1

Model-Based Experimental Analysis of a Fixed Bed

Reactor for Catalytic SO2 Oxidation

J.C. Schneberger1), H. Arellano-Garcia1), G. Wozny1)

1) Chair of Process Dynamics and Operation, Berlin Institute of Technology

Sekr. KWT-9, Str. des 17. Juni 135, D-10623 Berlin, Germany

S. Krkel2)

2) DFG Research Center MATHEON, Rudower Chaussee 25, D-12489 Berlin, Germany

H. Thielert3)

3) ThyssenKrupp Uhde GmbH, Friedrich-Uhde-Str. 15, D-44141 Dortmund, Germany

In this work, a developed approach is proposed which aims at computing experimental layouts, setups

and controls in order to optimize the statistical reliability of parameter estimates from the resulting

experimental data. Furthermore, since the optimum experimental design depends on the values of the

parameters, a sequential design is proposed. This represents an interaction between proposed

experiments (e.g. inlet temperature and concentration), their evaluation, parameter estimation, and new

designs information such as reactor layout, sampling points and temperature measurement positions.

This follows an iterative procedure until all parameters achieve a prescribed statistical quality. Due to

the fact that the overlaying optimization problem contains a highly nonlinear objective function, a hybrid

optimization framework is proposed in order to overcome the problem of local minima. To show the

efficiency of the developed framework, it has been applied to the contact reaction. This reaction is a

2

basic step in the sulfuric acid production and takes place in a catalytic fixed bed reactor called the

contact horde. In summary, this work presents a systematic approach based on nonlinear experimental

design as an efficient tool for the validation of kinetic models.

Keywords: Experimental Design, Catalytic Fixed Bed Reactor, Hybrid Optimization, Experimental Set-

Up, Parameter Accuracy, Model Validation.

1. Introduction

Mathematical models are widely used to represent individual unit operations as well as entire

processes and consequently their behavior. They can be applied for simulation, process design and/or

improvement of existing understanding about specific phenomenon. On the other hand, the accurate and

reliable representation of the basic phenomena strongly relies upon model structure and the parameter

values, which are to be adjusted so as to correspond with the real process. Therefore, experimental data

are required in order to estimate the model parameters and, thus, to validate the developed model.

However, the estimation error depends on the design of the experiments. In particular for nonlinear

systems, the decision on the experiments to be executed is decisive on account of cost and time. This

issue represents a compromise between experimental attempt and data quality. Methods for the

determination of the data quality, i.e. the model parameter accuracy, were first developed for linear

systems (see e.g. [1]) and then expanded to nonlinear problems [2,3]. The descriptiveness of these

properties allows the so called experimental design to perform experiments that improve these

characteristics [4,5]. The application of these methods is especially useful, when experiments are

aligned to a high financial and time effort.

In this work, the methods of nonlinear optimal experimental design are applied to the process of sulfur

dioxide oxidation. Since sulfuric acid is worldwide one of the most produced chemicals [6], the reaction

has been subject of many experimental studies. This is also the reason why plenty of catalysts are

commercially available and there are still new catalysts entering the market. On the other hand, the

experimental effort for new catalyst materials represents an expensive and challenging task since high

3

temperatures are required for this particular reaction and corrosive and toxic gases are involved, which

not only form sulfuric acid but also corrode almost every steel type when condensing. This makes an

optimal experimental design necessary. In a first step, the experimental set-up can be designed by

applying the method of dynamic experimental design to a steady state problem. Instead of the time the

space is considered as independent variable. Experiments are designed leaving the position of a

measurement as a free variable. With a repetition of this procedure the most desired measurement

positions can be determined. Then, the optimal control factors, e.g. inlet temperature, pressure and

concentrations can be calculated for each experiment so as to reach maximal model parameter accuracy,

i.e. maximal model quality. Finally, it is possible to determine the number of experiments necessary to

reach the desired or required parameter accuracy. The rest of the paper is structured as follows. First, the

sulfuric acid production process is introduced pointing out the importance of an accurate model for the

sulfur dioxide conversion. The model used for the fixed bed reactor is derived including its solution

techniques and gradient calculation. Afterwards, the approach employed for experimental design is

presented followed by the introduction of the developed hybrid optimization framework. At last the

results for some scenarios of the experimental procedure are given and discussed in detail.

