Design of optimal multiphase reactors exemplified on the hydroformylation of long chain alkenes

Chemical Engineering Journal 188 (2012) 126–141

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Design of optimal multiphase reactors exemplified on the hydroformylation of long chain alkenes Andreas Peschel a , Benjamin Hentschel b , Hannsjörg Freund a,∗ , Kai Sundmacher a,b a b

Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106 Magdeburg, Germany Otto-von-Guericke University Magdeburg, Process Systems Engineering, Universitätsplatz 2, 39106 Magdeburg, Germany

a r t i c l e

i n f o

Article history: Received 7 September 2011 Received in revised form 26 January 2012 Accepted 28 January 2012 Keywords: Multiphase reactor Design Optimization Hydroformylation Process intensification Elementary process functions

a b s t r a c t In this work, a methodology for the design of optimal multiphase reactors is proposed and illustrated on the hydroformylation of 1-octene using a biphasic ionic liquid system with TPPTS modified Rh catalyst. The applied three level design methodology is apparatus independent and thus able to validate existing and generate innovative reactors. On the first design level, the optimal heat and mass flux profiles, which provide the best route in the thermodynamic state space, are determined by solving a dynamic optimization problem. In case of the investigated hydroformylation system, a selectivity increase of 11.7% is obtained by optimal profiles for the 1-octene, hydrogen, carbon monoxide, and heat flux. On the second design level, the minimum required (kL a)-value and the design variables, which are suited to establish the desired flux profiles, are determined. It is shown that both the space time yield and the selectivity of the hydroformylation process depend strongly on the intensity of gas–liquid mass transfer. The optimal design variable profiles and (kL a)-value of level 2 are approximated by a suited reactor set-up on the third level. By use of static mixers, advanced cooling, and discrete 1-octene dosing, the selectivity of the derived technical reactor is 9.1% higher compared to an optimized reference case. In summary, the proposed methodology is suited for the design of tailor-made superior multiphase reactors. For the example process, namely the hydroformylation of long chain linear alkenes, a new reactor concept was derived, leading to a significantly higher selectivity compared to conventional reactor concepts. © 2012 Elsevier B.V. All rights reserved.

1. Hydroformylation of long chain alkenes The hydroformylation is one of the industrially most important homogeneously catalyzed reactions. The aldehydes with a chain length above C4 are used for plasticizers, detergents, and surfactants. Due to biodegradability of unbranched surfactants, the market for linear long chain aldehydes strongly rises and hence linear aldehydes are more valuable than iso-aldehydes. Since the separation of n- and iso-aldehydes is very energy intensive and requires high capital investment, a high n/iso-ratio is very important. Beside a high selectivity towards the linear aldehyde, the recovery of the expensive Rh catalyst is of major importance for the economics of the process. The state-of-the-art hydroformylation process for short chain aldehydes (≤C4), namely the Ruhrchemie-Rhône Poulenc Process, cannot be applied to long chain alkenes since the alkenes are

∗ Corresponding author. Tel.: +49 391 6110275; fax: +49 391 6110634. E-mail address: [email protected] (H. Freund). 1385-8947/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cej.2012.01.123

not soluble in the aqueous catalyst phase (e.g. [1]). An alternative prevalent process for short chain aldehydes, the low pressure UCC process [2], was adapted for higher alkenes. In this process, solvent and catalyst are mainly recovered by extraction, which requires energy intensive recovery of the catalyst from the extracting phase and involves high investment costs [3,4]. Due to the catalyst separation problem, industrial processes for the production of long chain aldehydes are still mainly based on Co-systems as catalysts (for more information refer to [1]). Since Rh based catalyst systems with state-of-the-art ligands provide much higher selectivities and space time yields (STY), large efforts are put into the development of Rh based processes for the hydroformylation of long chain alkenes. In order to overcome the catalyst recycle and reactant solubility problem and obtain an energetically efficient process, several innovative concepts are currently under investigation in academia. Those are thermomorphic solvent systems (TMS, e.g. [5]), micellar solvent systems (MSS, e.g. [6,7]), perfluorinated solvents (e.g. [8]), biphasic ionic liquid (IL) systems (e.g. [9,10]), supported ionic liquid phase (SILP) catalysts (e.g. [11]), supported aqueous phase catalysts (SAPC) (e.g. [12,13]), and polymer attached ligands where the

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141

Nomenclature Latin symbols A cross sectional area, m2 a specific exchange area for flux, m2 /m3 C concentration, kmol/m3 cp heat capacity, kJ/(kmol K) EA activation energy, kJ/kmol h specific enthalpy, kJ/kmol component flux, kmol/(m2 s), or molar flux into/out j of fluid element, kmol/s (kL a) product of mass transport coefficient and specific exchange area based on total reactor volume, s−1 (kL a)liq product of mass transport coefficient and specific exchange area based on total liquid volume (IL + org), s−1 L reactor length, m ˙ m mass flow, kg/s M molecular weight, kg/kmol n amount of substance in fluid element, kmol n˙ molar flow, kmol/s p pressure, MPa q heat flux, kW/m2 R radius (tube), m, or gas constant, kJ/(kmol K) rj reaction rate, kmol/(m3IL s) T temperature, K t residence time, s S selectivity STY space time yield, kmol/(m3 s) superficial velocity of gas or liquid, m/s us ui interstitial velocity, m/s V volume, m3 X conversion x molar fraction liquid phase molar fraction gas phase y z axial coordinate, m Abbreviations and dimensionless numbers set of all components (COM = COMG ∪ COML) COM COMG set of gas components (COMG = {H2 , CO}) COML set of liquid components (COML = {tAlk, inAlk, nAld, isoAld, nAlc, isoAlc, Dec, IL}) DIS set of distributors (level 3) (DIS = {1, . . ., 31}) DoF degree of freedom NR set of reactions set of phases NP SEG set of segment (level 3) (SEG = {1, . . ., 3}) s.t. subject to Greek symbols heat transport coefficient, kW/(m2 K) ˛  phase fraction  stoichiometric coefficient  density, kg/m3 residence time, s  Indices c f i IL j g k liq

coolant final/outlet value component index ionic liquid phase reaction index gas phase phase index total liquid phase

max min org s 0

127

upper bound lower bound organic phase solid phase inlet

Components tAlk 1-octene inAlk 2-octene n-nonanal nAld isoAld iso-nonanal nAlc n-nonanol isoAlc iso-nonanol Dec decane IL [Bmim][PF6 ]

