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Chemical Engineering Science 66 (2011) 6453–6469

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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces

Analysis and optimal design of an ethylene oxide reactor ¨ Freund a,n, Kai Sundmacher a,b Andreas Peschel a, Florian Karst a, Hannsjorg a b

Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany Process Systems Engineering, Otto-von-Guericke University Magdeburg, Universit¨ atsplatz 2, 39106 Magdeburg, Germany

a r t i c l e i n f o

abstract

Article history: Received 16 May 2011 Received in revised form 30 August 2011 Accepted 31 August 2011 Available online 12 September 2011

In this work, a recently proposed multi-level reactor design methodology (Peschel et al., 2010) is extended and applied for the optimal design of an ethylene oxide reactor. In a first step, the optimal reaction route is calculated taking various process intensification concepts into account. The potential of each reaction concept can be efficiently quantified, which is the economic basis for the design of advanced reactors. Based on these results, a promising concept is further investigated and a technical reactor is designed. As an extension to the design method, reactor design criteria for external and internal heat and mass transfer limitations are directly included in the optimization approach in order to design the catalyst packing. The derived reactor concept is investigated with a detailed 2D reactor model accounting for radial concentration and temperature gradients in addition to a radial velocity profile. The example considered in this work is the production of ethylene oxide which is one of the most important bulk chemicals. Due to the high ethylene costs, the selectivity is the main factor for the economics of the process. A membrane reactor with an advanced cooling strategy is proposed as best technical reactor. With this reactor design it is possible to increase the selectivity of the ethylene epoxidation by approximately 3% compared to an optimized reference case. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Chemical reactors Optimization Design Process intensification Elementary process functions Ethylene oxide

1. Ethylene oxide: industrial process and intensification aspects 1.1. Industrial production Since ethylene oxide (EO) is a bulk scale product with an annual production of about 19 million tons in 2010, cost efficiency in the production is very important. Even a selectivity increase in the range of 1% has a large effect on the economy of the process since ethylene makes up for approximately 68% of the total cost of the EO process (Baerns et al., 2006). The reaction network of the silver catalyzed ethylene oxide production is shown in Fig. 1. It consists of the desired partial oxidation of ethylene (E) to ethylene oxide (reaction 1), the total oxidation of ethylene to carbon dioxide (reaction 2), and the consecutive oxidation of EO to carbon dioxide (reaction 3). In general, ethylene is oxidized over a promoted Ag-catalyst either using air (air based process) or pure oxygen (oxygen based process). In addition to the different process technologies, different types of Ag-catalysts can be used. In both processes, tube bundle reactors with a constant cooling temperature are commonly used.

n

Corresponding author. Tel.: þ49 391 6110275; fax: þ49 391 6110634. E-mail address: [email protected] (H. Freund).

0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.08.054

Due to explosion hazards, the E, O2 and EO concentrations as well as the conversion are limited. In addition, the used reactor types cannot provide optimal component concentration and temperature profiles along the reactor length. This may give rise to problems with an axial hot spot as well as ethylene and oxygen depletion along the channel, which results in a lower reactor productivity. For a general process description and more details, refer to the literature (e.g. Rebsdat and Mayer, 2007). 1.2. Process intensification aspects for ethylene oxide reactors In order to increase the EO production selectivity, many reactor types such as membrane reactors or micro-reactors have been investigated. The influence of different dosing options (ethylene or oxygen) for the air based process was experimentally studied by Lafarga and Varma (2000) using a fixed bed membrane reactor. They compared the performance to a classical fixed bed reactor and their results indicate that concentration manipulation along the reaction channel can increase the selectivity. In order to investigate the potential of micro-reactors for the EO process, Kestenbaum et al. (2002) constructed several microreactors and compared their performance to industrial EO reactors. While the space time yield (STY) and the conversion were in the same range as the conventional reactors, the selectivity was

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much lower compared to the industrial process. An important aspect when using micro-reactors is that the reaction mixture can lie within the explosive range, which may allow for higher ethylene and oxygen inlet and higher EO outlet concentrations. Increasing the EO outlet concentration can improve the productivity of the plant. However, in the connecting pipes the composition will also be in the explosive range and hence such a reactor will require additional safety measures in the plant. In addition, even in a micro-reactor the maximum safe diameter might be exceeded as shown by Fischer et al. (2009). Therefore, the gas mixture must never be within the explosive regime for the EO process. Besides these approaches to design new micro- or membrane reactors, also other reaction routes to EO were investigated. ¨ Berndt and Brasel (2009) worked on a completely new reaction route, where the epoxidation of olefins is performed using ozone in the gas phase. Lee et al. (2010) presented a concept for the EO production based on gas-expanded liquids similar to the HPPO process for propylene. For both reaction concepts, the process efficiency must account for the production of ozone or hydrogen peroxide, respectively, and hence these concepts are not directly comparable to the established process. To sum up, selectivity optimization for the EO reactor is of major industrial importance. Different trends for temperature, pressure, and component concentrations on the selectivity can be observed depending on the used catalyst. As shown by various

Ethylene Oxide Partial Oxidation (r1)

(CH2)2O

ΔHr1 = -106.7 kJ/mol

Ethylene

+ 2.5·O2

Consecutive Oxidation (r3)

2. Design of an optimal ethylene oxide reactor

ΔHr3 = -1216.3 kJ/mol

C 2H 4

Total Oxidation (r2) ΔHr2 = -1323 kJ/mol

2·CO2 + 2·H2O

Fig. 1. Simplified macroscopic reaction scheme.

Level 2

Level 3

Detailed Design

The state-of-the-art reactor design methods can be classified into heuristics (e.g. Hanratty and Joseph, 1992; Schembecker et al., 1995; Jacobs and Jansweijer, 2000), attainable region methods (e.g. Glasser et al., 1987; Hildebrandt and Glasser, 1990; Feinberg and Hildebrandt, 1997), and rigorous optimization approaches such as superstructure optimization (e.g. Balakrishna and Biegler, 1992; Kokossis and Floudas, 1994; Lakshmanan and

Model

Decisions

Level 1

authors, process intensification concepts such as membrane reactors or micro-reactors seem to be an interesting option for the production of EO. However, no study was performed which considers all PI measures in a systematic manner. Hence, it can be concluded that the question on how an optimal EO reactor has to be designed and what would be the benefit compared to the standard design has not yet been answered satisfactorily. Therefore, this contribution is intended to design an improved EO reactor, which performs optimal from both the reaction concept and the technical point of view. The selectivity of the reactor is maximized taking component dosing and removal concepts, advanced temperature control, the dimensioning of specific exchange areas for heat and mass transport, the choice of catalyst, and the choice of the catalyst packing into account. Since most of the reaction kinetics available from the literature are only valid for the air based process, this study focuses on the air based EO process. This work extends successfully our recently published reactor design methodology (Peschel et al., 2010) to a selectivity problem taking more advanced process intensification methods into account and providing more insight into the modeling based on elementary process function concept. In addition, design criteria for external and internal mass and heat transport are considered directly in the optimization based design framework. These criteria—to the best of our knowledge—are not yet rigorously considered in any published reactor design method. Furthermore, the derived reactor design is validated by optimizing a 2D reactor model. Such an optimization can also not be found in the literature and can be considered as proof of principle. The followed design approach considers non-idealities of the technical approximation in order to derive best possible technical reactors and not only design reactors based on simplified models.

Balance equations Reaction kinetics Thermodynamics Intrinsic bounds

Results

• Optimal route • Potential of intensification concept

• Integration & enhancement concept

• • • •

• Schematic reactor concept, catalyst support • Transport mechanisms • Control variables

• Mass & energy transport • Exchange areas • Reaction eng. bounds

• Best reactor concept • Best control variables • Losses due to limited mass & energy transport

• Approximation concept

• Balance equations for all supporting phases

• Best technical approximation • Losses due to non-ideal control variable profiles

• Type of model for detailed investigation and design

• 2D or 3D model • pseudo-homogeneous, heterogeneous

• Losses due to non-ideal flow field, radial gradients, diffusion effects

Fig. 2. Generalized decision structure for the development of an optimal reactor.

A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

reaction system and determines how the selectivity changes with temperature, pressure, and concentration of reactants and reaction products. From the many investigated catalysts for the EO process, the kinetics published by Petrov et al. (1986), Stoukides and Pavlou (1986), Al-Saleh et al. (1988), Borman and Westerterp (1995), Schouten et al. (1996), and Lafarga et al. (2000) are compared in order to identify the best catalyst from this selection. Referring to Fig. 3, the selectivities of the investigated systems are quite different and also the reaction rates (only r1 shown) vary strongly. The comparison is performed at typical inlet conditions for the component partial pressures (pE ¼ pO2 ¼ pCO2 ¼ 1:5 bar, pEO ¼ pH2 O ¼ 0 bar) and a wide temperature range. The qualitative trends and the differences between the various reaction rates are the same for other operating conditions. As can be observed in Fig. 3 the catalyst investigated by Al-Saleh et al. (1988) exhibits by far the highest differential selectivity and hence this catalyst system with the according reaction kinetics is chosen for the further investigations.