2. Process Description

All established processes for the production of sulfuric acid operate with the intermediate product

sulfur dioxide. This can be produced by the combustion of brimstone, sulfur ore or hydrogen sulfide.

The whole process is depicted in Fig. 1.

The further oxidation after the combustion to sulfur trioxide takes place in fixed bed reactors and is

catalyzed by a vanadium pentoxid containing catalyst. Due to the strong heat production, the reaction

beds are separated allowing a quenching with air. They are allocated together in the contact tower. The

sulfur dioxide is converted to sulfur trioxide, which can then be absorbed in sulfuric acid increasing its

concentration. The input of make-up-water allows the steady state production of sulfuric acid with a

desired concentration.

4

The main task concerning the design of the process considered is the determination of the number of

beds, their length, and the amount of quench air as well. With these steps, on the one hand, the

conversion of the sulfur dioxide is determined, which is important in order to hold waste gas directives.

On the other hand, the main stream amount is fixed which leads to smaller or larger apparatus size.

Thus, in order to find an optimal design and fulfill the restrictions, a reliable reactor model is essential.

The use of non suitable models in the design phase may result in critical situations, such as e.g. the

destruction of the catalyst or the violation of toxic gas concentrations in the waste gas outlet.

Figure 1. Flowsheet of a simplified sulfuric acid production process (ChemCADTM).

The present work is part of an ongoing industrial cooperation project, in which new operation

conditions are to be validated for wet catalysis sulfuric acid processes. However, due to the inlet

condition changes in the contact tower and new available catalysts, the reaction rate models have to be

validated further.

5

3. Reactor Model

In order to describe the behavior of the process unit, a model for a fixed bed plug flow reactor has

been used. In Fig. 2, the control volume is shown.

Figure 2. Control volume for a fixed bed plug flow reactor.

The steady state model equations include the component balances in Eq. (1), where the concentrations

are replaced by the molar flow rates Fi. The gas phase volume is given as a function of the reactor

diameter D, the catalyst density K, and the void fraction , respectively. The reaction rate for each

component i is calculated as the product of its stoichiometry coefficient i and the overall reaction rater

as follows,

( ) D4

1rdzdF

Kii pre-n= . (1)

The energy balance in Eq. (2) is not only a strong function of the reaction rate, but also of the

temperature dependency of the heat of reaction RxHD and the molar heat capacityi,Pc of the components,

( )

( )

pre-D-

=i,Pi

KRx

cF

D4

1Hr

dzdT

. (2)

The pressure drop along the reactor is modeled using the Ergun equation in Eq. (3). This is necessary

because the pressure drop in the fixed bed can not be neglected at all, and the overall reaction rate

depends on the pressure,

( ) ( )

p

+

he-

e

e-

p

r

-=

D4

MF75.1

D)1(150

1

DD4

MF

dzdP ii

PPGas

ii . (3)

6

Where DP is the catalysts particle diameter, Gas and are the gas phase density and viscosity, and Mi

the components molar weights, respectively. Making use of the laws of stoichiometry, the component

balances in Eq. (1) can be reformulated to Eq. (4) by considering the conversion, 0SOSO

0SO 222

F/)FF(X -= of

sulfur dioxide over the reactor length

( )0SO

K

2F

D4

1r

dzdX

pre-

=

. (4)

The reaction, which that takes place over the vanadium catalyst,

SO2 + O2 SO3, (5)

can be described in (6) with the kinetic equation given by Eklund [7] as a function of the component

partial pressures Pi, where KP is the equilibrium constant related to the reactant partial pressures.

-

++-=

2

PSO

SOO

SO

SO

k

321

KP

PP

P

Pk)Tln(k

Tk

expr2

3

2

3

2

. (6)

Eq. (6) can be reformulated as a function of the conversion with the initial component ratios i:

( )

-

+Q-

n-Q

n-Q

+Q-

=

2

P

SO

Oi

OO

SO KX1

XP

X

X

XX1

kr 3

2

22

3

. (7)

The parameters to be estimated during the model identification and validation procedure are the kinetic

constants k1, k2 and k3, while the equilibrium constant KP can be calculated from property data and is

considered as known. However, it should be noted, that the structure of Eq. (6) is an empirical one. It

corresponds to the generic equation for power law reaction rates,

-=-= -+ iiii

b

iP

f

i

b

i

f

iP

K1

PkPkPkr . (8)

The exponent values found by Eklund are given in table 1. The same coefficients are reported by Fogler

[8]. Carberry reports a slightly different reaction rate [9]. Nevertheless, they all are empirical and should

be treated with care. Here, the model published by Eklund, Eq. (6), is considered as the true model.

