs is achieved after19 iterations. In fact after iteration 7, only a marginal decrease in the objective function is obtained. This can be observed in Figure 22, which shows the evolutions of the cost and the number of plant evaluations with respect to the number of RTO iterations. The same procedure was applied in different scenarios. In all cases, convergence to a neighborhood of the simulated plant optimum is reached. No assumptions on uncertain model parameters were needed. The value of the catalyst dosing, partial pressure and reaction temperature at the nominal optimal point depends on the simulated mismatch. For all the scenarios, the optimum total 51



Iterative Optimization by Modifier Adaptation

95 TR [°C]



90 Model−based Optimum 85 0

2

4

6

8

10 12 RTO Iteration

14

yCO [−]

16

18

20

22


0.65 0.6


0.55 0.5 0.45 0

Initial Operating Point 2

4

6

8

10 12 RTO Iteration

14

16

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22

2.5 Catalyst Dosing [ppm]

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

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2



1.5 1


0.5 0 0

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6

8

10 12 RTO Iteration

14

16

18

20

22

Figure 21: Results of simulation studies of iterative optimization using MAWQA algorithm: Trajectory of the optimization variables as a function of the number of RTO-iterations

52


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920

900

Iterative Optimization by Modifier Adaptation

880

860

Cost (Euro/kmol)

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


840



820

800

780

760

740

0

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14

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RTO Iteration

Figure 22: Results of simulation studies of iterative optimization using MAWQA algorithm: Trajectory of the plant operating cost as a function of the number of RTO-iterations pressure was at the maximum of 30 bar.

7.5

Proof of concept: Validation in the Miniplant

The RTO scheme described above was validated in the continuously operated miniplant for the hydroformylation of 1-dodecene in a thermomorphic solvent system. Preliminary experiments showed a very long settling time associated with changes in catalyst dosing. Therefore, the catalyst dosing was not used as an optimized input for these experiments. Furthermore, according to the simulation results all experiments were performed at the maximum total pressure of 30 bar. As decision variables, the reaction temperature and the fraction of carbon monoxide in the feed gas are considered (u = {TR , yCO }). The other operating conditions are the same as the ones that were used in the experiments for model validation (see table 3) with an initial catalyst concetration of 0.05% mol and a catalyst dosing of 2 ppm. The optimization problem (50) was reformulated as the maximization of the yield of tridecanal (Ytridecanal ) subject to the model equations (represented by the map G) and input 53



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bounds (ulb , uub ):

min Ytridecanal

(50a)

s.t. H(u) = 0

(50b)

u

ulb ≤ u ≤ uub .

(50c)

Figure 23 displays the yield of tridecanal over time. The reference line with a yield of 70 % corresponds to the operating point reported by Zagajewski et al.. 46 The experiment started with the same operating conditions as the reference case but at a total pressure of 30 bar and a reactor temperature of 95 ◦ C (nominal model-based optimal point). A yield of 73 % was obtained, this represents a significant improvent in comparison to the previous operating point. This can be explained by an increase in the concentration of hydrogen and carbon monoxide in the reaction medium as a consequence of the higher operating pressure. As a result, the reaction rate for the hydroformylation reaction is favored. After 15 hours, the iterative RTO scheme is employed and generates a sequence of operating points which drives the process to a final yield of more than 76 %. Figure 24 shows the trajectory of the manipulated variables, the first two operating points are introduced to the process in order to estimate the derivatives by using finite differences. Afterwards, the algorithm iteratively improves the operating point of the process. From the initial reactor temperature of 95 ◦ C, the proposed scheme drives the process to the maximum of 105 ◦ C. The experiment starts with an equimolar composition of the feed gas, while at the end of the experiment the optimal molar fraction of CO is computed as 0.58. An increase in the yield of branched tridecanal from less than 1.0 % (at the nominal operating point) to 3.3 % by the end of the experiment was observed. However, this is overcompensated by the higher increase in the yield of tridecanal of more than 6.6 %.

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Figure 23: Yield of tridecanal during iterative optimization with MAWQA

Figure 24: Trajectory of the manipuated variables: reactor temperature and molar fraction of carbon monoxide in the feed gas.