catalyst is retarded by a membrane [14]. In addition, the solventfree hydroformylation [15] and the hydroformylation using scCO2 [16] are current research topics. The objective of this contribution is the design of an optimal reactor for the hydroformylation of long chain alkenes. For this purpose, we have chosen the biphasic ionic liquid system with TPPTS modified Rh as catalyst as investigated by Sharma et al. [10]. The catalyst is present in an IL phase, while reactants and products are present in the organic phase. The IL system ensures a simple and efficient catalyst recycling and the octene solubility in the catalyst phase is significantly improved compared to aqueous systems [10,17]. 2. Optimal design of multiphase reactors Most of the existing reactor design approaches such as the attainable region method or superstructure based approaches have been developed for homogeneous systems. For a comprehensive discussion of the design methods, refer to [18]. In the field of multiphase reactor design methodologies, only few approaches are published (e.g. [19–21]). Krishna and Sie [19] developed a strategy for the selection of multiphase reactors. It is intended to choose from a set of suited reactors from a multitude of existing apparatuses depending on the existing phases (gas/solid, gas/liquid, gas/liquid/solid) and the specific requirements of the investigated reaction system. Mehta and Kokossis [20] proposed reactor configurations for non-isothermal, multiphase systems. The flow pattern (ideally distributed or ideally back-mixed) for the gas and liquid phase can be chosen and in each reactor segment the phases are in equilibrium. Kelkar and Ng [21] proposed a screening method for non-isothermal, multiphase reactors. It is based on a generic reactor model, a sensitivity analysis, and knowledge based heuristics. Despite the fact that these approaches can support the decision of the contacting pattern, of the exchange area, and at the end of the reactor, they can hardly give rise to new apparatuses. The methods are intended to choose from existing reactors [19], are often based on ideal PFTR and CSTR reactors [20], or use a construction set of different reactor parts [21]. In addition, a reactor can be regarded as optimal if the reactant concentrations, temperature, and pressure profiles are in accordance with their optimal profiles. Since these state variables can only be influenced by outer fluxes, the reactor must provide the optimal flux profiles over the reaction coordinate. For this purpose, an adapted reactor design with optimized profiles for the design variables such as the exchange areas is required. The concept of elementary process functions [22] is based on the same idea and provides a framework how to systematically derive intensified

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fluid element 1-octene flux (jtAlk)

heat flux (q)

organic phase IL

IL

IL IL IL

CO flux (jCO)

H2 flux (jH2)

Fig. 1. Material and energy envelope of the balanced fluid element.

processes without an extensive comparison of numerous design options. Here, a fluid element is tracked on its way through the reactor and manipulated by optimal flux profiles. For single phase systems a design methodology for optimal reactors based on this idea was recently presented [18,23,24]. In this publication, we extend the method for the optimal design of multiphase reactors. The main focus is on the optimal concentration and temperature profiles in the gas and liquid phase, and on the optimal gas–liquid mass transfer as well as for the heat transfer. Since the reactions occur in the liquid phase, the gas phase is considered as service phase which can be manipulated in order to provide optimal gas component fluxes into the reactive liquid phase. The method is exemplified on the hydroformylation of long chain linear alkenes. The methodology consists of three design levels where technical restrictions are added on each level. On the first level, the optimal route in state space is calculated without technical restrictions in order to ensure an unlimited design space. This route is obtained by optimal manipulation of a fluid element via energy and material fluxes (refer to Fig. 1). The flux profiles with respect to the reaction coordinate are the solution of a dynamic optimization problem that aims at providing the optimal reaction conditions at each time. Although the fluxes are not restricted by apparatus dependent limitations, the problem is still bounded by intrinsic limitations on this level, e.g., an upper limit for the temperature in order to avoid catalyst deactivation. Different integration and enhancement options are evaluated, and the potential of each option is quantified by comparison with an optimized reference case. The integration concept determines which fluxes are integrated together with the reaction flux in the reactor, e.g., heat flux and component fluxes. The enhancement concept defines which fluxes are actively manipulated in order to obtain the best route in state space. On the second level, the influence of limited mass and energy transport is investigated. The kinetic expressions for mass and energy transport are added, and in case of a multiphase system a sensitivity study with respect to the (kL a)-value is performed. From this study an optimal (kL a)-value with respect to reactor performance and costs can be chosen. In addition, it is investigated if the desired flux profiles are attainable by identifying control variables which can be manipulated by the reactor design and solving the according optimization problems. On the third level, the best possible technical reactor is derived based on the results of level 2. The control variable profiles are approximated and the desired (kL a)-value is realized by the choice of the gas-liquid mixing strategy. The proposed reactor design method is applicable to multiphase systems in general and is not limited to the hydroformylation example since it is rigorously based on balance equations, reaction kinetics, thermodynamic relationships, and system inherent

Fig. 2. Simplified reaction network.

boundaries. The best technical apparatus is derived from the optimal route in state space and not from the comparison of a large set of different reactor set-ups. Following this approach innovative reactor concepts in addition to the existing apparatuses can be obtained. With regard to the hydroformylation, two earlier publications demonstrate the potential of innovative reactor concepts. Wiese et al. [25] showed that the STY of the biphasic RuhrchemieRhône Poulenc Process for short chain alkenes can be significantly increased by applying a reactor which provides higher gas–liquid mass transfer rates compared to the industrial most dominant bubble column reactors. Enache et al. [15] compared a stirred tank batch reactor with a capillary flow reactor with sophisticated temperature control. The reaction rate was increased by a factor of 10 and also the selectivity was slightly increased. These two examples demonstrate the potential of an innovative reactor design for the hydroformylation, but the rigorous design of optimal hydroformylation reactors is still missing. 2.1. Level 1: Integration and enhancement concepts 2.1.1. Reaction system The product spectrum of the hydroformylation consists of the n- and iso-aldehydes, the n- and iso-alcohols, the alkane, internal alkenes, and high boiling components. The extent of the occurrence of the side reactions depends on the catalyst (Rh or Co), the ligand, the solvent, and the reaction conditions. The macroscopic reaction network considered in this work is shown in Fig. 2. The catalyst is present in several modifications (refer to Fig. 3), where each species has a different selectivity and activity. In order to obtain high amounts of n-aldehyde, the HRh(CO)L2 species is desired. The mono-substituted and unmodified species are considered to be highly active, but unselective with regard to the n/iso-ratio and the hydrogenation and isomerization reaction. The tri-substituted and HRh(CO)2 L2 species are regarded as inactive. The catalyst equilibrium depends on the reaction conditions (T, CH2 , CCO ) and hence also the selectivity. In addition to the catalyst equilibrium shown in Fig. 3, the catalyst agglomerates if the hydrogen concentration is too low. According to the Wilkinson cycle [26], a high CO concentration leads to the formation of an inactive species in the catalyst cycle. Therefore, a high CO concentration will reduce the activity of the hydroformylation system. Contrarily, hydrogen is most often assumed to have a linear influence on the reaction rate. Hence, a high H2 :CO-ratio will increase the activity of the hydroformylation system, which was also shown experimentally for CO:H2 ratios

HRh(CO)L3 (inactive)

-L +L

HRh(CO)L2 (selective)

+CO -CO

HRh(CO)2L2 (inactive)

-L +L

HRh(CO)2L (unselective)

Fig. 3. Catalyst equilibrium of the TPPTS modified Rh catalyst.