7

Table 1. Coefficients of the power law approach taken from [8].

The complete set of equations is discretized with the orthogonal collocation method using 3

collocation points and 5 intervals, which shows a good convergence behavior [10]. The model

parameters as well as the resulting algebraic equation system composed of 60 equations and variables

are summarized in Appendix A. In order to obtain a better convergence behavior, the state variables are

normalized by scaling them on their inlet values except for the conversion, which is by nature is in the

range between 0 and 1. Parameters are scaled in a logarithmic manner including them in the exponential

term in Eq. (6). The system is solved using a Newton-Raphson step. Then, the sensitivities of the state

variables [ ]TP,T,Xx = concerning the parameters T321 ]k,k,k[P = can be calculated using the Jacobian of

the equation system Jx and the internal numerical differentiation as follows,

P1

xPx

JJ --= . (9)

Where JP is the Jacobian of the equation system with respect to the parameters. Equation (9) gives the

exact derivation of the numerical solution with respect to the parameters [11]. The sensitivities are used

to calculate the covariance matrix which will then be used in the objective function of the later

introduced optimization problem in Eq.(12),

1

1M

T

Px

CPx

C

-

-

= . (10)

If the state variables x are measured directly, the covariance matrixes C can be calculated from Eq.

(10), where the sensitivities are weighted with the inverse variances of the measured variables CM, [12].

Therefore, independent additive normally distributed measurement errors are assumed. It has though to

be noticed that CM has a large influence on the result of the optimal experimental design. In this work,

three state variables are measured: conversion X, Temperature T and pressure P. The measurement

k1 [K] k 2 [1/ln(K)] k3 [-] fN2 fSO2 fO2 fSO3 bN2 bSO2 bO2 bSO3

-97782 -110.1 848.5 0 0.5 1 -0.5 0 -1.5 0 1.5

8

accuracy of these variables differs considerably. The errors resulting from the measurement procedure

can be taken from the devices, such as gas chromatograph, resistance thermometers, and pressure

gauges. In addition to these measurement errors, a value for the model inaccuracy has to be added,

which in many cases is larger than the measurement errors. This can be, in the case of the temperature

measurement, for instance, a penalty for the assumption adiabatic reactor. A real reactor is not

adiabatic, and thus, this assumption leads to a model inaccuracy. Both the total standard deviations and

the covariance matrix of the measured variables are given in Eq. (11). The high error for the temperature

measurement is a result of the uncertainty of the related properties, i.e. the heat capacities of the

components and the assumption of an adiabatic reactor.

=

s

s

s

==s=s=s

-

4000000

0250

001025.6

C,Pa200,K5%,5.2

4

2P

2T

2x

MPTX . (11)

The profiles for the state variables are depicted on the top of Fig. 3. They show the typical behavior of

an equilibrium limited reaction. After a certain distance z in the reactor, the reaction stops because the

equilibrium state is reached. An additional length represents a waste of catalyst and an unnecessary

increase of the pressure drop. However, it can also be used as a safety margin to assure the maximal

possible conversion, i.e. equilibrium conversion. On the bottom of Fig. 3 the sensitivities of the state

variables with regard to the parameters are shown. A change in the kinetic constants ki does not lead to a

change in the equilibrium conversion and temperature, because the term in brackets in Eq. (7)

approaches zero when reaching chemical equilibrium and the velocity constant k does not influence the

reaction rate anymore. Thus, the sensitivity trajectories approach zero when the equilibrium state is

reached. The pressure drop is a weak function of the reaction rate. The pressure drop is increased when

the forward reaction is faster in the front part of the reactor. Nevertheless, a change of 0.1 % of

parameter k1 leads to a change of 0.04 Pa in the pressure drop. This is out of the detectable range of the

pressure measurement. Therefore, the pressure measurement is not used in the parameter identification

procedure.

9

Figure 3. State variables conversion X, temperature T and pressure drop dP and their sensitivities for a sample test experiment.