Biegler, 1996), the dynamic optimization approach (e.g. Horn and ¨ Kuchler, 1959; Bilous and Amundson, 1956; Aris, 1960), or the systematic staging of reactors (Hillestad, 2010). However, all of these methods do not directly allow the design of advanced and innovative reactors. In addition to the classical reactor design approaches, process intensification options (Freund and Sundmacher, 2008) become more and more important for the design of tailor-made reactors, which are superior to standard reactors for the specific task. Here, a three step approach (Peschel et al., 2010) is used to develop a reactor design which ensures the optimal process conditions along the reaction coordinate. The general decision structure is shown in Fig. 2. On the first level, the optimal route is calculated considering the equations of change, reaction kinetics, thermodynamic limitations, and system inherent boundaries. This optimal route is obtained by balancing a fluid element and manipulate the fluid element along the reaction coordinate such that the reaction conditions are optimal all along the reaction coordinate. Here, different integration and enhancement options are compared with each other and the potential of every option is quantified by the comparison with an optimized technical reference case. On the second level, a schematic reactor set-up is chosen based on the result of the first level. The kinetic expressions for mass and energy transport are added and the influence of limited mass and energy transport is quantified. By choosing the schematic reactor set-up, bounds for the specific exchange areas and the catalyst density are defined and correlations for heat and mass transfer from the literature can be used. On the third level, the best possible technical reactor is derived by approximation of the control variable profiles and the design is validated taking non-idealities into account. Summarizing the applied reactor design method, all possible intensification options are screened with relatively simple models on the first level and only the most promising reactor set-ups are investigated in detail. This approach enables the model based development of an optimal technical reactor on the basis of the optimal route in state space taking a wide range of physical and chemical phenomena into account. The optimal reactor is designed following a structured, knowledge- and optimization based approach, which yields the best reactor considering the objective function, reaction engineering criteria, and decisions taken by the engineer.

2.2. Level 1: optimal route in state space On the first level, the potential of different integration and enhancement concepts is investigated. For this purpose, a reference case must be specified, which is a tube bundle reactor with constant cooling temperature as used in the industrial process. For the investigated catalyst, the temperature profile in the reactor for a constant cooling temperature is nearly uniform since the selectivity is high and the reaction rates are relatively low. Hence, the temperature and pressure for a fluid element with fixed inlet composition (xE ¼ xO2 ¼ xCO2 ¼ 0:075, xEO ¼ xH2 O ¼ 0) are directly optimized. The inlet mole fractions are the mean values of the industrial process range (Rebsdat and Mayer, 2007). For this study, we chose a STY of 0.27 mol/(m3 s) within a residence time of 30 s for all investigated reaction concepts. Specifying the STY and the residence time is better suited than comparing different integration concepts for a fixed conversion since the amount of reactants can vary depending on the initial composition and on the dosing of reactants. The optimal operation parameters for the reference reactor are T¼625.7 K and p¼20 bar. With this reactor a selectivity of up to 79.10% can be achieved. In Section 2.2.1 each investigated integration and enhancement concept is explained. Each case is a dynamic optimization problem, where the optimal route in state space with respect to the integrated and enhanced fluxes is determined in order to maximize the selectivity. The required model equations are presented in Section 2.2.2. Bringing the model equations and the investigated concepts together, the arising optimization problems are stated in

2.1. Catalyst selection Before the best reaction concept for the air based EO process can be derived, a suitable catalyst must be chosen. The choice of the catalyst has a major influence on the performance of the

0.9 Reaction rate [mol/kg/s]

10−1

Selectivity [−]

0.8 0.7 0.6 0.5 0.4 0.3 550

6455

560

570 580 590 Temperature [K]

600

10−2 10−3 10−4 10−5 10−6 10−7 550

560

570 580 590 Temperature [K]

600

Fig. 3. Catalyst comparison (Al-Saleh et al., 1988, ; Stoukides and Pavlou, 1986, ; Borman and Westerterp, 1995 (model 2 including long time deactivation factors), —; Lafarga et al., 2000,    ; Schouten et al., 1996 (model 3, tubular reactor), – –; Petrov et al., 1986, -  -). (a) Differential selectivity and (b) reaction rate r1.

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Section 2.2.3 before the results are presented and discussed in Section 2.2.4. As discussed in Section 3, the numerical solution approach can only yield locally optimal solutions and hence every solution must be taken as locally optimal.

2.2.1. Investigated integration and enhancement concepts For the EO production process, controlling the heat flux and applying dosing and removal strategies are of great interest to increase the EO selectivity. The different integration and enhancement concepts which are investigated are shown in Fig. 4. Here, all investigated cases from level 1 to level 3 are shown. In all cases the inlet composition and the system pressure are degrees of freedom for the optimization. We refer to all investigated cases as intensified air based processes even if ethylene or oxygen is dosed individually to the reaction mixture since the concentration range, the inert gas, the conversion, and the space time yield are chosen in the range of the industrial air based process. Considering assist oxygen (enrichment in oxygen) is an important aspect in the industrial application of process intensification. For safety reasons, the ethylene and oxygen concentration must not exceed the explosion limit. Since a mathematical formulation of the explosive range at elevated pressure and temperature for the multi-component mixture in the EO reactor is not available in the open literature, it is assumed that the mole fractions must always stay in the industrial range of the inlet conditions given in Table 2 in order to stay out of the explosive range. The concentration limit of each component is considered as an intrinsic bound and hence considered directly on level 1 (and all subsequent levels) in our approach. Case 1: Optimized inlet composition, isothermal. This case is similar to the reference case except that the inlet composition is optimized. This does not affect the reactor itself, but the technical process must be adjusted to meet the optimal inlet conditions and hence this case is regarded as an intensified concept. Case 2: Inlet composition and heat flux optimized (q). In addition to the optimization parameters of case 1, the heat flux is adjusted along the reaction coordinate in order to obtain an optimal temperature profile. This case yields the potential of an advanced heat flux profile. Case 3: Inlet composition, heat and ethylene flux optimized ðq, jE Þ. Besides the optimal heat flux profile, an optimal ethylene dosing profile is calculated. This case quantifies the potential of an advanced heat flux profile combined with an ethylene dosing strategy.

Case 4: Inlet composition, heat and oxygen flux optimized ðq, jO2 Þ. This case is similar to case 3, but here an optimal oxygen dosing strategy instead of an ethylene dosing strategy is applied. Case 5: Inlet composition, heat and CO2 flux optimized (q, jCO2 Þ. Here, an optimal removal strategy for CO2 is calculated combined with an advanced heat flux profile. Although CO2 in principle reduces both reaction rates owing to the adsorption term, the CO2 partial pressure has a large influence on the selectivity in case the STY is fixed. Due to the STY constraint the reaction temperature must be increased in case of higher CO2 partial pressure. This reduces the selectivity due to the higher activation energy of r2 compared to r1. Other removal strategies will not be investigated in this contribution, even if the in situ removal of EO is of great interest, since the consecutive oxidation of EO and the downstream absorption can be avoided. The influence of the EO removal on the selectivity is not investigated in the chosen reaction rate laws (Al-Saleh et al., 1988) since EO does not affect r1 and r2, and r3 was neither observed nor determined for this catalyst. The removal of CO2 can be considered as a general show case to exemplify the proposed methodology for product removal. Case 6: Inlet composition, heat, ethylene and oxygen flux optimized ðq, jE , jO2 Þ. This case investigates the potential of an optimal ethylene and oxygen dosing strategy combined with an optimal heat flux profile. Case 7: Inlet composition, heat, ethylene, oxygen and CO2 flux optimized ðq, jE , jO2 , jCO2 Þ. Heat, ethylene, oxygen, and CO2 flux are optimized in addition to the inlet composition for this case. This concept yields an upper bound for all intensified reaction concepts since all factors influencing the reaction rates, namely the mole fractions of E, O2, CO2, the pressure, and the temperature, are optimally manipulated. All degrees of freedom of the cases 1-6 are included in this case. Nevertheless, the cases 1-6 are of great interest since it may be sufficient to influence only a certain set of the mole fractions in addition to the temperature. Hence, the potential of each set must be known in order to decide which concept should be further investigated. In addition, the results of this case answer the question whether a reactor with CSTR characteristics can approximate the optimal profiles. In principle, optimal pressure profiles and apparent catalyst density profiles—which can be obtained by catalyst dilution or by realizing different void fractions using different catalyst packings—are of interest for the optimal reactor design. In case of EO, the optimal pressure is always at the upper boundary since a high pressure helps to fulfill the required STY and increases the

Objective: Selectivity

Level 1

Heat and reaction flux integrated Case1: isothermal

Case 2: q

Heat, reaction and component fluxes integrated

Case 3: q, jE

Level 2

Tc, Per

Level 3

Techn. Appr.

Case 4: q, jO2

Case 5: q, jCO2

Case 6: q, jE , jO2

Detailed Model Fig. 4. Decision structure for the development of an optimal EO reactor.

Case 7: q, jE , jO2, jCO2

A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

selectivity. In addition, the apparent catalyst density is also always at its upper limit specified by the lower limit for the void fraction since the heat flux can be ideally controlled, and hence a high catalyst density helps to fulfill the required STY. 2.2.2. Model equations of level 1 The component mole balance for every component is given by the following equation: NR X dni 1e ¼ si  ji þ rp   Vgas  ni,j  rj dt e j¼1

ð2:1Þ

The case selection variables si are used in order to obtain a comprehensive and clear representation in the balance equation. The value of si depends on the investigated case explained in Section 2.2.1, and it can either be zero if the flux is not considered or one if the flux is considered as optimization function. The void fraction depends on the catalyst packing and may vary in a wide range depending on the type and shape of the catalyst support. On the first level a constant void fraction is assumed ðe ¼ 0:4Þ. The chosen value represents a typical technical void fraction in case of a randomly packed bed with uniform spheres. The reaction rates along with the reaction rate constants and adsorption constants are given by Eqs. (2.2)–(2.4) and the parameters are summarized in Table 1. rj ¼

n n kj pEE,j pOO2,j 2

1 þKj pCO2

,

j ¼ 1; 2

Table 2 Limits for the mole fractions used for optimization, typical inlet conditions of the air based process (Rebsdat and Mayer, 2007). Component

xL

xU

Typical inlet conditions (% v/v)

E O2 EO CO2 H2O N2

10  5 10  5 0 0.05 0 0

0.1 0.08 0.1 0.1 0.1 1

2–10 4–8 0 5–10 0 72–89

Table 3 Attributes and selectivity of each reaction concept investigated on level 1. Case

Attributes (DoF)

si

Selectivity (%)