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141

between 0.8 and 1.2 by Haumann et al. [7]. However, if the hydrogen concentration is too high, the hydrogenation of the reactants and products will be promoted. Despite these promising findings, controlling the H2 :CO-ratio in the design of hydroformylation reactors has not been considered so far. The isomerization reaction indirectly promotes the reaction to the iso-aldehyde via the formation of internal alkenes. Based on the experimental results of Sharma et al. [10] it can be concluded that hardly any iso-aldehyde is directly formed from the terminal alkene in the investigated hydroformylation system. Since the isomerization depends on the 1-octene concentration, controlling the reactant concentration is of major importance for minimization of the iso-aldehyde formation. The discussed dependencies can also be observed in the reaction kinetics (Eqs. (A.5)–(A.9)). Referring to Eq. (A.5), CO has an inhibiting effect on the formation of the linear aldehyde and the optimal CO concentration depends on the alkene concentration. Hydrogen has a positive influence on the formation of the linear aldehyde since the isomerization reaction is not promoted by hydrogen. However, a high hydrogen concentration will also lead to the increased formation of alcohols. The reaction rates are obtained from Sharma et al. [10] and are extended by the temperature dependencies using additional information from the literature [3,10,13,27]. In order to describe the conditions in the hydroformylation reaction system more realistically, the consecutive formation of alcohols is taken into account since alcohols are observed in the product spectrum of the adapted UCC process for higher alkenes [3]. With respect to the consecutive reactions this work should be regarded as a case study since no kinetics of the alcohol formation are available in the open literature and therefore several assumptions had to be made. 2.1.2. Investigated reaction concepts In order to calculate the optimal route in state space, a dynamic optimization problem must be solved. The problem is subject to the component mass balances, the energy balance, the reaction kinetics, thermodynamic relationships such as the solubility of the reactants in the IL, and bounds on temperature, pressure, STY, and conversion (X). The objective is to maximize the n-aldehyde selectivity since it is the desired product. The investigated reaction concepts are summarized in Fig. 4 and explained below with their according optimization problems (OP1 ref), (OP1 CSTR), and (OP1 int). Since the best conversion for the reaction system is not

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known, a conversion study is performed on this level. In order to have comparable conditions for all investigated set-ups, the same amount of 1-octene must be converted in each case (ntAlk,tot and X must be the same). All equations are presented in Appendix A and the numerical solution approach is described in Appendix E. At the first level of the reactor design approach, an optimal, apparatus independent reaction concept, which is a benchmark for the derivation of an optimal reactor, is calculated. For this purpose, the flow field is assumed to be ideal at this stage. Later the influence of the flow field on the reactor performance can be considered in a detailed analysis on level 3. In order to determine the potential of such an intensified reaction concept, a comparison with standard reaction concepts such as the limiting cases of an ideally distributed and an ideally back-mixed system is carried out. Reference case As reference case an isothermal and isobaric batch reactor (corresponds to a continuously operated reactor with PFTR characteristics in steady state) is chosen. It is assumed that the system operates with synthesis gas (pH2 : pCO = 1) and that the gas concentration in the liquid is defined by its saturation limit. The total pressure is a degree of freedom, but it is bounded by an upper limit. Obj =

max S

T,p,Vfrac ,

s.t. Component balances and phase distribution: Eqs. (A.1)–(A.4) Reaction kinetics: Eqs. (A.5)–(A.9) Constitutive equations: Eqs. (A.10) and (A.11) Solubility and gas composition: Eqs. (A.12)–(A.15), pH2 = pCO Intrinsic bounds: Eqs. (A.16) and (A.17) System bounds: Eqs. (A.18)–(A.20) Recycle condition: Eq. (A.21) Conversion: Eq. (A.23)

(OP1 ref)

(OP1 ref)

All additional required parameters, a discussion of the details of the modeling approach and the according assumptions can be found in Appendix A. Intensified case In the intensified case, an optimal heat flux profile as well as optimal H2 , CO, and 1-octene flux profiles into the fluid element are calculated in order to maintain the optimal reaction conditions in terms of T, CH2 , CCO , and CtAlk over the entire reaction coordinate. Controlling the 1-octene flux is meaningful due to the isomerization reaction, which reduces the selectivity towards the linear aldehyde significantly at high 1-octene concentrations. In addition, the optimal gas phase composition will change along the reaction coordinate and hence the fluxes of H2 and CO should be

Fig. 4. Decision structure for the development of an optimal hydroformylation reactor.

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0.8

0.015 Reference case

0.7

Intensified case

0.012

CSTR STY [kmol/m3/h]

0.6

S [−]

0.5 0.4 0.3

0.009

0.006

0.2 0.003 0.1 0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

X [−]

0.4

0.6

0.8

1

X [−]

(a) Selectivity vs. conversion

(b) Space time yield vs. conversion

Fig. 5. Results level 1 – selectivity and space time yield.

controlled. The upper limits for the gas and octene concentrations in the IL are still defined by the solubility. In addition, a minimum STY for each conversion is enforced, which is defined by the according STY of the reference case (refer to Fig. 5(b)). Obj =

S max T (t), pH2 (t), pCO (t), jtAlk (t), ntAlk,0 , Vfrac , 

(OP1 int)

s.t. Component balances and phase distribution: Eqs. (A.1)–(A.4) Reaction kinetics: Eqs. (A.5)–(A.9) Constitutive equations: Eqs. (A.10) and (A.11) Solubility: Eqs.. (A.12)–(A.15) Intrinsic bounds: Eqs. (A.16) and (A.17) System bounds: Eqs. (A.18)–(A.20) Recycle condition: Eq. (A.21) STY: Eq. (A.22) Conversion: Eq. (A.23)

(OP1 int)

CSTR In order to compare the results obtained for the intensified case to an ideally back-mixed system, a CSTR model is formulated and optimized additionally. In case no consecutive reactions occur, a CSTR is a well suited approximation of the dosing profiles obtained in the intensified case [28]. However, considering consecutive reactions the CSTR performs worse compared to the intensified case. The CSTR operates at the outlet conditions and hence the balance equations simplify. All additional required equations are given at the end of Appendix A. The inlet concentration of 1-octene is defined by Eq. (A.19). The temperature, the volume fraction of the IL, and the partial pressures of H2 and CO are degrees of freedom. In contrast to the reference case, it is assumed that the partial pressures of the gases (i.e. H2 and CO) can be adjusted individually. Obj =

max SCSTR

T,p,Vfrac ,

s.t. Component balances: Eq. (A.25) Reaction kinetics: Eqs. (A.5)–(A.9) Constitutive equations: Eqs. (A.26)–(A.28), (A.11) Intrinsic bounds: Eqs. (A.16) and (A.17) System bounds: Eqs. (A.29) and (A.30) Recycle condition: Eq. (A.31) Conversion: Eq. (A.32)

(OP1 CSTR)

(OP1 CSTR)

2.1.3. Results The results of the calculations of level 1 are summarized in Fig. 5. The attainable selectivities (Fig. 5(a)) and the according STY (Fig. 5(b)) for each reaction concept depending on the conversion are shown.