4. Nonlinear Optimal Experimental Design

For the nonlinear optimal experimental design, a nonlinear constrained optimization problem has to be

solved. In this case, the optimization variables are the experimental design variables, namely, the inlet

conditions, and the measurement positions for the state variables. The resulting optimization problem is

defined as follows:

( )

( )( )

( ) bw,qa0x,w,q,p,xG

0x,w,q,p,x,xF

.t.s

Cmin

s

s

w,q

Y

=

=

j

(12)

10

=

)max(eig(C):E

det(C):D

NPtrace(C)

:A

(C)with NP1

j

where q are the experimental conditions (i.e. inlet temperature, concentrations etc.), w the

measurement position weights, which are indicating whether a measurement is taken (w=1) or not

(w=0), and xS the discrete state variables, respectively. F represents the model equations, and G the

nonlinear constrains with the state variables x and the parameters p, while defines the bounds of the

optimization variables. NP denotes the number of parameters to be estimated. The objective function in

Eq. (12) is a function, j , of the covariance matrix C of the parameter estimation problem. There are

different criteria which can be minimized, see e.g. [2]. In Fig. 4, these criteria are interpreted graphically

for the estimation of two parameters making use of a linearization of the confidence region.

Figure 4. Graphical interpretation of the optimization criteria (linearization).

The inner point gives the estimated value of the parameters p1 and p2. The ellipsoid surrounding this

point gives the linearized confidence region for a certain probability. According to this, the real values

of the parameters are inside the ellipsoid with a probability of e.g. = 95%. The aim of the experimental

design is to shrink this ellipsoid to an acceptable size. Therefore, the D-Criterion minimizes the

determinant of C. This can be interpreted as the ellipsoids surface area. This criterion can lead to very

large stretched confidence regions with a high correlation of parameters, but to a small surface area. Due

11

to the structure of the approach for the velocity constant k in Eq. (6), the parameters k1 to k3 are strong

correlated. Thus, working with the D-Criterion will not lead to improvements of the model accuracy.

The A-Criterion minimizes the area of the circle formed with the average main axis length of the

ellipsoid. It is calculated as the trace of C. The advantage of this criterion is a low computational effort

since the trace calculation is quite fast compared to the other criteria. The E-Criterion represents the

length of the longest main axis of the ellipsoid. Its value is given from the largest eigenvalue of C. This

includes an eigenvalue calculation, and a discontinuity. In any case, the result is similar to the A-

Criterion for strong correlated parameters. Due to these facts, in this work, the A-Criterion is chosen, i.e.

minimizing the trace of C. The optimization problem is solved with an approach based on SQP with a

tailored derivative computation. For the derivative calculation procedure of the of the different objective

functions, we refer to the work in [13]. Here, the commercial solvers SNOPT and NPSOL [14] from

TomlabTM are used in a MatlabTM environment. It has been found that NPSOL is more robust and faster-

converging. The model equation system is solved sequentially and not simultaneously leading to an

optimization problem that contains few free variables but a dense Jacobian. In open literature, NPSOL is

recommended for such problems, see ref. in [15].

5. Hybrid Optimization Framework

The objective function of the considered optimization problem in (12) is highly nonlinear.

Furthermore, the functions of the covariance matrix exhibit several local minima within the considered

region (see Fig. 9). However, the surface of the objective function shows a trend towards a certain

direction. This tendency can be exploit using the stochastic search algorithm PSO (particle swarm

optimization),[16]. We refer to [17] for a detailed description of the PSO algorithm. It simulates

particles which move over the surface of the objective function.

The particles share their actual and historical information (i.e. the values of the objective function) and

change the direction of their movement towards the best particle position. The main problem with this

algorithm is that it is very difficult to find any minima, local or global, since it is not using gradient

12

information. This leads to huge convergence times. To overcome this, the algorithm is combined with a

gradient-based optimization algorithm. The key idea is that each particle should move to the next

optimum using gradient information before it shares information with the other particles (see Fig. 5).

Due to the particle movement, the initial points of the gradient based algorithm can be in the constrained

region. This requires a robust algorithm such as NPSOL that can handle these kinds of problems [18].

An illustrative example pointing out the advantage of the proposed algorithm can be found in Appendix

B.

Figure 5: Hybrid optimization procedure.

6. Results

6.1 Experimental Set-Up

To determine the best experimental set-up, a specific scenario is given. As shown in Fig. 6, the

experimental reactor is divided into 5 sections. In all of these sections the temperature is measurable.