Reference Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

T, p T, p, xi,0 T(t), p, xi,0 T(t), p, xi,0, jE(t) T(t), p, xi,0, jO2 ðtÞ T(t), p, xi,0, jCO2 ðtÞ T(t), p, xi,0, jE ðtÞ, jO2 ðtÞ T(t), p, xi,0, jE ðtÞ, jO2 ðtÞ, jCO2 ðtÞ

si ¼ 0 8i A COM si ¼ 0 8i A COM si ¼ 0 8i A COM si ¼ 0 8i A COM\fEg, sE ¼ 1 si ¼ 0 8i A COM\fO2 g, sO2 ¼ 1 si ¼ 0 8i A COM\fCO2 g, sCO2 ¼ 1 si ¼ 0 8i A COM\fE, O2 g, sE ¼ sO2 ¼ 1 si ¼ 0 8i A COM\fE, O2 , CO2 g, sE ¼ sO2 ¼ sCO2 ¼ 1

79.10 81.35 81.48 82.35 81.73 81.99 82.71

Case 7

j ¼ 1; 2

ð2:3Þ

xi ¼

  Tads,j , T

j ¼ 1; 2

ð2:4Þ

Vgas ¼

The energy balance is written in terms of temperature and is simplified by assuming no technical work, negligible influence of the pressure change and of the dosed components on the temperature change (refer to Eq. (2.5)). The heat flux into and out of the fluid element qtot is the total heat flux. While the second term on the right hand side is bounded, qtot is assumed to be an unrestricted control function. Hence, every temperature profile can be obtained on level 1 and the temperature profile can directly be taken as an optimization function on this level. 0 0 11 NCOM NR X X n  cp dT 1 e @hi ¼ @qtot þ rp  ð2:5Þ   ni,j  rj AA Vgas dt e i¼1 j¼1 The gas volume is described by the ideal gas law. This yields the total gas volume and the components partial pressures using the definition for the total amount of substance and the component mole fractions. n¼

NCOM X

82.84

ð2:2Þ

  EA,j , kj ¼ k0,j exp  RT Kj ¼ K0,j exp

6457

ni

ð2:6Þ

ni n

Parameter

Reaction 1

Reaction 2

k0,j

6:275  106 mol=ðkgp s Pa1:1 Þ

1:206  107 mol=ðkgp s PaÞ

EA,j nE,j nO2 ,j K0,j Tads,j

74:9 kJ=mol 0.6 0.5 1.985  102 Pa  1 2400 K

89:8 kJ=mol 0.5 0.5 1.08  102 Pa  1 1530 K

nRT p

pi ¼ xi  p

ð2:8Þ ð2:9Þ

The initial conditions are chosen to be in the typical range for the air based ethylene oxide process according to Table 2. In addition to E, O2, CO2, and N2, some small amounts of EO, H2O and CH4 are present. The upper and lower mole fraction of each component used for the optimization are also given in Table 2. The inlet mole fractions of E, O2, and CO2, as well as the inlet pressure, and the inlet temperature are optimization variables in all calculations. Due to numerical reasons during the optimization calculations a lower limit on the ethylene and oxygen mole fraction has to be assigned. The inlet mole fractions of EO and H2O are set to zero and the inlet mole fraction of N2 is calculated from the summation constraint. The total pressure is in the range of 10–20 bar and the temperature range is 550–630 K. The space time yield is related to the catalytic channel and is given by the following equation: nEO,f nEO,0 STY ¼ R ¼ 0:27 mol=ðm3 sÞ tf Vgas dt t0

ð2:10Þ

e

i¼1

Table 1 Model parameters for the reaction rates (Al-Saleh et al., 1988).

ð2:7Þ

A general formulation of the selectivity considering ethylene dosing is used as objective function: S¼ 

nEO,f nEO,0  Rt nE,0 þ t0f jE dt nE,f

ð2:11Þ

2.2.3. Optimization problems of level 1 The full optimization problems which must be solved on level 1 are stated in (OP1). In order to compare the case stated above in a comprehensive manner, the optimization functions, the optimization

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A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

Initial conditions: xi ðt ¼ 0Þ ¼ xi,0

parameters (together referred to as degree of freedom (DoF)), and the integrated component fluxes are summarized in Table 3.

Residence time: t ¼ 30 s

Obj ¼ max S

STY : Eq: ð2:10Þ

ðOP1Þ

DoF

Intrinsic bounds: T L r T r T U , pL r p r pU , xLi r xi r xU i

s:t: Component balances: Eq: ð2:1Þ

Case selection: si

The bounds for temperature, pressure, and STY are stated in the model description. The bounds for the composition are given in Table 2 and the parameters of the reaction rates in Table 1.

Reaction kinetics: Eqs: ð2:2Þ2ð2:4Þ Constitutive equations: Eqs: ð2:6Þ2ð2:9Þ

620

0.08

2

600

case 6 (q, jE, jO ) 2

xi [−]

590

0.06 0.04

580

0.02

570 560

0 20

30

0

10

0.1

2.4

0.08

0.08

2.32

0.06

0.06

2.24

0.04

0.04

2.16

0.02

0.02

2.08 2

0 0

10

20

30

0

10

t [s]

20

30

t [s] 2.3

0.1

2

0.08

2.2

0.08

−3

0.06

2.1

0.06

−8

0.04

2

0.04

−13

0.02

1.9

0.02

−18

1.8 30

0

0 0

10

20

xi [−]

0.1

jO2/nin 10−3 [1/s]

xi [−]

30

0.1

0

−23 0

10

t [s]

20

30

t [s] 12

0.1

2.5

0.08

10

0.08

1.8

0.06

8

0.06

1.1

0.04

6

0.04

0.4

0.02

4

0.02

−0.3

2

0

0 0

10

20 t [s]

30

xi [−]

0.1

ji/nin 10−3 [1/s]

xi [−]

20 t [s]

xi [−]

xi [−]

t [s]

jE/nin 10−3 [1/s]

10

jCO2/nin 10−3 [1/s]

0

ji/nin 10−3 [1/s]

T [K]

0.1

case 2 (q) case 3 (q, jE) case 5 (q, jCO )

610

8i A COM

according to Table 3

−1 0

10

20

30

t [s]

Fig. 5. (a) Temperature profiles. (b)–(h) Mole fractions and dosing profiles (—, E; – –, O2;    , EO; -  -, CO2; thin lines, mole fractions; thick lines, dosing profiles). (a) Temperature profiles, (b) Case 1: isothermal. S¼ 81.35%, (c) Case 2: q. S¼ 81.48%, (d) Case 3: q, jE. S ¼82.35%, (e) Case 4: q, jO2. S¼ 81.73%, (f) Case 5: q, jCO2. S ¼81.99%, (g) Case 6: q, jE, jO2. S¼ 82.71%, and (h) Case 7: q, jE, jO2, jCO2. S ¼82.84%.

A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

2.2.4. Results The results for all investigated reaction concepts of level 1 are presented in Fig. 5 and the selectivities are summarized in Table 3. The optimal temperature of the isothermal case with optimized inlet composition (case 1) is 592.7 K. In this case, the ethylene and oxygen inlet mole fractions are at their upper bound U (xE,0 ¼ xU E ¼ 0:1, xO2 ,0 ¼ xO2 ¼ 0:08) while the CO2 inlet mole fraction is at its lower bound (xCO2 ,0 ¼ xLCO2 ¼ 0:05). The high inlet fractions of ethylene and oxygen in combination with the low inlet fraction of CO2 enable a lower temperature level compared to the reference case. This reduced temperature level and the direct influence of the higher ethylene mole fraction increase the selectivity by 2.25% compared to the reference case. In Fig. 5(a) the temperature profiles of selected cases where the heat flux is optimized are shown. For the sake of clarity, the temperature profile of case 4 ðq, jO2 Þ is not shown in Fig. 5(a) since it is almost the same as in case 3 ðq, jE Þ. In case 7 ðq, jE , jO2 , jCO2 Þ the optimal temperature profile is constant at 569.5 K and it is also not shown for clarity. From case 2–7, the mole fraction profiles and the flux profiles of the manipulated components are presented in Fig. 5(c)–(h). In addition, the selectivity of each case is given in the caption. Referring to case 2, it can be concluded that an optimal temperature profile exists for the conventional EO reactor, which is not constant. The optimal temperature profile increases continuously starting from 570.1 K to 616.7 K. This temperature rise counterbalances the lower reaction rates due the decreasing ethylene and oxygen partial pressure and the increasing CO2 partial pressure with increasing residence time. However, the selectivity increases only by 0.13% comparing case 1 with case 2. Hence, it can be concluded that realizing a temperature profile in the reactor is probably not worthwhile if additional investment costs arise. The optimal ethylene dosing profile with the resulting component mole fraction profiles of case 3 are shown in Fig. 5(d). The ethylene dosage makes up for the consumed ethylene so that the ethylene mole fraction is always at its upper bound. The temperature profile is similar to case 2 except that the temperature level is lower. Besides the direct influence of ethylene on the selectivity (compare reaction order of ethylene in r1 and r2), the lower temperature level contributes to the selectivity increase of 3.25% compared to the reference case. Referring to Fig. 5(e) belonging to case 4, the optimal oxygen dosing strategy keeps oxygen at its upper boundary. The optimal temperature profile is almost the same as in case 3. The selectivity improvement is lower than in the ethylene dosing case. Hence, dosing ethylene is more worthwhile to investigate than dosing oxygen. In case the produced CO2 can be removed from the reaction mixture in situ (case 5), the selectivity can be increased by 2.89%. Due to the inhibiting effect of CO2 on the catalyst, a lower CO2 partial pressure allows a temperature reduction while still matching the required STY. As shown in Fig. 5(f), the optimal CO2 removal strategy starts with a high CO2 inlet mole fraction and continuously removes CO2 from the reaction mixture until the lower CO2 limit is reached. Afterwards, CO2 is kept at its lower limit. This strategy is advantageous over starting with a CO2 inlet mole fraction at its lower limit and keeping the CO2 mole fraction constant at its lower limit since the former strategy keeps the mole fractions of ethylene and oxygen high up to a residence time of 3 s. This effect increases the selectivity more than always staying at the lower limit for the CO2 mole fraction, which would yield a selectivity 0.10% lower. Such a result can hardly be obtained by intuition or heuristics, which demonstrates that our model based approach is advantageous for the design of optimal reactors.