For the reference case, the selectivity reaches a maximum at a conversion of 90% followed by a rapid decrease for higher conversions. For a decreasing 1-octene concentration the rate of the parallel reaction decreases faster than the rate of the hydroformylation and hence the selectivity increases up to a certain conversion (refer to Fig. 6(c), differential selectivity). However, larger residence times in the reactor, which are necessary to obtain higher conversions, promote the consecutive reaction and thus the selectivity reduces. In the intensified reaction concept the optimal trade-off between parallel and consecutive reaction is obtained over the entire residence time since the 1-octene concentration, H2 , and CO concentration as well as the temperature are optimally controlled. At low conversions the intensified case and the CSTR yield almost the same selectivity and STY. Due to the minor importance of the consecutive reaction at low residence times, the optimal 1-octene profile matches the characteristics of an ideally back-mixed system which is in accordance to the results of our earlier studies [28]. At higher conversion, selectivity and STY of the CSTR are significantly reduced and hence a CSTR is not able to approximate the intensified reaction concept anymore. Applying the intensified reaction concept the selectivity can be maintained close to its maximum up to a conversion of 80%. For the choice of the desired conversion in the reactor plant wide considerations have to be taken into account. A higher conversion will lead to lower separation costs due to lower recycle flows. Hence, there is a trade-off between the selectivity and the conversion. In order to reasonable satisfy this trade-off, we have chosen a conversion of 80% for the further investigations. The profiles for the concentration of the liquid components in the pseudo-liquid phase (IL + organic phase), the reaction rates, the differential selectivity, and the flux profiles are shown in Fig. 6 for the reference case (left hand side) and for the intensified case (right hand side). For a better comparison, all fluxes are normalized with their respective maximum value, and the reaction rates are normalized with the maximum rate observed in the discussed case. All normalized variables are summarized with the according reference values in Appendix D. Fig. 6(a) illustrates the liquid phase concentration profiles for the reference case. The 1-octene profile is in accordance with a typical reactant profile in a batch process (or reactor with PFTR characteristics). The optimal pressure and temperature are both at their respective maximum value (not shown in figure). In order to obtain saturated conditions for H2 and CO in the liquid phase and a constant temperature, non-trivial flux profiles are required as shown in Fig. 6(e). On the right hand side of Fig. 6, the profiles for the intensified case are shown. As can be seen from the comparison of Fig. 6(a)

1.4

0.8

1.2

0.7

1.0

0.6 3

C [kmol/mliq]

3

C [kmol/mliq]

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141

0.8 0.6 0.4

tAlk inAlk nAld isoAld nAlc isoAlc

0.4 0.3 0.2

0.3 0 0

0.5

131

0.1 20

40

60

80

100

0 0

120

20

40

60

t [s]

80

100

120

140

t [s]

(a) Reference: Concentrations

(b) Intensified: Concentrations

1.0

1.0

0.9

0.9

0.8

0.8

r

0.7

0.7

r

0.6

r

Sdiff r

[−], Sdiff [−]

0.6 0.5

rel

0.4

0.5

2 3 4

r

5

0.4

r

r

rel

[−], Sdiff [−]

1

0.3

0.3

0.2

0.2

0.1

0.1

0 0

20

40

60

80

100

0 0

120

20

0.9

0.8

0.8

−q j

0.7

0.7

jCO

0.6

0.6 [−]

1.0

0.9

rel

0.5

60

80

100

120

140

(d) Intensified: Reaction rates, Sdiff

1.0

j

tAlk

H2

0.5

j

j

rel

[−]

(c) Reference: Reaction rates, Sdiff

40

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 0

20

40

60

80

100

120

t [s]

(e) Reference: Fluxes

0 0

20

40

60

80

100

120

140

t [s]

(f) Intensified: Fluxes

Fig. 6. Results level 1 – reference (l.h.s.) and intensified case (r.h.s.) (X = 0.8).

and (b), both cases differ significantly. The 1-octene concentration starts at the lowest value allowed by the recycling condition (Eq. (A.21)) and decreases to approximately 0.2 kmol/m3liq followed by an increase up to 0.75 kmol/m3liq caused by the dosing of 1-octene (refer to Fig. 6(f)). After a residence time of 80 s the 1-octene flux vanishes and hence 1-octene is consumed to its outlet value defined by the conversion. It should be noted that the dosing profile of 1-octene is optimized as a continuous and free control function, i.e. any dosing profile of 1-octene is allowed, and is not a priori restricted to three

segments as it is done by other design approaches [29]. Within the systematic staging of reactors [29] a different model formulation, which directly assumes a reactor geometry, is used and the reactor is segmented a priori. Hence, no apparatus independent route can be determined, and a sub-optimal reactor might be derived. The 1octene flux is restricted to be positive since a selective withdrawal of 1-octene from the system is not reasonable being a reactant component, and also hardly feasible. The unconventional dosing profile of 1-octene can be explained with the fact that the same amount of 1-octene must be converted

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1.0

0.8

3

C [kmol/mliq]

0.6 0.5

normalized control profiles [−]

0.7

tAlk inAlk nAld isoAld nAlc isoAlc

0.4 0.3 0.2

jtAlk,rel

0.8

Tc,rel

0.7

p

0.6

pCO,rel

H2,rel

0.5 0.4 0.3 0.2

0.1 0 0

0.9

0.1 20

40

60

80 t [s]

100

120

140

0 0

160

20

40

60

80 t [s]

100

120

140

160

(b) Control variables: Tc, pi

(a) Concentrations

C [kmol/m3]

0.15 H equil.

organic phase 0.1

2

H2

IL phase

CO equil. CO

0.05

0 0

20

40

60

80 t [s]

100

120

140

160

(c) Gas component concentrations Fig. 7. Results level 2 (X = 0.8, (kL a)liq = 1 s−1 ).

as in the reference case. In the intensified case, the reactant can be present at the beginning and in addition it can be dosed over the residence time. The minimum amount of 1-octene present at the beginning is defined by the recycle condition Eq. (A.21), which defines that the unconverted amount of 1-octene is recycled with the organic solvent (refer to Fig. 9). At the beginning of the reaction, the optimal alkene dosing profile increases the selectivity of r1 over r2 since r2 has a higher order dependency on the 1-octene concentration than r1 (compare Eqs. (A.5) and (A.6)). However, 1-octene is not kept at a low concentration all the time since the consecutive reactions become increasingly important for longer residence times and higher naldehyde concentrations. The dosing profile of 1-octene causes kinks in the 1-octene concentration and subsequently in all other fluxes. Since the optimal heat flux profile aims to maintain the temperature at its upper limit, it is approximately proportional to the rate of the dominant reaction (r1 ), which depends on the 1-octene concentration. The optimal CO concentration is inversely proportional to the 1octene concentration (refer to Eq. (A.5)), and hence CO is adjusted accordingly. Since H2 is correlated to CO by the maximum pressure constraint (Eq. (A.17)), the H2 flux is also linked to the CO flux. The realization of the optimal CO and H2 fluxes will probably require a varying H2 :CO-ratio in the gas phase, which is part of the investigations on level 2. The optimal temperature and pressure of the system are always at their respective upper boundaries given by Eqs. (A.16) and (A.17), respectively. Regarding the ionic liquid phase, the optimal H2

concentration is much higher than the optimal CO concentration even if the fluxes are almost the same (refer to the equilibrium concentration in Fig. 7(c) for level 2). This leads to the conclusion that the selectivity can be enhanced further by a higher temperature and H2 concentration in the ionic liquid phase. The latter can either be achieved by increasing the total pressure, changing the gas composition, or by concepts such as of gas expanded liquids (e.g. [30], [31]). 2.2. Level 2: Heat and mass transport On level 2, the influence of heat and mass transport is studied and suited transport mechanisms and control variables, which make the desired fluxes attainable, are identified. The heat flux is realized by indirect cooling and it is controlled by adapting the cooling temperature Tc along the reaction coordinate. The 1-octene flux is provided by ideal dosing into the liquid phase. Later, this can be approximated by a membrane, by a perforated tube, or by discrete dosing points. The H2 and CO fluxes into the liquid phase depend on the (kL a)value and the respective partial pressure. In order to control the individual gas component fluxes, the partial pressures are controlled by ideal manipulation of the gas phase. A mass balance for the gas phase is not solved on this level, instead the partial pressure profiles are directly optimized. This ideal treatment of the gas phase is chosen in order to reduce the model complexity and to focus on the gas-liquid mass transfer. By performing the proposed (kL a)-study, the influence of limited mass transfer between the