Additionally, the concentration of the reactor outlet can be measured. Since concentration

13

measurements are expensive and time consuming (e.g. gas chromatography), only two additional

concentration measurements are allowed. It is to be determined at which positions these measurements

are the most informative. In order to answer this question, 10 optimal experiments are designed leaving

the measurement weights, w, of the position 1 to 4 as free variables in Eq. (12). Furthermore, a

constraint has to be included which explicitly assures that the sum of these variables must be equal to 2.

The optimal positions are related to the experiment they have been determined for. This is the reason

why a set of experiments is designed. In 9 of these, the position 3 and 4 are found to be the best (Fig. 6).

This can be interpreted as follows: the concentration profile can be moved through the reactor with the

other design variables, i.e. the inlet conditions. The latter positions are of advantage because of the final

concentration measurement, which is fixed at the reactor outlet. This leads to optimal experiments with

a concentration change close to the reactor outlet.

Figure 6. Experimental set-up of the PFR system, FCR: Flow control and registration. PCR: Pressure control and registration. TCR: Temperature control and registration. TIR: Temperature indication and registration. QIR: Quality indication and registration, here the conversion X.

14

6.2 Optimal Single Experiment

In this section, one optimal experiment is designed. Here, the design variables are the inlet conditions

of the experimental reactor, i.e. the Reynolds number Re, the inlet concentrations xSO2 and xO2, as well

as the inlet temperature Tin. They can be influenced with the controls shown in Fig. 6. The optimal

single experiment is compared to a reference experiment derived intuitively. Based on the linear

experimental design, it can be concluded that the optimum of a constrained optimization problem has to

lie on the bound of the constraints. The optimal bounds can be identified easily: the temperature and the

educts concentrations are to be maximal in order to get a high conversion. Moreover, the total flow rate

represented by the Reynolds number has to be minimal in order to maximize the temperature increase.

Table 1 summarizes the derived inlet conditions which concern the Reynolds number Re, the mole

fractions xi of sulfur dioxide and oxygen, the inlet temperature Tin as well as their bounds.

Table 1: Bounds and optimal values of the design variables.

Inlet Condition Re [-] xSO2 [mol/mol] xO2 [mol/mol] Tin [C]

Lower bound 300 0.002 0.002 400

Upper bound 600 0.080 0.100 500

Reference value 300 0.080 0.100 500

Optimal value 300 0.080 0.100 447

As shown in Fig. 7, due to the nonlinearity of the system the reference experiment is not optimal. The

conversion reaches the equilibrium very fast and no further parameter information can be extracted from

the flat profiles. The optimal single experiment, i.e. the solution of the optimization problem given in

Eq. (12), results in a conversion profile, in which the equilibrium conversion is reached at the end of the

reactor allowing an informative concentration measurement in the position 5 (see Fig. 6). Moreover, Fig.

7 shows the profiles of the two experiments the intuitive and the single optimal as well as the confidence

15

ellipsoid for their corresponding parameters k1 and k2. The confidence region is calculated based on the

covariance matrix C with a multivariable normal distribution sampling method. For a better comparison,

the parameters are scaled to unity and only the two parameters k1 and k2 are considered. The two

additional confidence regions (k1-k3 and k2-k3) appear similar. With the optimal single experiment, the

spreading of the parameter values can anyway be reduced considerably.

Figure 7. Trajectories and confidence region of an intuitive experiment (top) versus an optimal single

experiment.

It should be noted that both the reference and the optimal experiment may cause problems in the

experimental implementation. In both cases the temperature inside the reactor crosses the bound value of

T=600 C. This is the highest temperature allowed for the vanadium pentoxid catalyst [19]. Thus, in this

work, the maximal temperature in the reactor is limited to Tmax=550C. This constraint is included in the

optimization problem leading to a new optimal single experiment. The resulting trajectories and the

confidence region are given in Fig. 8.

16

Figure 8. Optimal single experiment with a restricted reactor temperature.

The new optimal decision variables are given in table 2, and the resulting values for the objective

function and the parameter spreading in table 3, respectively. The last column in table 3 represents the

standard deviation including the correlation of the parameters.

Table 2: Optimal values of the design variables in the temperature constrained single experiment.