55



8

Conclusions

In this work, an iterative Real-time Optimization scheme which is able to deal with model and measurements uncertainties has been applied to the hydroformylation of 1-dodecene in a thermomorphic solvent system in miniplant scale. A complete model of the miniplant was developed and validated by using experimental data. The model combines a rigorous kinetic model of the reactor with simplified empirical expressions for the description of the catalyst recovery section. The results showed a good agreement between the predictions and the experimental data. In order to deal with model uncertainty, the robust MAWQA scheme was used, which is based on the estimation of the plant gradients by quadratic approximation of the actual plant map. Additional modules for steady state identification (SSI) and data reconciliation (DR) were integrated into the control software in order to provide robustness to measurements errors. The different modules were validated with actual data from the process and then the complete scheme was applied on-line to maximize the yield of the target product tridecanal. The results showed an improved yield of tridecanal not only with respect to the previous operation but also with respect to the optimal operating point that was computed based on the nominal model. As a consequence, the results of this work motivate the application of the methodology to other chemical processes even in the early stages of development where only models with limited prediction capabilities are available. The validation of the method in the continuously operated miniplant scale motivates its implementation to larger scale plants as an effective method to improve plant operations despite model and measurement inaccuracies. This work was limited to the case where steady-state measurements are considered for the adaptation of the nominal optimization problem. In many industrial processes, the transitions between steady states take place slowly; which is particularly true in the case of processes with internal recycles. A possible approach to overcome these limitations which is 56


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currently being considered is the estimation of the steady state gradient (or the steady-state itself) in the transient phase. This can be realized by reformulating the gradient estimation problem as a time-varying parameter estimation, which enables the use of estimation techniques such as recursive extended least squares (RELS) algorithms. 66,67 Another possibility is the use of system identification methods, 68 their applicability to the presented case study has already been demonstrated in simulation studies. 69

Acknowledgement This work was supported as part of the Collaborative Research Center: "Integrated Chemical Processes in Liquid Multiphase Systems" (SFB/Transregio 63 InPROMPT) by the Deutsche Forschungsgemeinschaft (DFG). Furthermore, the authors gratefully acknowledge the support of Umicore N.V. for providing the rhodium catalyst precursor Rh(acac)(CO)2 .

Supporting Information Available Supporting information regarding the model aparameters is included. This information is available free of charge via the Internet at http://pubs.acs.org/

References (1) Darby, M. L.; Nikolau, M.; Jones, J.; Nicholson, D. RTO: An overview and assessment of current practice. J. Process Control 2011, 21, 874–884. (2) Mandur, J. S.; Budman, H. M. Simultaneous model identification and optimization in presence of model-plant mismatch. Chem. Eng. Sci. 2015, 129, 106–115. (3) Chachuat, B.; Srinivansa, B.; Bonvin, D. Adaptation strategies for real-time optimization. Comput. Chem. Eng. 2009, 33, 1557–1567.

57



(4) Yip, W. S.; Martin, T. E. Designing plant experiments for real-time optimization systems. Contr. Eng. Pract. 2003, 11, 837–845. (5) Marchetti, A.; Francois, G.; Faulwasser, T.; Bonvin, D. Modifier Adaptation for Realtime Optimization - Methods and Applications. Processes 2016, 4, 1–35. (6) Gao, W.; Engell, S. Iterative set-point optimization of batch chromatography. Comput. Chem. Eng. 2005, 29, 1401–1409. (7) Tatjewski, P. Iterative Optimizing Set-Point Control- The Basic Principle Redesigned. 15th Triennal World Congress IFAC. 2002. (8) Marchetti, A.; Chachuat, B.; Bonvin, D. Modifier-Adaptation Methodology for Realtime Optimization. Ind. Eng. Chem. Res. 2009, 48, 6022–6033. (9) Roberts, P. D. An Algortihm for Steady-State System Optimization and ParameterEstimation. International Journal of Systems Science 1979, 10, 719–734. (10) Marchetti, A.; Chachuat, B.; Bonvin, D. A dual modifier adaptation approach for realtime optimization. J. Process Control 2010, 20, 1027–1037. (11) Navia, D.; Briceno, L.; de Prada, C. Modifier-Adaptation Methodology for Real-Time Optimization Reformulated as a Nested Optimization Problem. Ind. Eng. Chem. Res. 2015, 54, 12054–12071. (12) Gao, W.; Wenzel, S.; Engell, S. A reliable modifier-adaptation strategy for real-time optimization. Comput. Chem. Eng. 2016, 91, 318–328. (13) Bunin, G.; Vuillemin, Z.; Francois, G.; Nakato, A.; Tsikonis, L.; Bonvin, D. Experimental real-time optimization of a solid fuel cell stack via constraint adaptation. Energy 2012, 39, 54–62. (14) Navia, D.; Villegas, D.; Cornejo, I.; de Prada, C. Real-time Optimization for a laboratory-scale flotation column. Comput. Chem. Eng. 2016, 86, 62–74. 58