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141

133

−3

0.74 ΔS

max

6.2

0.72 S [−]

x 10

= 1%

STY [kmol/m3/h]

0.73

6.5

0.71

desired (k a) −range L

liq

0.70

5.9

5.6

5.3

0.69 0.68 −1 10

0

1

10 (k a) [s−1] L

10

5 −1 10

liq

STYmin

0

10 (k a) [s−1] L

(a) Selectivity depending on (kLa)liq

1

10

liq

(b) STY depending on (kLa)liq

Fig. 8. Results level 2 – (kL a)-study (X = 0.8).

gas and the liquid phase is investigated and a minimum desired (kL a)-value is identified. The resulting optimization problem for each investigated (kL a)-value is stated in (OP2). All equations that are additionally required to the equations of level 1 are given in Appendix B. The inlet amount of H2 and CO (nH2 ,liq,0 , nCO,liq,0 ) in the liquid phase are calculated from saturation conditions at the inlet. Obj =

S max Tc (t) , To , pH2 (t) , pCO (t) , jtAlk (t) , ntAlk,0 , Vfrac , 

s.t. Component balances and phase distribution: Eqs. (A.1)–(A.4) Energy balances: Eqs. (B.1)–(B.3) Reaction kinetics: Eqs. (A.5)–(A.9) Transport kinetics: Eqs. (B.4)–(B.7) Constitutive equations: Eqs. (A.10) and (A.11) Solubility: Eqs. (A.12)–(A.15) Intrinsic bounds: Eqs. (A.16) and (A.17) System bounds: Eqs. (A.18)–(A.20) Recycle condition: Eq. (A.21) STY: Eq. (A.22) Conversion: X=0.8 G/L mass transfer: (kL a) according to investigated case

(OP2)

(OP2)

2.2.1. Results In Fig. 7(a) the concentration profiles of the liquid phase are shown. The concentration profiles look qualitatively similar to the profiles of level 1, but due to the limited gas–liquid mass transfer the residence time increased slightly and the maximum and minimum concentration of 1-octene slightly shifted as well. The control variables for the fluxes, namely the cooling temperature, the partial pressure of H2 and CO, and the dosing flux of 1-octene are presented in Fig. 7(b) with the definition of the normalized variables given in Appendix D. The optimal cooling temperature profile (refer to Fig. 7(b)) provides the optimal fluid temperature (Tmax ) over the entire residence time and hence the required heat flux is completely attainable. It is calculated for a specific heat exchange area based on the liquid phase of aliq = 20 m2 /m3 . For this exchange area a maximum temperature difference between cooling and fluid temperature of 18.8 K arises. From this result it can already be concluded that there is no need to investigate reactor concepts with a very high heat exchange area (e.g. a micro reactor) for this reaction system. The total gas pressure is always at its maximum. In Fig. 7(c) the concentration of the gas phase components in the organic phase and in the ionic liquid are shown for (kL a)liq = 1 s−1 . The equilibrium concentrations depend on the temperature and the individual

gas pressure and are also plotted in the diagram. Since the partial pressures of the gases are not constant, the equilibrium concentrations are also not constant. Even for this high (kL a)liq -value, it can be observed that the gas concentrations in the liquid are still not at their equilibrium values. Fig. 8 illustrates how selectivity and STY depend on the gasliquid mass transfer. For high (kL a)liq -values (≈10 s−1 ) almost the maximum potential of the reaction system with respect to selectivity and STY identified in level 1 can be achieved. At a (kL a)liq of 0.6 s−1 , the minimum required STY defined by the reference case of level 1 (refer to Fig. 8(b)) is reached. For lower (kL a)liq -values the required STY can only be obtained by higher 1-octene concentrations in the reactor. The selectivity decreases significantly already for (kL a)liq ≤ 1 s−1 . Assuming a maximum acceptable selectivity loss of 1%, the (kL a)liq -value should be above 1 as indicated in Fig. 8(a). From the results of level 2 it can be concluded that the chosen control variables (Tc , pi , direct dosing) are suited to attain the optimal profiles and besides, high gas-liquid mass transfer rates ((kL a)liq ≥ 1 s−1 ) are required for a good performance of the investigated hydroformylation reaction system. 2.3. Level 3: Technical approximation In order to obtain the best technical reactor, the optimal control variable profiles of level 2 have to be approximated by a technically feasible design. In addition, a gas–liquid mixing strategy has to be chosen which can provide the desired (kL a)-value and plug flow characteristics. In summary, the catalog of requirements for the best technical reactor is given by: • High gas–liquid mass transfer rates. • Design can be manipulated to approximate the optimal control variable profiles. • Plug flow characteristics. For the investigated system, bubble columns are not a promising option due to the low gas–liquid mass transfer rates. However, it should be noted that the design of bubble columns is very flexible, and therefore they are still attractive for gas–liquid systems with lower reaction rates. Since several mixing strategies exist which fulfill the defined catalog of requirements, the gas–liquid mixing strategy with the lowest energy consumption with respect to the provided (kL a)value is the preferred reactor concept. For this purpose, Nagel et al.

134

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141

Fig. 9. Best technical reactor configuration.