Inlet Condition Re [-] xSO2 [mol/mol] xO2 [mol/mol] Tin [C]

Optimal value 300 0.080 0.100 421

Table 3: The optimization criteria and the related scattering of the parameter values.

Experiment A D E k1 [%] k2 [%] k3[%] Max [%]

Reference 0.9342 2.00 10-6 2.803 92.1 98.7 98.9 167.4

Optimal 0.0021 9.08 10-8 0.006 4.2 4.7 4.7 7.9

Constrained 0.0058 1.36 10-7 0.017 6.8 8.0 7.9 13.2

It can be concluded that with only one single optimal experiment, no acceptable parameter accuracy can

still be reached. Thus, additional experiments have to be designed. In the sequel of this work, the

designed experiments are restricted to the temperature bound. Results for an unrestricted experimental

design have been reported in [20].

6.3 Multiple Experiments

17

Based on the presented solution strategies, it is now possible to design a set of multiple experiments.

For this propose, a sequential procedure is recommended since the estimated values of the parameters

can change with each performed experiment, and thus, leading to a different optimal experiment in the

next step. In this study, the parameter values are assumed to be constant from experiment to experiment,

due to the fact that no real experiments are performed so far. Anyhow, in the case of real performed

experiments, there will be a change of parameter values and a sequential proceeding becomes necessary.

In Fig. 9, the surface of the objective function for the fifth experiment is plotted for the two decision

variables: the inlet temperature Tin, and the Reynolds number, Re. The striped region on the right picture

of Fig. 9 represents the infeasible combinations of these variables.

Figure 9: Objective function (A-Criterion) for the 5th designed experiment w.r.t. the two decision

variables by fixed inlet concentrations at xSO2=0.05 and xO2=0.1.

The problem concerning the local minima can be seen on the left of Fig. 9, where the surface is

plotted. The objective function is highly non-convex. Thus, a tailored optimization algorithm is

required. To overcome the issue of ending up in a local optimum, the proposed hybrid optimization

algorithm is applied. In this case study, 5 particles and 5 iterations are used leading to 25 calls of the

optimization problem.

18

As a goal for the planning procedure, certain parameter accuracy can be assigned. Moreover, based on

the sequential experimental design, it can be determined how many experiments are required to reach

the requested accuracy. This gives an idea of the load of experimental work that has to be carried out in

order to reach the given objective. However, this is only an approximation. The optimal experiments and

parameter accuracies will change with each new parameter estimation, which has to be done after each

performed experiment, since the parameters converge to their true values. Besides, based on this

procedure the evolution of the confidence region can be monitored, and thus, it can be decided from

which point on further experiments will not increase the parameter accuracy significantly.

Figure 10: Progress of the optimization criteria according to the number of experiments.

19

Figure 11: Improvement of the confidence region w.r.t. the number of experiments.

In Fig. 10, the progress of the parameter values accuracy is shown and the horizontal line there

delineates the 2 % standard deviation. Fig. 11 visualizes the shrinking of the confidence ellipsoid for the

parameters k1 and k2. After ten experiments, curves which represent the optimization criteria, are getting

quite flat. This means that no significant improvement of the parameter value accuracies can be reached

with more experiments. In Fig. 12 can also be noticed that some high informative experiments are

repeated with an increasing number of experiments (e.g. experiments 4, 5, 8 and 12).

Figure 12: First 15 optimal experiment settings and their trajectories.

20

These recurring experiments are optimal because they reduce the measurement error in the repeated

experiment. In Fig. 12, the first 15 optimal experiment settings and their corresponding trajectories are

plotted. After 14 experiments, the standard deviation including the correlation of the parameters is lower

than 2 % (see Fig. 10). This could also be an objective of the experimental design procedure.

Nevertheless, lower deviation values imply higher experimental effort. On the other hand, this value is

not satisfactory, in particular when it is compared to Fig. 3, where a change of only 1 % of k3 results in a

change of up to 3 % in the conversion. One way to overcome this problem and to reduce the model

inaccuracy is to change the model structure. In order to select the responsible parameter, the analytical

derived confidence ellipsoids for the single experiment are calculated with a 95% confidence region

using a function stated in Appendix C (see also Fig. 13). The parameters k2 and k3 are highly correlated

leading to the largest ellipsoid in Fig. 13. The correlations of k1-k2 and k1-k3 are nearly the same. Thus,

it can be concluded that k2 or k3 could be eliminated. Mathematically it makes no difference which

variable is eliminated. Anyhow, from a physical point of view it is recommended to take out the

parameter k2, because this reduces the approach for the rate constant k in Eq. (7) to the well- known

Arrhenius equation.