6459

If optimal ethylene, oxygen, and heat fluxes are provided (case 6), the ethylene mole fraction is always kept at its upper boundary while the oxygen mole fraction is increased from xO2 ðt0 Þ ¼ 0:05 to its upper boundary in the first 4 s (refer to Fig. 5(g)). Such a strategy keeps the CO2 mole fraction low at the beginning and is advantageous over a strategy where both the ethylene mole fraction and the oxygen mole fraction are constant at their upper boundaries. In case 7 ðq, jE , jO2 , jCO2 Þ the ethylene and oxygen mole fractions are always kept at their upper bound, while the CO2 mole fraction is kept at its lower bound (refer to Fig. 5(h)). Since these mole fractions are all constant, the optimal temperature is also constant (T¼596.5 K). In case the profiles for the temperature and all influenced mole fractions are constant, these profiles could be approximated by a completely back-mixed reactor. However, the reactant mole fraction profiles must not be at their upper boundaries for this case since the ethylene and oxygen concentration in the feed are even higher, which is not allowed due to the explosion hazard. In addition, the CO2 mole fraction profile should not be at its lower boundary since the feed would have to contain even less CO2 in this case and that contradicts the assumed concentration bounds. The selectivity gain compared to the reference case is 3.74%. Case 7 has the highest potential for the selectivity increase, but it is also most complicated to realize. The removal of CO2 could in principle be realized by absorption with an amine solution or a potassium carbonate solution. However, these solvents are not in liquid state at reaction conditions, which makes an in situ absorption impossible with these solvents. Hence, either intermediate absorption must be applied or other solvents (such as ionic liquids) allowing an in situ absorption at the reaction conditions must be used. A detailed investigation which solute may be used at the reaction temperature and designing a reactor with integrated extraction of CO2 exceeds the scope of this paper and hence all cases including CO2 removal are not further investigated in this contribution. Nevertheless, the methodology identifies the upper limit for the selectivity which can be obtained by such a reaction concept. From the results of level 1 it can be observed that the exact knowledge of the explosive range is very important for the optimal operation of the reactor. The optimal operation conditions are often at the upper boundaries for the ethylene and oxygen mole fractions defined by the explosion limit and this indicates that higher ethylene and oxygen mole fractions will further increase the overall EO selectivity. In addition, decreasing the CO2 mole fraction will increase the selectivity for the same STY and residence time. However, the rate law is not applicable at CO2 levels below the chosen limit and hence the lower bound on the CO2 mole fraction is necessary to obtain reliable results. Summing up, it can be concluded that applying advanced concentration and temperature control strategies has high potential for improving the EO selectivity. Hence, it is interesting to investigate if this potential can also be exploited in case the optimal flux profiles are approximated in a technical reactor. Taking the explosion limits into account, ethylene and oxygen must be dosed using separate channels. Hence, the dosing of only one of the components is much easier. In this contribution, case 3 ðq, jE Þ will be further investigated since it seems to be the most promising case from a selectivity and a reaction engineering point of view. 2.3. Level 2: reactor concept, transport mechanisms, and control variables Based on the case studies of level 1, only the most promising cases are further investigated. It is determined which transport mechanisms and control variables are suited to make the desired flux profiles attainable. For this purpose, a schematic reactor

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A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

Coolant

Ro

Reaction Channel

Ri

Membrane Ethylene supply

addition, a maximal temperature difference between the cooling media and the fluid and a maximum pressure difference between the ethylene dosing side and the fluid are enforced. These criteria limit the mechanical and thermal stress on the construction material and hence are important for safety reasons. The model equations are presented in Section 2.3.1 and the arising optimization problem is stated in Section 2.3.2.

z Fig. 6. Reactor design for case 3 (q, jE) (longitudinal section view).

design is proposed and control variables which can be changed by the reactor design are identified. The losses due to the limited mass and energy transport are quantified by comparing the results of level 1 and level 2. In principle, several mechanisms for the realization of heat and component fluxes exist. Here, for the sake of simplicity the heat flux is controlled by changing the cooling temperature along the reaction coordinate and the distributed dosing of ethylene is provided via a membrane. At constant pressure on the ethylene supply side the permeance of the membrane is adjusted along the reaction coordinate in order to control the ethylene flux in an optimal manner. Alternatively, the exchange areas for both fluxes or the pressure on the ethylene supply side could be controlled. It should be noted that the cooling temperature and the permeance of the membrane can still be ideally manipulated within the specified bounds on the second level. The proposed schematic reactor design is shown in Fig. 6. At this point, the choice of the schematic reactor set-up includes the choice of the catalyst support geometry. Here, a fixed bed reactor with a randomly packed bed is chosen since it offers the highest catalyst density compared to other catalyst support concepts. A high catalyst density is required in order to meet the required STY—a result which can already be obtained by a sensitivity analysis on the first level. The design is optimized in a wide design range for all optimization variables, such as the inlet temperatures and exchanges areas. Meaningful bounds for the inner and outer tube diameter (here: 0:5 cm r Di r3 cm, 4 cm r Do r 10 cm) yield the bounds for the specific exchange areas for heat and mass transfer. In case of a fixed bed reactor with a conventional catalyst packing, an average void fraction dependent on the inner and outer tube diameter and on the catalyst particle diameter (here: 2 mmrDp r 10 mm) must be specified. If additional catalyst supports and reactor concepts, for example coated wall catalyst and micro-reactors, should be investigated, the void fraction correlation and the bounds for the geometric design variables must be adjusted. However, including different catalyst supports would only make sense in case higher exchange areas to catalyst volume ratios are required. In order to describe the heat transport, a suited heat transport coefficient model for the one-dimensional heat transfer is used (Li and Finlayson, 1977). Ethylene is provided via a porous stainless steel membrane with a wide range of the permeance (here: 0 mol=ðPa s m2 Þ rPer r 102 mol=ðPa s m2 ÞÞ. Such a membrane was successfully used for EO reactors in a previous work reported in the literature (Al-Juaied et al., 2001). It is assumed that the ethylene flow through the porous membrane is purely convective and hence driven by the pressure difference of both sides. Criteria for external heat and mass transfer (Mears, 1971a, 1971b) as well as for internal heat transfer (Anderson, 1963) and mass transfer (expressed by the Thiele modulus ¨ (Thiele, 1939) or the Damkohler number (e.g. Emig and Klemm, 2005)) are considered. These criteria must be applied in order to ensure a reasonable reactor design and to use the proposed pseudo-homogeneous model with a high catalyst efficiency. In

2.3.1. Model equations of level 2 In addition to the equations of level 1, the energy balance, the momentum balance, the transport equations, and the equations describing the specific exchange areas are required on this level. The model is kept in the substantial formulation, however, the local form of the balance equations can also be used if desired. For the energy balance the same assumptions are taken as on level 1, but heat exchange with the cooling side is considered. Since it is not obvious that the desired heat flux is attainable, the transport kinetics for the heat fluxes are included and the heat exchange with the cooling side is controlled via the cooling temperature profile. 0 0 11 NR X X n  cp dT 1e NCOM @ @ ¼  Ao  qo þ rp  ð2:13Þ   hi ni,j  rj AA Vgas dt e i¼1 j¼1 The specific exchange area based on the gas volume for the exchange with the cooling channel and the exchange with the ethylene supply channel are referred to as Ao and Ai, respectively. Ao ¼

Ai ¼

4

Do



ð2:14Þ

e D2o D2i 4

Di



ð2:15Þ

e D2o D2i

The heat capacity of the mixture is modeled assuming a linear mixing rule and individual heat capacities depending linearly on the temperature according to Eq. (2.16). The coefficients Ai and Bi are fitted to the higher polynomial functions (Shomate equation) from NIST chemical web book (Linstrom and Mallard, 2010), since a linear function is by far easier to handle for the optimization solver than higher order polynomials. In addition, the component heat capacities depend almost perfectly in a linear manner on the temperature in the investigated temperature range. cp,i ¼ Acp,i þ Bcp,i  T n

ð2:16Þ

The individual component enthalpies are calculated using Eq. (2.17), where Fi is also fitted to the values given in the literature (Linstrom and Mallard, 2010) ðT n ¼ T=1000, Tref ¼ 298:15 KÞ. hi ðTÞ ¼ Acp,i  T n þ

Bcp,i  ðT n Þ2 þ Fi 2

ð2:17Þ

In case of a randomly packed fixed bed reactor with spherical particles of uniform size, the Ergun equation with the parameters stated in Eq. (2.19) approximates the pressure drop (Ergun, 1952). Under steady state conditions, the operator equation (2.18) can be used to convert the Ergun equation into the substantial formulation as shown in Eq. (2.19). dp @p ¼ vi dt @z

ð2:18Þ

! dp mð1e Þ2 vs rð1e Þ v2s ¼  150 þ1:75 dt e D2p e 3 Dp e 3

ð2:19Þ

vs ¼ vs,0

A0  p0  T  n A  p  T0  n0

ð2:20Þ

A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

The inlet velocity vs,0 is chosen as degree of freedom. The dynamic viscosity is assumed to be constant ðm ¼ 2:52  105 kg=ðm sÞÞ. This value was calculated for a typical inlet mixture at 500 K and 20 bar using Aspen Plus. The heat transport between the reaction channel and the cooling channel is described by an one-dimensional model according to Eqs. (2.21) and (2.22) (Li and Finlayson, 1977). It is assumed that the heat transfer resistance is completely on the reaction channel side. qo ¼ ao  ðTTc Þ

ao ¼ 2:03  Re0:8 Dp 

Table 4 Heat capacity coefficients. Component

Acp,i ðJ=ðmol KÞÞ

Bcp,i ðJ=ðmol K2 ÞÞ

Fi ðJ=molÞ

E O2 EO CO2 H2O N2

21.07 27.73 20.14 31.01 29.74 27.08

82.76 7.68 110.45 27.27 10.95 5.65

42.26  8.70  63.98  404.13  251.14  8.33

ð2:21Þ   6  Dp  exp  Do Do

l

with

r  vs  Dp r7600 20 r ReDp ¼ m

ð2:22Þ

The heat conductivity of the mixture is assumed to be constant for all calculations ðl ¼ 3:91  102 W=ðm KÞÞ and was obtained using Aspen Plus for a typical inlet composition at 500 K and 20 bar. The mass transport through the membrane is assumed to be purely convective and is described by Eq. (2.23), based on the work of Lafarga and Varma (2000) and Al-Juaied et al. (2001), who investigated the transport kinetics for the flux across the membrane using the dusty gas model. Their results fit qualitatively to the experimental observations, but a simpler model for the crossmembrane transport yields similar results. This observation justified the simple transport kinetics assumed in this work. jE ¼ Ai  Per  ðpM pÞ