[32,33] derived correlations for the required energy in dependence of the (kL a)-value for several gas–liquid reactors. Grosz-Röll et al. [34] compared the energy demand of stirred tank reactors, jet loop reactors, and static mixers and found that static mixers offer the highest (kL a)-value for a given energy demand. Furthermore, it was shown that the residence time behavior of static mixers is approximately plug flow like (e.g. [35]) and that static mixers can be used for gas–liquid–liquid systems [25]. Other concepts such as the capillary flow reactor, which was successfully demonstrated for gas-liquid-solid reactions [36] and also for the hydroformylation [15], are not taken into account for the technical realization since the energy demand in dependence of the (kL a)-value is not known. Hence, these reactor types can hardly be compared. In addition, it was shown on level 2 that the required heat exchange area is small and hence no reactor concept with high heat exchange areas is required. From the control function profiles of level 2 (refer to Fig. 7(b)) it is evident that a reactor with three segments seems to be adequate to approximate the desired profiles. In the first segment, additional gas is dosed into the reactor at three discrete dosing points. Since H2 and CO are approximately equally consumed, synthesis gas (H2 :CO=1) can be used to maintain the H2 :CO–partial pressure ratio approximately constant. By using perforated pipe distributors (as shown in [37]) the gas is equally distributed over the pipe radius. The cooling temperature is approximated using a co-current heat exchanger since the optimal cooling temperature rises almost linearly. The optimal profiles in the second segment are approximated by dosing additional synthesis gas (H2 :CO = 1) and 1-octene in combination with a counter-current heat exchanger. Here, 20 discrete dosing points are used for the gas as well as for the 1-octene. Again perforated pipe distributors are used to obtain a good radial mixing at the dosing points. The third segment features the same principle design as the first segment, except that 8 dosing points are selected. The number of dosing points is chosen in order to provide nearly equidistantly spaced dosing over the entire reactor length. In principle, the axial location of the dosing points can be optimized, which might yield a reactor with even better performance. However, including the choice of position for every dosing point increases the number of degrees of freedom on level 3 significantly and complicates the model formulation. Hence, the maximum potential of distributing the dosing point in an optimized manner is obtained by comparing the objective function value of levels 2 and 3 first with fixed dosing points. In case the losses due to the fixed dosing points and the approximated control variable profiles are small, there is no need to change the derived reactor set-up. The proposed reactor configuration is shown in Fig. 9. The dosing of gas and 1-octene is only shown symbolically and the static mixer elements are represented by crossed boxes. To model the shown reactor set-up, additional balance equations for the gas and coolant phase as well as for the dosing points

are required. The static mixer investigated by Heyouni et al. [38] is selected since correlations to calculate the local (kL a)-value and pressure drop for these static mixers are available. Here, the local formulation of the balance equations is used since the local velocities of the gas and liquid phase are required to calculate the (kL a)-value and pressure drop, and also the dimensions of the reactor are of interest. In contrast to the previous levels where optimal control problems were solved, in level 3 we now deal with a parameter optimization problem since all phases (liquid, gas, and coolant) and all fluxes are rigorously modelled. The according optimization problem is formulated in (OP3). All required equations are presented in Appendix C. The inlet streams of H2 and CO (n˙ H2 ,liq,0 , n˙ CO,liq,0 ) in the liquid phase are calculated from saturation conditions at the inlet. Obj =

max S Rh , Lh , Tc,0,h , Kc,0,h , T0 , n˙ liq,0 , n˙ gas,0 , n˙ tAlk,liq,0 , n˙ tAlk,org,d , n˙ gas,org,d , pH2 ,0 , pCO,0 , Vfrac

s.t. Component balances and phase distribution: Eqs. (C.1)–(C.4), (A.2), and (A.3) Component balances distributor: Eqs. (C.5) and (C.6) Energy balances: Eqs. (C.7)–(C.11), (B.2), and (B.3) Energy balances distributor: Eq. (C.12) Momentum balances: Eq. (C.14) Reaction kinetics: Eqs. (A.5)–(A.9) G/L mass transport: Eq. (C.13) Constitutive equations: Eqs. (C.16)–(C.23), (A.11) Solubility: Eqs. (A.12)–(A.15) Intrinsic bounds: Eqs. (A.16) and (A.17) System bounds: Eqs. (C.24)–(C.26) Recycle condition: Eq. (C.27) STY: Eq. (C.28) Conversion: Eq. (C.29) Coolant side: Eq. (C.30)

(OP3)

(OP3)

2.3.1. Results The selectivity of the optimized reactor set-up is 71.2%. Due to the non-ideal control profiles and the pressure drop, this value is slightly below the selectivity of level 2. However, it is remarkably higher than the selectivity which can be obtained with the reference reactor set-up or a CSTR on level 1 (refer to Fig. 5(a)). In addition, it should be noted that the reference case and the CSTR are calculated with ideal gas–liquid transfer and the losses due to the limited (kL a)-value are not yet considered. The results for the concentration profiles, the amount of dosed 1-octene and synthesis gas, the fluid and cooling temperature, the total pressure, and the gas composition for the optimized reactor are shown in Fig. 10. In Fig. 10(a), the profiles for the liquid phase concentrations are given. Due to the discrete dosing of 1-octene (refer to Fig. 10(b)), the concentration of 1-octene occurs as zig–zag profile in segment 2. More discrete dosing points will smooth this profile and will also

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141

135

1.0 tAlk

0.7

0.7

nAlc

0.6

0.4

isoAlc

rel,i,h

[−]

isoAld 0.5

n

liq

0.8

nAld

0.6 C [kmol/m3 ]

syngas tAlk

0.9

inAlk

0.3

0.5 0.4 0.3

0.2 0.2 0.1 0

0.1 0 0

20

40

60 z [m]

80

100

0

120

20

40

60

80

100

120

z [m] (b) Discrete dosing: n˙ gas,add , n˙ tAlk,add

(a) Concentrations 1.0 370

0.9

Segment 1 Segment 2

Segment 3

0.8 350

0.6

T, Tc [K]

p

rel

[−], y [−]

0.7

0.5 0.4 0.3

prel

0.2

yH2

0.1

yCO

0

0

20

40

60 z [m]

80

100

120

T Tc

330

310

290

0

20

40

60

80

100

120

z [m] (d) Temperatures: T, Tc

(c) Gas composition, pressure Fig. 10. Results level 3 (X = 0.8).

improve the reactor performance since the ideal profile can be better approximated. However, the qualitative trend is in accordance to levels 1 and 2. As shown in Fig. 10(c), the gas composition cannot be ideally controlled using synthesis gas with a composition of H2 :CO = 1. However, the dosing of synthesis gas can keep the gas phase composition almost constant. Using another gas composition will improve the gas composition profile and hence the reactor performance, but from an economic point of view this does not seem to be worthwhile. In addition, the gas dosing maintains the volumetric gas fraction high, which improves the gas–liquid mass transfer. At the inlet the gas fraction is at its maximum, which was defined based on the values given in the literature [37] for static mixers. The pressure drop, which was neglected on levels 1 and 2 so far, is relatively high (refer to Fig. 10(c)). It reduces the STY and selectivity of the system and can also be regarded as a loss caused by the technical realization. The mean (kL a)liq -value obtained in the technical reactor is 0.75 s−1 , which is below the desired value of 1 s−1 . The (kL a)-value can be further enhanced by a higher fluid velocity, however, in this case the pressure drop will also increase. Since the superficial fluid velocity (ug,s,0 = 0.25 m/s) is not at its bounds, the (kL a)-value and pressure drop are in an optimized balance. The approximation of the cooling profile is very good and almost the optimal temperature profile is obtained (refer to Fig. 10(d)). Hence, the chosen cooling strategy is absolutely appropriate for the temperature control. The optimization results for the decision variables which cannot be obtained from Fig. 10 are given in Table 1.