Page 58 of 67

Page 59 of 67 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


(15) Behr, A. Angewandte homogene Katalyse; WILEY-VCH: Weinheim, 2008. (16) Lummiss, J. A. M.; Oliveira, K. C.; Pranckevicius, A. M. T.; Santos, A. G.; dos Santos, E. N.; Fogg, D. E. Chemical Plants: High-Value Molecules from Essential Oils. J. Am. Chem. Soc. 2012, 134, 18889–18891. (17) Dreimann, J.; Lutze, P.; Zagajewski, M.; Behr, A.; Gorak, A.; Vorholt, A. Highly integrated reactor-separator systems for the recycling of homogeneous catalysis. Chem. Eng. Process. 2016, 99, 124–131. (18) Hernández, R.; Engell, S. Modelling and iterative real-time optimization of a homogeneously catalyzed hydroformylation process. Comput.-Aided Chem. Eng. 2016; pp 1–6. (19) Gao, W.; Hernández, R.; Engell, S. A study of Explorative Moves during Modifier Adaptation with Quadratic Approximation. Processes 2016, 4, 36–52. (20) Forbes, J. F. Model structure and adjustable parameter selection for operations optimization. Ph.D. thesis, McMaster University, 1994. (21) Skogestad, S. Self-optimizing control: the missing link between steady state optimization and control. Comput. Chem. Eng. 2000, 24, 569–575. (22) Jäschke, J.; Skogestad, S. NCO-tracking and self-optimizing control in the context of real-time optimization. J. Process Control 2011, 21, 1407–1416. (23) Bonvin, D.; Srinivasan, B. On the role of the necessary conditions of optimality in structuring dynamic real-time optimization schemes. Comput. Chem. Eng. 2013, 51, 172–180. (24) Krstic, M.; Wang, H. Stability of extremum seeking feedback for general dynamic systems. Automatica 2000, 36, 595–601.

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(25) Roux, G. A. L. L.; Santoro, B. F.; Sotelo, F. F.; Teissier, M.; Joulia, X. Improving Steady State Identification. Comput.-Aided Chem. Eng. 2008, 25, 459–464. (26) Savitzky, A.; Golay, M. J. E. Smoothing and differentiation of data by simplified least square procedures. Anal. Chem. 1964, 36, 1627–1639. (27) Bendat, J. S.; Pierson, A. Random Data: Analysis and Measurement Procedures, 2nd ed.; John Wiley and Sons: New York, 1986. (28) Korbel, M.; Bellec, S.; Jiang, T.; Stuart, P. Steady state identification for the online data reconciliation based on wavelet transform and filtering. Comput. Chem. Eng. 2014, 63, 206–218. (29) Cao, S.; Russell, R. An efficient method for on-line identification of steady state. Process Control 1995, 5, 363–374. (30) Bhat, S. A.; Saraf, D. N. Steady-state Identification, Gross Error Detection, and Data Reconciliation for Industrial Process Units. Ind. Eng. Chem. Res. 2004, 43, 4323–4336. (31) Camara, M.; Quelhas, A.; Pinto, J. Performance Evaluation of Real Industrial RTO Systems. Processes 2016, 4, 139–158. (32) Mobed, P.; Maddala, J.; Rengaswamy, R.; Bhattacharyya, D.; Turton, R. Data Reconciliation and Dynamic Modeling of a Sour Water Shift Reactor. Ind. Eng. Chem. Res. 2014, 53, 19855–19869. (33) Rafiee, A.; Behrouzshad, F. Data reconciliation with application to a natural gas processing plant. J. Nat. Gas Sci. Eng. 2016, 31, 538–545. (34) Romagnoli, J.; Sanchez, M. Data Processing and Reconciliation for Chemical Process Operation; Academic Press: San Diego, 1999. (35) Wang, D.; Romagnoli, J. A. A Framework for Robust Data Reconciliation Based on a Generalized Objective Function. Ind. Eng. Chem. Res. 2003, 42, 3075–3084. 60


Page 60 of 67

Page 61 of 67 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


(36) Frey, G. D.; Daembkes, G. 75 Jahre Oxo-Synthese: 75 Years of Oxo Synthesis; KlartextVerlag: Essen, 2013. (37) Kohlpainter,

C.;

Fisher,

R.;

Cornils,

B.