ð2:23Þ

In case of an annular tube with randomly packed spherical catalyst particles of uniform size, the average void fraction depends on the outer tube diameter, on the inner tube diameter, and on the particle diameter according to Eq. (2.24). Here, the radius dependent void fraction eðrÞ is defined in Eq. (B.33). RR 2 o r eðrÞdr ð2:24Þ e ¼ Ri 2 2 ðRo Ri Þ In order to ensure a reasonable reactor design and to be able to use the proposed pseudo-homogeneous model with a high catalyst efficiency, reactor design criteria on the external and internal heat and mass transfer have to considered in the design stage. All criteria are tracked over the whole reaction coordinate and must be fulfilled at all times. The criteria defined by Mears for external mass (Mears, 1971b) and heat (Mears, 1971a) transfer are taken into account according to the following equations, respectively: PNR

6461

0:3  bi  ci,s

j ¼ 1 ðni,j  ni,j  rj Þ  rp  Dp

Z 1,

0:3  R  T 2  a Z1 PNR E A  ð j ¼ 1 ðDHr,j  rj Þ  rp  Dp

i ¼ E,O2

ð2:25Þ

ð2:26Þ

The criteria for internal and external mass transfer must be checked for both reactants, i.e. ethylene and oxygen. In Eq. (2.25), ci,s is the surface concentration of the component calculated from the ideal gas law and ni,j is the reaction order. To calculate the transport coefficient bi the correlation from Thoenes and Kramers (1958) for a packed bed of spherical particles is used according to:    0:33 1:9  Dm,i vs  Dp  r 0:5 m bi ¼ ð2:27Þ m Dp r  Dm,i It is assumed that the molecular diffusion coefficients Dm,i can be described by the method of Wilke and Lee (1955) assuming binary diffusion in nitrogen. Here, mean values in the temperature range of

500–600 K at 20 bar are used ðDm,E ¼ 2:49  106 m2 =s, Dm,O2 ¼ 3:15  106 m2 =sÞ. The heat transfer coefficient a is related to the mass transfer coefficient b by Eq. (2.28) (De Acetis and Thodos, 1960) assuming a Lewis number close to unity. Here, the lower b value is chosen in order to ensure a conservative design.

b

a

¼

0:7  M Z1 m3 K=J r  cp

with b ¼ min bi i ¼ E,O2

ð2:28Þ

The Anderson criterion (Anderson, 1963) is considered to ensure isothermal catalyst pellets (Eq. (2.29)). For the heat transfer criteria the arithmetic mean value of both activation energies E A is used. EA  ð

PNR

3  R  T 2  lp

j ¼ 1 ðDHr,j

 rj Þ  rp  D2p

Z1

ð2:29Þ

¨ In order to ensure a catalyst efficiency above 95% the Damkohler criterion (e.g. Emig and Klemm, 2005) is applied according to the following equation: ! PNR 2 0:5 qffiffiffiffiffiffiffiffiffiffiffi j ¼ 1 ðni,j  rj Þ  rp  Dp r 1, i ¼ E,O2 ð2:30Þ DaII,i ¼ 4  Deff ,i  ci,s Here, the effective diffusion coefficient inside the pellet Deff ,i is calculated assuming no Knudsen diffusion influence, an inner void fraction of 0.44, and a tortuosity of 3. To limit the mechanical and thermal stress, criteria for a maximum temperature difference ðDTmax ¼ 20 KÞ (Eq. (2.31)) between the cooling and the fluid side and for a maximum pressure difference ðDpmax ¼ 1 barÞ between both sides of the membrane (Eq. (2.32)) are applied. TðtÞTc ðtÞ r DTmax

ð2:31Þ

pM,0 p0 r Dpmax

ð2:32Þ

2.3.2. Optimization problem of level 2 The full optimization problem which must be solved on level 2 is stated in (OP2). For temperature, pressure, composition, and STY the same bounds apply as on level 1. The substance properties required for the calculation of cp,i and hi are given in Table 4. Obj ¼

max

T c ðtÞ,PerðtÞ,x i,0 ,p, Do ,Di ,Dp ,T0 ,vs,0

S

s:t: Component balances: Eq: ð2:1Þ Reaction kinetics: Eqs: ð2:2Þ2ð2:4Þ Energy balance: Eqs: ð2:13Þ2ð2:17Þ Momentum balance: Eqs: ð2:18Þ2ð2:20Þ Transport kinetics: Eqs: ð2:21Þ2ð2:23Þ Catalyst support: Eq: ð2:24Þ Design criteria: Eqs: ð2:25Þ2ð2:32Þ Constitutive equations: Eqs: ð2:6Þ2ð2:9Þ

ðOP2Þ

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A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

Initial conditions: xi ðt ¼ 0Þ ¼ xi,0 , Tðt ¼ 0Þ ¼ T0 Residence time: t ¼ 30 s STY: Eq: ð2:10Þ Intrinsic bounds: T L r T r T U , pL r p rpU , xLi r xi r xU i L

U

DLk rDk rDU k

Design bounds: Per rPer r Per , Case selection: si ¼ 0 8i A COM\fEg, sE ¼ 1

8i A COM

for k ¼ i,o,p

2.3.3. Results In Fig. 7 the optimal permeance and optimal cooling profile as well as the resulting temperature, reactor design criteria, ethylene flux and mole fraction profiles are summarized. Referring to Fig. 7(b) the temperature increases from T¼570.4 K to T¼597.6 K which is qualitatively the same profile as in case 3 on level 1. The temperature difference between the fluid and the cooling is always below the maximum allowed temperature difference of 20 K. The optimal outer tube diameter is Do ¼4.07 cm. Referring to Fig. 7(c) the ethylene and oxygen inlet mole fractions are at their upper bound, while the CO2 inlet mole fraction is at its lower bound. Due to the shown permeance profile (refer to Fig. 7(a)) the ethylene mole fraction is always kept at its upper boundary. The optimal ethylene pressure on the membrane side is pE,M ¼ 20:78 bar and the inner tube diameter is at its lower bound ðDi ¼ 0:5 cmÞ. The particle diameter is at its lower bound ðDp ¼ 2 mmÞ, which is the optimal trade-off between pressure drop, void fraction and catalyst efficiency. The combination of the inner and outer tube as well as particle diameter yields a void fraction of e ¼ 0:424, which is slightly higher than the assumed void fraction on level 1. At the

inlet the velocity is 0.085 m/s, which gives rise to a small and almost linear pressure drop of Dp ¼ 0:06 bar. Referring to Table 5, the optimal reactor design never reaches the bounds for the reactor design criteria. The minimal values for the external heat and mass transfer as well for the internal heat transfer are always above their minimal value of 1. The maximum value for the internal mass transfer criterion is always below the value where pore diffusion becomes limiting. Here, oxygen is the more limiting component since it depletes along the reaction coordinate. The profiles of the reactor design criteria, which are close to their bounds, are shown in Fig. 7(d). The internal heat and the external mass transfer are far away from being limiting, and hence the according criteria are not shown. The maximum selectivity of such a reactor set-up is 82.33%. In case a degree of freedom is at its bound (e.g. the particle diameter), a sensitivity analysis can be used to investigate whether it is worthwhile to relax this bound. The relaxation of such a bound refers to a different reactor concept or to the shift of the design space, for example by advanced materials or miniaturization. Since the difference in the selectivity between level 2 and level 1 is merely 0.01%, it can be concluded that the cooling temperature and the permeance of the membrane are suitable control variables to obtain the desired fluxes and no design bounds need to be shifted. In addition, the chosen catalyst packing is optimal since it features the highest catalyst density compared with wall coated reactors, monolithic reactors, or foam like catalyst support structures, and no limitations on the overall heat and mass transport occur. In case severe limitations of the heat and mass transport occur, a reactor design with higher specific exchange areas is

600 Fluid Coolant

590 3.55 T [K]

Permeance. 107 [mol/(Pa m2 s)]

3.6

580 570

3.5 560 550

3.45 0

10

20

30

0

10

2.3

0.08

2.2

0.06

2.1

0.04

2

0.02

1.9

0

1.8 10

20 t [s]

30

Mass and heat transport criteria [−]

0.1

0

20

30

t [s]

jE/nin⋅ 10−3 [1/s]

xi [−]

t [s] 4

3 External heat Internal mass E Internal mass O 2

2

external heat transport criterion

1 internal mass transport criteria

0 0

10

20

30

t [s]

Fig. 7. Results level 2 ((c): —, E; – –, O2;    , EO; -  -, CO2; thin lines, mole fractions; thick line, dosing profile). (a) Permeance, (b) temperature, (c) mole fractions and ethylene dosing, and (d) design criteria.