2.4. Summary Within the followed design methodology, the reactor design task is divided into a sequence of three optimization problems. On the first level, an apparatus independent optimal reaction concept is determined. This optimal reaction concept is not yet a reactor, but defines the benchmark for all possible reactors. On the second level, the influence of limited mass and energy transport arising from a chosen reactor concept is quantified. It may be the case that a certain loss in the objective must be accepted in order to derive an adequate reaction concept. On the third level, the optimal control variable profiles are approximated by a technical reactor set-up and again the influence of the technical approximation is quantified. In case the chosen reactor set-up does not approximate the optimal reaction concept well, the objective function value will not come close enough to the benchmark value calculated on level 1. In this

Table 1 Optimized design and operating variables. Decision variable

Segment

Value

Tube radius, R [m] Segment length, L [m] Inlet cooling temperatures, Tc,0 [K] Coolant constant, Kc [K/kW] Inlet temperature, T0 [K] Total liquid inlet flowrate, n˙ liq,0 [kmol/s] Total gas inlet flowrate, n˙ gas,0 [kmol/s] Vfrac

S1–S3 S1–S3 S1–S3 S1–S3 S1 S1 S1 S1

0.11, 0.11, 0.10 5.28, 71.99, 47.14 316.4, 298.4, 302.6 0.1487, −0.0317, 0.0549 373.0 0.0501 0.0196 1.5

136

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141

Table 2 Result comparison level 1–3.

Appendix A. Model equations of level 1

Level

Characteristics and insight gained

Selectivity [%]

Reference case Level 1

Large potential for optimal reactor design All fluxes are required, CtAlk defines optimal value of CCO , best conversion from overall process point of view (kL a)liq ≥ 1 s−1 desired, pH2 :pCO ≈ 2 3 segments are adequate to approximate the profiles, static mixers are well suited to provide (kL a)-value

62.1 73.8

Level 2 Level 3

72.8 71.2

case, a different reactor set-up regarding the phase mixing or different control variables are necessary. This will lead to an iterative design process, but the result of level 1 for the optimal reaction concept will not change and will still be the benchmark for every other reactor design. As summarized in Table 2, the losses due to limited mass and energy transport (compare levels 1 and 2) and due to non-ideal control profiles (compare levels 2 and 3) are small. The overall losses due to the non-ideal control profiles, the limited gas–liquid transfer, and the pressure drop sum up to only 2.6%. Therefore, it can be concluded that the proposed reactor design using three segments is well suited and can only be slightly improved, e.g., by an increasing number of dosing points and a flexible gas composition of the dosed gas. Since this improvement does not seem to be worthwhile to realize, no iteration in the design process is necessary and the best technically possible reactor is designed. Designing an optimal reactor using just one optimization problem instead of the propose sequence of optimization problems, leads to excessive enumeration of different reactor set-ups, since the multitude of existing technical apparatuses for gas–liquid reactions is very large. The requirements on the gas–liquid mass transfer and on the optimal reaction concept must be known in order to reduce the number of possible optimal reactors. However, this information is provided by the proposed three level approach, for example level 2 gives rise to the adequate choice of gas–liquid contactor. Thereby, excessive enumeration of possible reactors is avoided.

Sharma [44] showed that the direct transport from the gas phase into the dispersed IL phase can be neglected compared to the transport from the gas phase into the continuous organic phase and subsequently into the IL phase. In addition, the transport of gas and 1-octene from the organic phase into the IL phase is much faster than the reaction [44]. Hence, a pseudo-homogeneous model of the liquid phase is used where the organic phase and the IL phase are in equilibrium. However, between the gas and liquid phase the mass transport is rigorously considered. In order to reduce the model complexity, the temperature profile is directly taken as an optimization profile for the intensified case of level 1 since the heat flux is unrestricted and hence every possible temperature profile is attainable. Component mass balance and org/IL phase distribution The mass balance for each component is given by Eq. (A.1). It is assumed that decane is not soluble in the IL phase and vice versa, which is in good agreement with the experimental findings [10]. On level 1, Eq. (A.1) has only to be solved for liquid phase components (i  COML) since the concentrations of the gas phase component are directly calculated using the according solubility. On level 2, Eq. (A.1) for H2 and CO in the liquid phase is additionally required (i  COM). dni,liq dt

Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft (DFG) (TRR 63) and by the International Max Planck Research School for Analysis, Design and Optimization in Chemical and Biochemical Process Engineering (Magdeburg, Germany) is gratefully acknowledged.



i,j · rj · VIL

(A.1)

j ∈ NR

The distribution of the liquid phase components between organic and IL phase are calculated using Eq. (A.2) combined with Eq. (A.4). The phase equilibrium can only be approximated since equilibrium data is hardly available and no parameters for activity coefficient models or equation of state models are available. For alkenes and aldehydes the K-values are mean values obtained from the literature [10] (KtAlk = KinAlk = 1.00 × 10−2 , KnAld = KisoAld = 1.61 × 10−3 ). It is assumed that the alcohols can be described with the K-values of the aldehydes. The K-values of the gas components are considered to behave like the inverse of the Henry coefficients stated in Eq. (A.3). Ki =

wi,IL , wi,org

Ki =

−1 Hi,IL xi,IL · Ctot,IL = −1 , xi,org Hi,org

2.5. Conclusion The proposed reactor design approach is suited to design multiphase reactors with complex reaction networks. The methodological approach is based on the realization of the optimal flux by a suited technical reactor. It includes existing as well as innovative reactor concepts as demonstrated for the investigated case study of the hydroformylation. The approach is superior to conventional reactor selection heuristics since it is able to yield non-intuitive results which originate from the strongly coupled nonlinear interaction of complex multiphase systems. In general, it can be applied to all gas-liquid systems where the gas phase is considered as service phase. By using the proposed (kL a)-study, the influence of limited gas-liquid transfer can be investigated and a very specific catalog of requirements for the selection of the best suited gas–liquid reactor is obtained.

= ji +

i = tAlk, inAlk, nAld, isoAld, nAlc, isoAlc

ni,liq = ni,IL + ni,org ,

i COMG

i COML

(A.2)

(A.3) (A.4)

Reaction rates The reaction rates for the hydroformylation of the terminal and internal alkene, and for the isomerization are obtained from Sharma et al. [10]. The model is extended by taking the temperature dependency of the reaction rates into account. The activation energy of r1 is taken from Sharma et al. [10], while the activation energy of the isomerization reaction is obtained by regression of data from Disser et al. [13]. The activation energy of the formation of isomeric aldehyde is assumed to be the same as for the n-aldehyde formation, which is based on the observations of Bernas et al. [27] for short chain aldehydes. In order to model the hydrogenation of the aldehydes, several assumptions need to be taken. Based on first principles, the alcohol formation is assumed to linearly depend on the aldehyde, the H2 , and the catalyst concentration in the IL phase. The hydrogenation of the n- and iso-aldehyde are assumed to have an identical activation energy and frequency factor. Based on the selectivity towards the aldehydes of the adapted UCC process for higher alkenes (S = 95%) [3], it is assumed that 5% of the converted alkene reacts to alcohols. Taking the reference case stated in Peschel et al. [28] and assuming

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141 Table 3 Parameters of reaction rates.

Table 5 Parameters of Henry coefficients.