Aqueous

Biphasic

Catalyst:

Ruhrchemie/Rhone-Poulenc Oxo Process. Appl. Catal., A 2001, 221, 219–225. (38) Cornils, B.; Herrmann, W.; Eckl, R. Industrial aspects of aqueous catalysis. J. Mol. Catal. A: Chem. 1997, 116, 27–33. (39) Kämper, A.; Warrelman, S. J.; Reiswich, K.; Kuhlmann, R.; Franke, R.; Behr, A. First iridium-catalyzed hydroformylation in a continuously operated miniplant. Chem. Eng. Sci. 2016, 144, 364–371. (40) Cornils, B.; Herrmann, W.; Horvath, I.; Leitner, W.; Mecking, S.; Oliver-Bourbigou, H.; Vogt, D. Multiphase Homogeneous Catalysis; Wiley-VCH: Weinheim, 2005. (41) Brunsch, Y.; Behr, A. Temperature controlled catalyst recycling in homogeneous transition-metal catalyst: minimization of catalyst leaching. Angew. Chem. Int. Ed 2013, 52, 1586–1589. (42) Vogl, C.; Paetzold, E.; Fisher, C.; Kragl, U. Highly selective hydroformylation of internal and terminal olefins to terminal aldehydes using a rhodium-BIPHEPHOS-catalyst system. J. Mol. Catal. A: Chem. 2005, 232, 41–44. (43) McBride, K.; Gaide, T.; Vorholt, A.; Behr, A.; Sundmacher, K. Thermomorphic solvent selection for homogeneous catalyst based on COSMO-RS. Chem. Eng. Process. 2015, 99, 97–106. (44) Markert, J.; Brunsch, Y.; Munkelt, T.; Kiedorf, G.; Behr, A.; Hamel, C.; SeidelMorgenstern, A. Analysis of the reaction network for the Rh-catalyzed hydroformylation of 1-dodecene in a thermomorphic multicomponent solvent system. Appl. Catal., A 2013, 462, 287–295. 61



(45) van Leeuwen, P. W. N. M.; Claver, C. Rhodium Catalyzed Hydroformylation; Kluwer Academic Publishers: New York, 2002. (46) Zagajewski, M.; Behr, A.; Sasse, P.; Wittman, J. Continuously operated miniplant for the rhodium catalyzed hydroformylation of 1-dodecene in a thermomorphic multicomponent solvent system (TMS). Chem. Eng. Sci. 2014, 115, 88–94. (47) Zagajewski, M.; Dreimann, J.; Behr, A. Verfahrensentwicklung vom Labor zur Miniplant: Hydroformylierung von 1-Dodecen in thermomorphen Lösungsmittelssystemen. Chem. Ing. Tech. 2014, 86, 449–457. (48) Zagajewski, M. Planung, Bau und Betrieb einer Miniplant zur kontinuerlichen Hydroformylierung von 1-Dodecen in thermomorphen Mehrkomponenten Lösungsmittelsystemen. Ph.D. thesis, Technische Universität Dortmund, Faculty of Biochemical and Chemical Engineering, 2015. (49) van Elk, E. P.; Borman, P. C.; Kuipers, J. A.; Versteeg, G. F. Modelling of gas-liquid reactors - stability and dynamic bahaviour of a hydroformylation reactor, influence of mass transfer in the kinetics controlled regimen. Chem. Eng. Sci. 2001, 56, 1491–1500. (50) Kiedorf, G.; Hoang, D.; Müeller, A.; Jörke, A.; Markert, J.; Arellano-Garcia, H.; SeidelMorgenstern, A.; Hamel, C. Kinetics of 1-dodecene hydroformylation in a thermomorphic solvent system using rhodium-biphephos catalyst. Chem. Eng. Sci. 2014, 115, 31–48. (51) Hentschel, B.; Kiedorf, G.; Gerlach, M.; Hamel, C.; Seidel-Morgenstern, A.; Freund, H.; Sundmacher, K. Model-Based Identification and Experimental Validation of the Optimal Reaction Route for the Hydroformylation of 1-Dodecene. Ind. Eng. Chem. Res. 2015, 54, 1755–1765. (52) Vogelpohl, C.; Brandenbusch, C.; Sadowski, G. High-pressure gas solubility in multi-