A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

6463

Table 5 Results comparison between level 2 and level 3. Decision variables

Level 2

Level 3 (1D)

Level 3 (2D)

Do (cm) Di (cm) Dp (mm) L (m) T0 (K) Tc,0 (K) Kc ðK=WÞ TM,0 (K) n_ 0 ðmol=sÞ Per ðmol=ðPa m2 sÞÞ n_ M,0 ðmol=sÞ Inlet composition (xE, xO2 , xCO2 , xN2 )

4.07 0.5 2 – 570.4 551.1 – – – – – 0.1, 0.08, 0.05, 0.77

4.37 0.84 2 5.88 570.9 550.9 4.0  10  2 550.9 4.9  10  2 8.95  10  7 0.1 0.1, 0.08, 0.05, 0.77

5.21 2.94 2 4.58 573.9 558.3 4.7  10  2 550.0 3.9  10  2 5.42  10  8 0.1 0.0973, 0.08, 0.05, 0.7727

Design criteria External heat (min) External mass (min) Internal heat (min) Internal mass (max) Selectivity (%)

3.16 132.8 658.9 0.83 82.33

3.03 140.5 650.6 0.81 82.32

2.56 41.5 613.7 0.57 82.22

required. However, such a reactor design will suffer from lower catalyst densities and hence will yield lower selectivities for the desired STY and residence time. Therefore, for the chosen catalyst it can be concluded that alternative catalyst supports and/or a micro-reactor concept is not necessary for the selective production of EO.

2.4. Level 3: best technical reactor 2.4.1. Derivation of a technical approximation On the third level, technical approximations based on the profiles of the best control variables are developed. In case different ways how to approximate the control variables are derived, the different set-ups are compared using simple models before a chosen set-up is further investigated using more detailed models. In the investigated case, a technical approximation of the control variable profiles based on existing apparatuses can directly be proposed. The cooling temperature profile can be approximated using a co-current heat exchanger. How the permeance profile can be approximated depends strongly on the manufacturing technology of the porous membrane used. However, a membrane with a constant permeance is a reasonable approximation since the optimal permeance profile on level 2 varies by less than 3.4%. On this level, the change in the cooling temperature is determined in a rigorous manner by solving the energy balance for the coolant. On the ethylene supply side, the mass, energy, and momentum balances are solved. The heat transport between the reaction channel and the cooling channel is determined by the heat transfer resistance on the gas side. In case of the heat transfer between the ethylene supply side and reaction channel, the heat transfer resistance of both sides are considered. Due to the additional balance equations for the coolant and for the ethylene supply side, the degrees of freedom of the optimization problem are reduced. On this level, the flux profiles cannot be optimized anymore. Now, only the design variables which are listed with their according optimal values in Table 5 are degrees of freedom. Hence, the solution of the optimization problem of level 3 cannot be better—with respect to local optimality—than the solution of (OP2). On level 3, the balance equations are written in the local formulation and are listed in Appendix A.

The according optimization problem (OP3) is stated below: Obj ¼

max

DoF refer to Table 5

n_ EO,f n_ EO,0 n_ E,0 n_ E,f

ðOP3Þ

s:t: Equations of changes: ðA:1Þ2ðA:3Þ Reaction kinetics: Eqs: ð2:2Þ2ð2:4Þ Transport kinetics: Eqs: ð2:21Þ2ð2:23Þ, ðA:16Þ; and ðA:17Þ Catalyst support: Eq: ð2:24Þ Design criteria: Eqs: ð2:25Þ2ð2:32Þ Constitutive equations: Eqs: ð2:9Þ; ðA:4Þ2ðA:6Þ Initial conditions: xi ðt ¼ 0Þ ¼ xi,0 , Tðt ¼ 0Þ ¼ T0 , Tc ðt ¼ 0Þ ¼ Tc,0 , TM ðt ¼ 0Þ ¼ TM,0 , n_ M ðt ¼ 0Þ ¼ n_ M,0 Z L 1 Residence time: e dz ¼ 30 s z ¼ 0 vs n_ EO,f n_ EO,0 ¼ 0:27 mol=ðm3 sÞ STY: pðR2o R2i ÞL Intrinsic bounds: T L r T r T U , pL r pr pU , xLi r xi r xU i L

U

Design bounds: Per r Per rPer ,

DLk r Dk r DU k

8i A COM

for k ¼ i,o,p

Case selection: si ¼ 08iA COM\fEg, sE ¼ 1 Coolant side: Eqs: ðA:7Þ and ðA:8Þ Membrane side: Eqs: ðA:9Þ2ðA:15Þ For temperature, pressure, composition, STY, and permeance the same bounds apply as on the previous levels. The bounds for the coolant constant and all additional required substance properties are stated in the model description. The heat capacity and enthalpy of ethylene on the membrane side are calculated with the same models and parameters as before. The selectivity is 82.32% and it is still much higher than the selectivity of the reference case. The difference in the objective between level 2 and level 3 is caused by the non-ideal control variable profiles. Since the difference between level 2 and level 3 is only 0.1%, the technical approximation is reasonable. In case the losses due to the non-ideal control profiles are severe, a different technical approximation should be developed. The results of the derived technical approximation are shown in Fig. 8(a)–(c). In order to validate the results obtained by the one-dimensional model on level 3, a more detailed reactor model accounting

A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

600

580

xi [−]

T [K]

0.1

Fluid Coolant

590

570 560

0.08

2

0.06

1.9

0.04

1.8

0.02

1.7

0

550 1

2

3 z [m]

4

5

1

2

3 z [m]

4

5

6

600 Coolant Fluid T((Ro+Ri)/2)

590 External heat Internal mass E Internal mass O2

2

T [K]

3

external heat transport criterion

1

Membrane

580 570 560

internal mass transport criteria

0 1

2

3 z [m]

4

5

550 0

6

1

2

3

4

5

z [m]

0.1

1.57 1.56

0.06

1.55

0.04

1.54

0.02

1.53

0

1.52 0

1

2

3

4

0.1 xE [−]

0.08

jE⋅ 104 [mol / (m2 s)]

Mass and heat transport criteria [−]

1.6

6

4

xi [−]

2.1 jE⋅ 102 [mol / (m2 s)]

6464

0.098 0.096 2.5 2 r [cm]

5

4 2 z [m]

1.5 0

z [m] 0.2 Mass and heat transport criteria [−]

4

vs [m / s]

0.15 0.1 0.05 0

3 2 external heat transport criterion

1

External heat Internal mass E Internal mass O2

internal mass transport criteria

0 1.5

2 r [cm]

2.5

0

1

2

3

4

5

z [m]

Fig. 8. Results level 3 ((b), (e): —, E; – –, O2;    , EO; -  -, CO2; thin lines, mole fractions; thick line, dosing profile). (a) 1D: temperature, (b) 1D: mole fractions and ethylene dosing, (c) 1D: design criteria, (d) 2D: temperature, (e) 2D: mole fractions and ethylene dosing, (f) 2D: ethylene mole fraction, (g) 2D: inlet velocity profile, and (h) 2D: design criteria.

for radial temperature, concentration and porosity profiles is derived and used for optimization in the next step.

2.4.2. Detailed investigation of the proposed reactor design For the detailed investigation of the chosen reactor set-up a two-dimensional, pseudo-homogeneous model is used (refer to Appendix B). The model accounts for a radial temperature, concentration, and porosity profile. In order to investigate if a

non-ideal temperature profile has a severe effect on the selectivity, the radial temperature distribution needs to be considered. The radial porosity profile results in a radial distribution of the axial velocity, which allows to investigate the effect of by-passing of the dosed ethylene. Since the dosing of ethylene and the radial distribution of the axial velocity gives rise to a non-plug flow like profile of ethylene, also radial mole fraction profiles need to be considered to ensure a non-explosive composition everywhere in the reactor. The optimization model has

A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

the same degrees of freedom as the 1D reactor model and is stated in (OP4). The equations describing the ethylene supply side and the coolant side are kept one-dimensional since only the radial concentration and temperature gradients as well as the radial velocity distribution in the reaction channel are important to validate the results obtained with the one-dimensional model. All balance equations of the reaction channel are written in cylindrical coordinates: Obj ¼

max

S2D

ðOP4Þ

DoF refer to Table 5

s:t: Component balances: Eqs: ðB:1Þ2ðB:10Þ Reaction kinetics: Eqs: ð2:2Þ2ð2:4Þ Energy balance: Eqs: ðB:11Þ2ðB:26Þ Momentum balance: Eqs: ðB:27Þ2ðB:32Þ Catalyst support: Eq: ðB:33Þ Design criteria: Eqs: ð2:25Þ2ð2:32Þ Constitutive equations: Eq: ð2:9Þ Initial conditions: xi ðt ¼ 0Þ ¼ xi,0 , Tðt ¼ 0Þ ¼ T0 , Tc ðt ¼ 0Þ ¼ Tc,0 , TM ðt ¼ 0Þ ¼ TM,0 , n_ M ðt ¼ 0Þ ¼ n_ M,0 Residence time: Eq: ðB:35Þ STY: Eq: ðB:36Þ Intrinsic bounds: T L r T r T U , pL rp rpU , xLi rxi r xU i

Design bounds: Per L r Per rPer U , DLk r Dk r DU k

8i A COM

for k ¼ i,o,p

Coolant side: Eqs: ðA:7Þ and ðA:8Þ Membrane side: Eqs: ðA:9Þ2ðA:15Þ The equations for the case selections are directly implemented in the mass balance of the reaction channel (refer to Eq. (B.3)), and hence do not appear separately in (OP4). The equations of the heat and mass transport kinetics are given together with the component balances and energy balance, respectively. Referring to the results presented in Table 5, it can be observed that the inlet temperatures of the reaction, cooling, and membrane channel are very close to the temperatures predicted by the 1D model. Also, the pellet diameter, the coolant constant, and the ethylene flux of the membrane side are in accordance with the 1D model. However, the diameters of the inner and outer tubes vary significantly from the results obtained with the simpler model. The different channel design results in different values for the reactor length, the inlet mole flow and the permeance. In Fig. 8(d) the profiles for the fluid temperature in the middle of the reaction channel, the cooling temperature, and the membrane temperature are shown. The profiles are similar to the 1D case and the maximum radial hot spot in the fluid is DTradial,max ¼ 8:2 K. Fig. 8(e) shows the average mole fraction profiles of all components. These profiles are very similar to the results obtained by the 1D model, however, the inlet mole fraction of ethylene is only 9.73%. Due to the dosing of ethylene, the ethylene mole fraction shows a strong radial distribution with maxima of up to 10% at the inner tube wall (refer to Fig. 8(f)). This underlines the necessity of more detailed models especially when safety constraints have to be met. The by-pass flow is significant as can be observed from the radial velocity profile shown in Fig. 8(g). Similar as in the 1D model the reactor design criteria are not critical and never exceeded along the reaction channel (refer to Fig. 8(h) calculated with radially averaged values and to Table 5 where the extreme values of the design criteria are given considering the radial and axial profiles). From a reaction engineering point of view, the difference in selectivity between the one-dimensional and the two-dimensional

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model is due to the non-ideal temperature and concentration distribution as well as the non-ideal flow field. However, since both models are different, a monotonic decrease in the objective function cannot necessarily be expected from a mathematical point of view. In case the non-ideal radial profiles cause a large decrease in the objective function, a different catalyst packing which yields a better radial heat and mass transport as well as a lower amount of by-passing might improve the reactor performance. Such effects can be realized for example by foam like catalyst packings. Beside the radial hot spots and the high amount of by-passing, the selectivity calculated with the optimized 2D model is still 82.25%, and hence an industrial realization of the proposed reactor design seems to be worthwhile.