Figure 13: Analytical calculated confidence regions for the parameter combinations (k1, k2 or k3).

21

7. Conclusions

The application of the methods of nonlinear optimal experimental design to steady state space-variant

reaction systems gives fairly good results and enables new options in the planning of experiments. The

developed hybrid optimization framework improves the results of the experimental design because it

helps to overcome the problem of local minima. While a single experiment can be designed intuitively,

in the case of multiple experiments this is not possible. It is shown that the confidence region can be

reduced considerably with the optimal designed experiments. Furthermore, a control of the parameter

accuracy is possible, and thus, an estimation of the experimental effort can be determined.

Moreover, restrictions related to the state variables can be introduced. Based on this, it can be assured

that the reactor temperature will not exceed a critical value. The results show then an interesting change

in the decision variables and trajectories of the optimal single experiment. This constraint represents

also an important safety aspect, which must be considered in the laboratory work.

During the model validation process highly correlated parameters can be identified such that the

model structure can be improved in the sense of parameter identifiability, what also means model

reliability. The impact of parameters, which are identified with a low accuracy on a process model, is

demonstrated with the related sensitivities. From this point of view, desired model accuracy can be

defined and the related experimental effort can be estimated. Based on this, the important procedure of

model identification and validation can be accomplished in a more systematic way using the proposed

approach based on the methods of nonlinear optimal experimental design. correspondingly

Appendix A: Equation System and Jacobians

The differential equations (2)-(4) are discretized with the orthogonal collocation on finite elements. In

this work NC=3 collocation points are used in NI=5 intervals resulting in an algebraic equation system of

60 equations. The mass balance is given exemplarily in Eq. (A1) to (A4) for the first interval.

)1(X)1(F = (A1)

22

( )C0

SO

2K

NCICC N:2jF

D4

1r

zNL

)1N:1(X)1N:1,j(A)j(F2

=

pre-

-++=

(A2-A4)

Where A and zNC represent the collocation matrix A for three collocation points and the coordinate of

the last collocation point, respectively, see e.g. [21]. In the equations (A5), (A6) and (A7), Jx for the first

interval I=1, the resulting Jacobians Jx, and JP are given, correspondingly.

+

+

+

+

+

+

==

)1N(P

)3)1N((F

)2(X

)3)1N((F

)1(X

)3)1N((F

)1N(P)2(F

)2(X)2(F

)1(X)2(F

)1N(P)1(F

)2(X)1(F

1

C

CCC

C

C

1I,x

J (A5)

---

---

---

---

=

=

=

=

=

=

5

4

3

2

x

]111[000

0]111[00

00]111[0

000]111[

0000

Ix,

Ix,

Ix,

Ix,

1Ix,

J

J

J

J

J

J

(A6)

The Jacobian versus the state variables Jx shows a block structure, which enables a faster inverse

calculation. The elements below the blocks result from the consistency conditions which couple the

intervals. They can be eliminated reducing the equation system to 45 equations with 45 variables.

+

+

+

=

3

Ic

2

Ic

1

Ic

321

P

k

)3N)1N((F

k

)3N)1N((F

k

)3N)1N((F

k)1(F

k)1(F

k)1(F

J (A7)

23

In the Tables A1 and A2 all used parameter values and their corresponding units are given.

Table A1: Reactor and catalyst parameters.

D L Kr e DP

0.1 m 0.5 m 1082 kg/m 56.7% 6.9 mm

Table A2: Gas properties and further parameters.

cP h M i Gasr RxHD KP

DIPPR 107 [22],

ideal mixing

2Nh [22] [22] Ideal Gas Heat of

formations [22], cP

See Eq.(A8) [23]

2

1

P atm11.309-TK 11852

expK-

= (A8)

Appendix B: Illustrative Example and Validation of the Hybrid Optimization Algorithm

The proposed hybrid algorithm is tested here based on an objective function, f, which features several

local minima. The function in Eq. (B1) with a global optimum at X = [6.8, 8]T is shown on the left of

Fig. B1.