62


Page 62 of 67

Page 63 of 67 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


component solvent systems for hydroformylation. Part I: Carbon monoxide solubility. J. Supercrit. Fluids 2013, 81, 23–32. (53) Vogelpohl, C.; Brandenbusch, C.; Sadowski, G. High-pressure gas solubility in multicomponent solvent systems for hydroformylation. Part II: Syngas solubility. J. Supercrit. Fluids 2014, 88, 74–84. (54) Hentschel, B.; Peschel, A.; Freund, H.; Sundmacher, K. Simultaneous design of the optimal reaction and process concept for multiphase systems. Chem. Eng. Sci. 2014, 115, 69–87. (55) van Elk, E. P.; Borman, P.; Kuipers, J. A. M.; Versteeg, G. F. Modelling of gas-liquid reactors - implementation of the penetration model in dynamic modelling of gas-liquid processes with the presence of a liquid bulk. Chem. Eng. J. 2000, 79, 223–237. (56) Brinkmann, M.; Warnecke, H. J.; Prüss, J. Modellierung reaktiver Stoffaustauchprozesse. Chem. Ing. Tech. 1996, 68, 239–252. (57) Salmi, T. O.; Mikkola, J. P.; Warna, J. P. Chemical Reaction Engineering and Reactor Technology; CRC Press: London, 2009. (58) Fogler, H. S. Elements of Chemical Reaction Engineering; Pearson Education: New Jersey, 2016. (59) Jörke, A.; Triemer, S.; Seidel-Morgenstern, A.; Hamel, C. Kinetic Investigation Exploiting Local Parameter Subset Selection: Isomerization of 1-Decene using Rh-Biphephos Catalyst. Chem. Ing. Tech. 2015, 87, 713–721. (60) Merchan, V. A.; Wozny, G. Comparative Evaluation of Rigurous Thermodynamic Models for the Description of the Hydroformylation of 1-Dodecene in a Thermomorphic Solvent System. Ind. Eng. Chem. Res. 2016, 55, 293–310.

63



(61) Steimel, J. Model-based Conceptual Design and Optimization of Continuous Chemical Processes under Uncertainties. Ph.D. thesis, Technische Universität Dortmund, Faculty of Biochemical and Chemical Engineering, 2015. (62) Brunsch, Y. Temperaturgesteuertes Katalysatorrecycling für die homogen katalysierte Hydroformylierung langkettiger Alkene. Ph.D. thesis, Technische Universität Dortmund, Faculty of Biochemical and Chemical Engineering, 2013. (63) Müller, D. Development of Trajectories Under Uncertainty for a Hydroformylation Mini-plant. Ph.D. thesis, Technische Universität Berlin, Fakultät III - Prozesswissenschaften, 2015. (64) Rodrigues, D.; Srinivasan, S.; Bonvin, D. Variant and invariant states for chemical reaction systems. Comput. Chem. Eng. 2015, 73, 23–33. (65) Korpela, T.; Suominen, O.; Majanne, Y.; Laukkanen, V.; Lautala, P. Robust data reconciliation of combustion variables in multi-fuel fired industrial boilers. Contr. Eng. Pract. 2016, 55, 101–115. (66) Rodriguez-Blanco, T.; Sarabia, D.; Pitarch, J.; de Prada, C. Modifier Adaptation methodology based on transient and static measurements for RTO to cope with structural uncertainties. Comput. Chem. Eng. 2017, 106, 480–500. (67) Navia, D.; Puen, A.; Bergh, L.; Rodriguez-Blanco, T.; Sarabia, D.; de Prada, C. Modifier-Adaptation Based on Transient Measurements Applied to a Laboratory-Scale Flotation Column. Comput.-Aided Chem. Eng. 2017; pp 1729–1734. (68) Cadavid, J.; Hernandez, R.; Engell, S. Speed-up of Iterative Real-Time Optimization by Estimating the Steady States in the Transient Phase using Nonlinear System Identification. IFAC PapersOnLine. 2017; pp 11269–11274.

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(69) Gao, W.; Hernandez, R.; Engell, S. Real-time optimization of a novel hydroformylation process by using transient measurements in modifier adaptation. IFAC-PapersOnLine. 2017; pp 5731–5736.

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