3. Numerical solution approach For all calculations, the dynamic optimization problems are transferred to large NLP problems using the simultaneous approach. The problems were implemented in AMPL and solved using CONOPT 3.14 G on a PC with an Intel(R)Core(TM)2 Duo CPU E6850 with 3.00 GHz (calculation on a single CPU), a cache size of 4096 KB, a memory of 2 GB, and Ubuntu 10.04 as operating system. For the one-dimensional problems orthogonal collocation on finite elements (Logsdon and Biegler, 1989) is used as discretization method. For the two-dimensional reactor model on level 3, the axial coordinate is discretized using orthogonal collocation on finite elements, while the radial coordinate is discretized using the finite volume method with constant profiles on each finite volume. This approach proved to be more robust in the optimization especially with respect to mass balance errors. In general, the computing times depend on the number of equations, the degrees of freedom, the nonlinearity of the model equations, the starting values, and the model formulation. Due to these many influencing factors, the given numbers for the computing times should only be considered for orientation purposes. On the first level, the computing times are short since the model formulation is efficient. However, the computing time increases drastically for increasing degrees of freedom, for example the CONOPT time for case 3 is 3 s, while the CONOPT time for case 7 is 2824 s on the first level. The computing time for the 1D model on level 3 is in the range of 92 s. In this case, the number of equations is high due to the additional balance equations, but only few degrees of freedom exist. In order to find feasible starting points for the 2D reactor model, an advanced strategy is necessary where single blocks of equality constraints defined by the balance and flux equations are added to the optimization problem step-by-step. In case of the two-dimensional reactor model, the calculation times are in the range of 80 h, where most of the time is required for the pre-processing to obtain a feasible point. Here, a sensitivity analysis can also be used in order to reduce the degree of freedom and help the optimizer to converge. This solution approach can only yield locally optimal solutions as long as no global optimization solver such as BARON is used. Hence, even better solutions might exist. In order to examine if different local solutions can occur, each problem was solved several times using different starting values. For all optimization problems (OP1), (OP2), (OP3), and (OP4) no other local optima apart from the stated solutions are obtained. As can be seen from the computing times, the optimization based on detailed reactor models takes much longer compared to the screening stage. Hence, the computational effort is significantly reduced by the proposed methodology compared to a brute force approach where the results of detailed reactor models of every reactor set-up are compared.

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A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

4. Summary and conclusion As a result of the current work, an improved reactor for the ethylene oxide production, which can also be referred to as optimal technical reactor, was designed. The proposed reactor applies a co-current cooling strategy, which approximates the optimal temperature profile quite well, and distributed ethylene dosing via a membrane. The maximum selectivity of such a reaction concept is calculated to be 82.35% (level 1). In addition, it was shown by applying a two-dimensional reactor model that the maximum achievable selectivity in a technical reactor is 82.23%. Hence, the losses due to the non-ideal temperature and concentration distribution as well as the non-ideal flow field are not severe and the proposed reactor design still has a large selectivity optimization potential. The applied 1D model gave good results with respect to the overall reactor performance and the optimal control profiles. However, a more detailed model—such as the used 2D reactor model—was required to obtain validated results for the optimal design variables. A comparison to the industrial used reactor set-up using the same catalyst is done by the optimized reference case and it is shown that a selectivity increase of 3.3% is possible. The selectivity values reported in the literature for the industrial air-based EO process are approximately 80% (Rebsdat and Mayer, 2007). This indicates that the used catalyst is comparable to the industrial used catalysts and the increase in selectivity is realistic. In comparison to other reactor configurations published in the literature, the obtained selectivity with the derived reactor concept is much higher. Lafarga and Varma (2000) investigated different fixed bed membrane reactors. In agreement with the current work, they found that dosing ethylene yields the highest selectivity increase, however, the obtained selectivity is only 67%. Kestenbaum et al. (2002) achieved a selectivity of up to 69% using a micro-reactor set-up. The large discrepancy in the selectivities between the work of Lafarga et al. (2000a) and Kestenbaum et al. (2002) on the one hand and our work on the other hand is partly due to the fact that the former works were performed with highly active, but less selective catalysts. Zhou and Yuan (2005) optimized a conventional fixed bed reactor with a highly selective catalyst and obtained a selectivity up to 80.5%. Summing up, the used reactor design methodology (Peschel et al., 2010) is well suited to reliably predict the potential of various integration and enhancement concepts as well as to design advanced technical reactors. The method proved to be able to design optimal reactors for complex reaction systems. The approach is very useful to investigate the potential of dosing and removal of certain components and yields the best suited integration and enhancement concept for the investigated reaction system. The methodology provides the optimal flux profiles and the required relative mass and energy exchange areas. From the exchange areas it can be concluded that a micro-reactor is not required for the ethylene oxide production using the chosen catalyst. In addition, it gives a strong catalog of requirements which type of catalyst support is best suited for the reaction system. Here, reactor design criteria ensure the validity of the model and a robust reactor design.

Dm Dr EA h j k Kc L _ m M n n_ p Per q rj R r S si STY T t vi vs vs V x z

molecular diffusion coefficient, m2/s radial dispersion coefficient, m2/s activation energy, J/(mol K) specific enthalpy, J/mol component flux, mol/(m2 s), or molar flux into/out of fluid element, mol/s heat transfer coefficient (2D model), W/(m2 K) coolant constant, K/W reactor length, m mass flow, kg/s molecular weight, kg/mol molar amount in fluid element, mol mole flow, mol/s pressure, Pa permeance of porous membrane, mol/(m2 Pa s) heat flux, W/m2 reaction rate, mol/(kgp s) gas constant, J=ðmol KÞ, or radius, m radial coordinate, m selectivity, – selection variable indicating whether the component flux is considered or not, – space time yield, mol/(m3 s) temperature, K residence time, s interstitial velocity, m/s superficial velocity, m/s average superficial velocity (2D model), m/s volume, m3 mole fraction, – axial coordinate, m

Abbreviations, sets, and dimensionless numbers COM Da DoF E EO NCOM NR Pe Pr Re

set of components COM ¼ fE,O2 ,EO,CO2 ,H2 O,N2 g ¨ Damkohler number degree of freedom ethylene ethylene oxide number of components number of reactions Peclet number Prandtl number Reynolds number

Greek symbols

a b

e e l L

m meff ni,j r t

heat transfer coefficient (1D model), W/(m2 K) mass transfer coefficient, m/s void fraction, – average void fraction (2D model), – heat conductivity, W/(m K) effective radial heat conductivity (2D model), W/(m K) dynamic viscosity, kg/(m s) effective viscosity, kg/(m s) stoichiometric coefficient (component i, reaction j), – density, kg/m3 residence time, s

Suffices Nomenclature Latin symbols A specific exchange area for flux, m2 =m3gas c concentration, mol/m3 cp heat capacity, J/(mol K) D diameter, m

c gas f i j L M

coolant side gas phase (without catalyst fraction) final/outlet value reaction channel at r ¼ Ri , or component index reaction index lower bound membrane side

A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

o p s U 0

6467

the inlet mole flow through the membrane channel is bounded according to Eq. (A.10) with n_ LM,0 ¼ 0:01 mol=s and n_ U M,0 ¼ 1 mol=s.

reaction channel at r ¼ Ro catalyst particle surface conditions upper bound inlet value

dn_ M ¼ p  Di  Per  ðpM pÞ dz

ðA:9Þ

n_ LM,0 r n_ M,0 r n_ U M,0 Acknowledgments A.P. and F.K. thank the International Max Planck Research School Magdeburg for financial support. The authors gratefully acknowledge the discussion with Dr. R. Benfer, Dr. C. Großmann, and Dr. G. Theis (all BASF SE, Ludwigshafen, Germany) as well as the financial support of the BASF SE.