( ) ( )( )21222211 XXXXcosXXXsinf +++-= (B1)

On the right of Fig. B1, both the starting positions of the particles (squares) and the detected optima

(points) are denoted. Here, 5 particles and 5 iterations are used. It can be shown that although the start

positions are distant from the global optimum (squares on the bottom of Fig. B1), it can be found

anyway by the new framework. In this example, 10 local minima were detected. It is quite evident that

with this procedure it can not be assured that the global optimum of the objective function is found.

However, it helps the gradient based algorithm not to get stuck in a bad, i.e. high value, local minimum.

24

Figure B1: Test function for the hybrid optimization algorithm (left) and particle positions (right).

For the sake of comparison, in case of nonlinear experimental design problems, optimal experiments are

calculated with a random seed of 25 initial points taking over always the best optimum. The

development of the A-Criterion for two approaches is shown in Fig. B2. Occasionally, the same optimal

experiments are found and the final difference in the A-Criterion is only 3 10-5. Anyhow, this leads to a

mean increase in the parameter standard deviation of 0.2 % and another experiment would be necessary

to reach the desired accuracy. However, the authors are aware of that this is not a final proof of the

advantages of the proposed approach. It has been used straightforward in this work so as to get improved

results. A detailed examination of the approach on a set of selected optimization problems is under way.

25

Figure B2: Comparison of the hybrid approach with a random seed of initial values.

Appendix C: Analytic Calculation of the 2D Confidence Ellipsoid

The calculation of the 2D confidence ellipsoid is based on the probability density function for a

multivariable normal distribution, Eq. (C1).

The confidence region G can be calculated with the integral in Eq. (C2), assuming an ellipsoidal form,

which leads then to the inequality in Eq. (C3).

( ) ( )( )

-

p=a

--

2exp

2

det

2

)dim(

1 pCpCp

1T

p (C1)

( )a

-

p

--

21G

1

dpdp2

exp2

det pCpC 1T (C2)

a

--

-

2exp1

pCp 1T (C3)

26

The ellipsoid now is positioned on the bound of the confidence region and can be calculated as the

solution of the Eq. (C3). Due to the quadratic nature of the ellipsoid, Eq. (C3) has two solutions for p1.

They are given in Eq. (C4).

22

122

221

212 c

)1ln(c

c

cc

pcc

ppa-

-

--= , with

=-

2

11

cc

ccC (C4)

In order to exclude complex solutions of Eq. (C4 ), p1 has to be bounded. The allowed interval is given

in Eq. (C5). Following this, the ellipsoidal can be plotted (see Fig. 13).

-

a-+

-

a--

212

2

212

21 ccc

c)1ln(;

ccc

c)1ln(p (C5)

List of Notation

Process related variables

Description Unit

bi Exponent in the reaction rate -

cP,i Component heat capacity J/(mol K)

D Reactor diameter m

DP Particle diameter m

dP Pressure drop Pa

dz Differential length m

f i Exponent in the reaction rate -

F Total flow rate mol/s

Fi Component flow rate mol/s

F0i Initial comp. flow rate mol/s

DHRx Heat of reaction J/mol

k, k+, k - Velocity constants variable

k1, k2, k3 Constants in the extended Arrhenius equation variable

KP Equilibrium constant 1/sqrt(bar)

L Reactor length m

M i Components molar mass kg/mol

P Pressure bar

Pi Partial pressure bar

27

Reaction rate mol/(kg s)

Re Reynolds number -

T Temperature K or C

xi Component mol fraction -

X Conversion of SO2 -

z Length coordinate m

Void fraction -

Dynamic viscosity Pa s

i Initial comp. Ratio -

i Comp. stoichiometric coefficient -

Gas Gas phase density kg/m

K Catalyst density kg/m

Mathematical symbols

Description

A Collocation matrix

a, b Bounds

C Covariance matrix (C.m.)

CM C.m. of the measurement errors

c Element of the inverted C.m.

f Function value

F (Discretized) Differential equations

G Algebraic equations

JP, Jx Jacobians

NC Number of collocation points

NI Number of intervals

NP Number of parameters to be estimated

P Parameter vector

p, p1, p2 Model parameters

q Control

w Weight

x State variable

xS Discrete state variable

zNC Position of the last collocation point

Standard deviation

28

Function of the covariance matrix.

Constraint function

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