ðA:10Þ

Energy balance membrane side: The energy balance on the membrane side is simplified so that it only accounts for the heat exchange between ethylene supply channel and reaction channel. Bounds on the temperature of the membrane side are given by L U Eq. (A.12) with TM ¼ 490 K and TM ¼ 600 K. n_ M cp,M

dTM ¼ p  Di  qi dz

ðA:11Þ

L U rTM r TM TM

Appendix A. Model equations of level 3: one-dimensional case Component balances: NR X @n_ i ¼ 2pRi si ji þ pðR2o R2i Þ  ð1e Þ  rp  ni,j  rj @z j¼1

ðA:1Þ

NCOM X

ðn_ i  cp,i Þ

i¼1

0

@T @z

¼ @2pðRo qo þ Ri qi Þ þ pðR2o R2i Þð1e Þrp

NCOM X i¼1

0 @hi

NR X

11

ni,j  rj AA

n_ ¼

NCOM X

n_ i

Re0:25 M

for 3000 r ReM ¼

rM  vM  Di r 105 mM

ðA:14Þ

Constitutive equations membrane side: The velocity on the membrane side vM is calculated from the volume flow using the ideal gas law and the cross sectional area of the ethylene supply channel according to the following equation: 4  n_ M  R  TM pM  p  D2i

vM ¼

ðA:3Þ

Heat transfer membrane side: The heat transfer between ethylene supply channel and reaction channel depends on the heat transfer resistance on both sides and is given by the following equation: qi ¼

ðA:4Þ

Energy balance coolant side: The energy balance on the coolant side (Index c) is written in temperature form and simplified so that the change in the coolant temperature is proportional to the heat flux. The introduced coolant constant is chosen within meaningful bounds ð1K=W rKc r 1K=WÞ. A value of Kc smaller than zero accounts for the possibility of counter-current cooling, a value larger than zero for co-current cooling, and Kc ¼0 accounts for isothermal cooling, for example by an evaporating fluid. dTc ¼ Kc  p  Do  qo dz

ðA:7Þ

KcL rKc r KcU

ðA:8Þ

Mass balance membrane side: The mass balance on the membrane side (Index M) only has to account for an ethylene flow through the membrane and is written in terms of the molar ethylene flux. The membrane side consists of pure ethylene and

ðA:16Þ

On the ethylene side, the heat transfer coefficient is determined according to the turbulent flow through an empty pipe by the following equation (Gnielinski, 2006):

aM ¼ 0:0214  ðA:6Þ

ðA:15Þ

ai  aM  ðTTM Þ ai þ aM

ðA:5Þ

i¼1

n_ xi ¼ i n_

0:3164

ðA:13Þ

ðA:2Þ

Constitutive equations: n_  R  T vs ¼ p  pðR2o R2i Þ

cw  rM  v2M dpM ¼ dz 2  Di

j¼1

Momentum balance: ! @p mð1e Þ2 vs rð1e Þ vs ¼  150 þ 1:75 @z D2p e 3 Dp e 3

Momentum balance membrane side: For the momentum balance on the ethylene supply side, the pressure drop correlation for the flow through an empty pipe is used (Karst, 2006).

cw ¼

Energy balance:

ðA:12Þ

lM Di

4=5

1=4

 ðReM 110Þ  PrM

ðA:17Þ

The heat conductivity and the viscosity of ethylene are assumed to be constant (lM ¼ 5:86  102 W=ðm KÞ, mM ¼ 1:75  105 kg=ðm s)) and are obtained from Aspen Plus at T¼550 K and p ¼20 bar. The Prandtl number is also assumed to be constant ðPrM ¼ 0:724Þ. On the reaction side the heat transfer coefficient is calculated using Eq. (2.22) with Di instead of Do.

Appendix B. Model equations of level 3: two-dimensional case 2D component mass balance: The component mass balances are simplified by assuming no axial dispersion, no radial convection, and steady state conditions. NR X @ðri vs Þ 1@ ¼ ðr  Mi  jr,i Þ þ rp  ð1eÞ  Mi  ni,j  rj @z r @r j¼1

Mi  jr,i ¼ Dr,i

@ri @r

ðB:1Þ

ðB:2Þ

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A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

jr,i 9r ¼ Ri ¼



Per  ðpM pÞ,

i¼E

0

else

jr,i 9r ¼ Ro ¼ 0

ðB:4Þ

_i m Mi xi ¼ P _  NCOM m i i¼1

ðB:3Þ

ðB:5Þ

Dr ¼ Dbed þK1  Pem  Dm  fD ðrÞ 

v s,0 vs ðrÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dbed ¼ Dm  ð1 1eðrÞÞ

K1 ¼

v s,0  Dp Dm

1 3 1þ 0:5 8 Pem

8 2 > < Ro r K2  Dp fL ðrÞ ¼ > :1

ðB:22Þ

if Ro r o K2  Dp

ðB:23Þ

if Ro r Z K2  Dp

  Re K2 ¼ 0:44 þ 4  exp  70

ðB:24Þ

_ 0  Dp m

@z 0

ðB:11Þ

ðB:12Þ

L

 @T  k  aM ¼ i ðTM TðRi ÞÞ @r r ¼ Ri ki þ aM

ðB:13Þ

L

 @T  ¼ ko  ðTðRo ÞTc Þ @r r ¼ Ro

ðB:14Þ

The heat flux is calculated using the effective radial conductivity according to Eq. (B.15) (e.g. Tsotsas, 2006b). The average velocity ðv s,0 Þ and all properties depending on the fluid state are calculated at the inlet conditions for the LðrÞ-model. The heat conductivity of the particles ðlp ¼ 17:75 W=ðm KÞÞ is assumed to be constant and calculated at 550 K (Yaws, 1999). ðB:15Þ

ðB:25Þ

pðR2o R2i Þm

ki ¼ ko ¼ 0:17

l

  Dp  r  v s,0 0:79

Dp

m

for

r  v s,0  Dp r7600 m

20 r ReDp ¼

r  cp  vs @T

1 v s,0  Pe  l  fL ðrÞ  8 vs ðrÞ

ðB:21Þ

v s,0  r  Dp  cp lM

ðB:10Þ

2D energy balance: The energy balance is simplified using the assumptions of the LðrÞ-model resulting in the following equations:

L ¼ lbed þ

  1eðrÞ 10=9 eðrÞ

At the boundaries (r ¼ Ri and r ¼ Ro ) the fluid heat conductivity is used, which is in agreement with the LðrÞ-model. For the heat transport coefficients on the reaction channel side the according correlations for a 2D model must be used according to the following equation (Li and Finlayson, 1977):

if Ro r Z 0:44  Dp

Tðz ¼ 0Þ ¼ T0

ðB:20Þ

ðB:7Þ

ðB:9Þ

¼ rp  ð1eÞ 1   NCOM NR X X 1@ @T @hi rL ni,j  rj A þ  r @r @r i¼1 j¼1

lp l

B ¼ 1:25

Pe ¼

ðB:19Þ

Re ¼

!1

if Ro r o 0:44  Dp

B kp

ðB:6Þ

ðB:8Þ

8 2 Ro r > < 0:44  Dp fD ðrÞ ¼ > :1

M

kp ¼

Mi

The initial mass flow distribution ðri,0 vs,0 Þ is calculated by solving Eq. (B.27) for the velocity distribution at the inlet. The component densities ri,0 and the total density r0 at the inlet are calculated from the inlet conditions (T0, p0) and the inlet composition (xi,0). The dispersion coefficients (Eqs. (B.6)–(B.9)) are calculated in accordance to the models published in the literature (e.g. Tsotsas, 2006a, 2006b). In order to estimate the dispersion coefficient, the diffusion coefficient of ethylene in nitrogen is used for all components in Eq. (B.7). To simplify the calculations, the Peclet number is calculated with average inlet velocity.

Pem ¼

N ¼ 1

ðB:26Þ

2D momentum balance: The Brinkman equation (B.27) is used to calculate the radial profile of the axial velocity for the inlet conditions. In order to reduce the model complexity so that it can be handled by the optimization solver, the radial velocity distribution is only calculated for the inlet since the radial concentration and temperature gradients do not strongly influence the velocity distribution along the channel. The Reynolds number is defined in Eq. (B.25) and the void fraction profile in Eq. (B.33). To calculate the pressure drop from the Brinkman equation, the total mass balance according to Eq. (B.32) is required. The pressure gradient is assumed to be constant over the reactor length, which is in good agreement with the results of the 1D model.    meff @ @vs,0 ðrÞ @p 2 r ðB:27Þ  ¼ f  v ðrÞf  v ðrÞ þ 1 s,0 2 s,0 @z z ¼ 0 @r r @r f1 ¼ 150 

ð1eðrÞÞ2  m

f2 ¼ 1:75 

ð1eðrÞÞ  r0

ðB:29Þ

eðrÞ3  Dp

pðz ¼ 0Þ ¼ p0 ,

vs,0 ðr ¼ Ri Þ ¼ 0,

meff ¼ 2  m  expð2  103  ReÞ Z

ðB:28Þ

eðrÞ3  D2p

vs,0 ðr ¼ Ro Þ ¼ 0

ðB:30Þ ðB:31Þ

Ro

lbed ¼ l  kbed

ðB:16Þ

_ 0 ¼ 2p m

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi kbed ¼ 1 1eðrÞ þ 1eðrÞ  kc

ðB:17Þ

Radial void fraction profile: The radial void fraction for imperfect spherical particles is usually described using exponential functions. Here, a modified correlation for the void fraction distribution of an annular tube obtained from the literature



   kp 2 B kp 1 B þ 1 B1   ln kc ¼ N N2 kp 2 N B

ðB:18Þ

r ¼ Ri

vs,0 ðrÞ  r0  r dr

ðB:32Þ

A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469

Tsotsas (2006a) is used according to:    8 5  ðrRi Þ > > , 0:4  1 þ 1:36  exp > < Dp    eðrÞ ¼ 5  ðRo rÞ > > > , : 0:4  1 þ 1:36  exp Dp

Ro þRi Ri rr o 2 Ro þ Ri Z r ZRo 2 ðB:33Þ

Selectivity, residence time, and STY for the 2D case: The selectivity, residence time, and STY for the 2D case are given by the following equations: R Ro ME r ¼ Ri ðvs,f rEO,f vs,0 rEO,0 Þr dr  S2D ¼ R R ðB:34Þ RL o ðvs,0 rE,0 vs,f rE,f Þr dr þ Ri ME jE ðr ¼ Ri Þdz MEO z¼0

r ¼ Ri

t2D ¼

Z

L

ðR2o R2i Þ dz ¼ 30 s R Ro z¼0 r ¼ Ri vs r dr R Ro

STY2D ¼

r ¼ Ri ðvs,f

r

r

EO,f vs,0 EO,0 Þr 2 ðRo R2i ÞL  MEO

ðB:35Þ

dr

¼ 0:27 mol=m3 s

ðB:36Þ

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