Reaction 1: Hydroformylation of terminal alkene 2: Isomerization of terminal alkene 3: Hydroformylation of internal alkene 4: Alcohol formation from n-aldehyde 5: Alcohol formation from iso-aldehyde

EA [kJ/mol]

k0 [(kmol/(m3 ))−n ]

107.9 43.6 107.9 53.95 53.95

40.65 × 10 , n = 3 11.16 × 106 , n = 1 33.51 × 1018 , n = 3 1.233 × 108 , n = 2 1.233 × 108 , n = 2 18

an activation energy of half of that of the n-aldehyde formation, the frequency factor of the alcohol formation was estimated. r1 = k0,1 · exp r2 = k0,2 · exp r3 = k0,3 · exp r4 = k0,4 · exp r5 = k0,5 · exp

 −E

A,1

 C tAlk,IL · CH2 ,IL · CCO,IL · Ccat,IL

A,2

R·T

 −E

A,3

R·T

 −E

A,4

R·T

 −E

A,5

   

R·T

 ni,k · Mi i  COML

i

i = A,i · T + B,i ,

−1 ∗ = Hi,org · pi , xi,org

4.6936 × 10 1.076 × 10−2

CO H2

AH,i,org [1/MPa]

BH,i,org [1/(MPa K)]

CO H2

6.3853 × 10−3 −7.8249 × 10−3

2.4456 × 10−5 4.3979 × 10−5

(A.6)

353 K ≤ T ≤ 373 K

(A.16)

1 bar ≤ p ≤ 60 bar

(A.17)

· CinAlk,IL · CH2 ,IL · CCO,IL · Ccat,IL

(A.7)

· CnAld,IL · CH2 ,IL · Ccat,IL

(A.8)

· CisoAld,IL · CH2 ,IL · Ccat,IL

(A.9)

k = IL, org i  COML

i  COMG

System bounds The total amount of 1-octene (ntAlk,tot ) must be the same for all cases and is chosen as scaling variable for the optimization. The initial amount of 1-octene (ntAlk,0 ) in addition to the amount added along the reaction coordinate must match the total amount, which is the same for all calculations.



tf

ntAlk,tot = ntAlk,0 +

jtAlk dt

(A.18)

t0

In the investigated system, decane is present as solvent in the organic phase. In order to ensure conditions comparable to the conditions at which the experiments for the reaction kinetics where conducted [10], the amount of decane with respect to the total amount of octene is specified according to Eq. (A.19). CtAlk =

ntAlk,tot ntAlk,tot · MtAlk /tAlk (373 K) + nDec · MDec /Dec (373 K)

i  COMG

pH2 + pCO = p

(A.19)

(A.10) (A.11)

The volume of organic solvent compared to the volume of IL is restricted to be in certain limits in order to ensure that the IL phase min = 1.5, V max = 5) (calculated is dispersed in the organic phase (Vfrac frac at 373 K) according to Eq. (A.20). min Vfrac ≤

VDec max ≤ Vfrac VIL

(A.20)

(A.13)

Recycle condition In order to capture the recycling of the solvent and unreacted alkene, the initial amount of alkene (ntAlk,0 ) must be greater than or equal to the amount of alkene at the end of the reaction (ntAlk,f ) (refer to the separation scheme shown in Fig. 9).

(A.14)

ntAlk,0 ≥ ntAlk,f

(A.12)

i  COMG

−1 = AH,i,k + BH,i,k · T, Hi,k

−8.6482 × 10−5 2.645921 × 10−6

Component

= 2 kmol/m3 ,

BH,i,IL [kmol/(m3 Mpa K)]

· CtAlk,IL · Ccat,IL

· CtAlk,IL

Solubility The concentration of H2 and CO in the IL and in the organic phase are limited by their solubility, which depends on the temperature and the according partial pressure. The solubility is modelled using Henry’s law based on the experimental data given by Sharma et al. [10], Srivatsan et al. [39], and Park et al. [40] and the parameters are summarized in Table 5. −1 ∗ Ci,IL = Hi,IL · pi ,

−2

Intrinsic bounds On all levels, bounds on the pressure and temperature range are enforced in order to ensure the validity of the reaction kinetics and the solubility model.

3 1 + KCO · CCO,IL

All reaction rate parameters are summarized in Table 3. The concentration of the TPPTS modified Rh catalyst in the IL phase is chosen constant Ccat,IL = 5 × 10−2 kmol/(m3IL ). The CO adsorption constant is KCO = 1.023 × 106 (kmol/(m3 ))−4 [10]. Constitutive equations The volumes of the organic and IL phase are calculated using a linear mixing rule and neglecting the influence of the gases and the catalyst (refer to Eq. (A.10)). The densities of all liquid phase components are considered as temperature dependent and given by Eq. (A.11). The parameters are fitted in the range 353 K ≤ T ≤ 373 K to data given in the literature [41,42] (Table 4). Vk =

AH,i,IL [kmol/(m3 MPa)]

Component

(A.5)

·

R·T

 −E

137

(A.15)

Table 4 Parameters for density calculation. Component

A,i [kg/(m3 K)]

B,i [kg/m3 ]

Decane 1-Octene 2-Octene n-Nonanal iso-Nonanal n-Nonanol iso-Nonanol [bmin][PF6 ]

−0.8597 −1.0240 −0.9869 −0.7925 −0.9777 −0.7610 −0.9704 −0.8052

990.1 1025.3 1022.6 1061.0 1165.0 1055.4 1125.4 1604.2

(A.21)

STY, X, and S In order to compare all investigated cases on a fair basis, a minimum space time yield depending on the conversion is defined. Here, the STY with respect to the liquid phase of the reference case is used as a lower bound for all intensified cases (refer to Fig. 5(b)). The conversion of 1-octene is given by Eq. (A.23). STY (X) =

nnAld,f − nnAld,0

 tf t0

X= S=

(VIL + Vorg )dt

ntAlk,tot − ntAlk,f ntAlk,tot nnAld,f − nnAld,0 ntAlk,tot − ntAlk,f

≥ STY min (X)

(A.22)

(A.23) (A.24)

138

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141

CSTR model The equations describing the CSTR are summarized in a comprehensive manner below.



n˙ i,f − n˙ i,0 = IL VR

i,j · rj ,

i  COML

(A.25)

j ∈ NR

The fraction of IL is calculated using Eq. (A.26) and VR is total liquid volume inside the reactor.

IL =

V˙ IL

(A.26)

V˙ IL + V˙ org

VR = (V˙ IL + V˙ org )

(A.27)

The volume flow of IL and organic phase is calculated by Eq. (A.28).

 n˙ i,k · Mi

V˙ k =

i

i  COML

,

k = IL, org

(A.28)

For comparison, the amount of solvent is connected to the amount of terminal alkene as in the reference case (Eq. (A.29)). The volume fraction of IL and organic phase at the inlet are also restricted by the same bounds as before (Eq. (A.30)). n˙ tAlk,0

CtAlk =

n˙ tAlk,0 · MtAlk /tAlk (373 K) + n˙ Dec · MDec /Dec (373 K)

= 2 kmol/m3

(A.29)

V˙ Dec max ≤ Vfrac V˙ IL

min Vfrac ≤

(A.30)

In case of a CSTR, the conversion and the selectivity are given by Eqs. (A.31), (A.32), and (A.33), respectively. n˙ tAlk,0 ≥ n˙ tAlk,f X=

(A.31)

n˙ tAlk,0 − n˙ tAlk,f

(A.32)

n˙ tAlk,0

SCSTR =

n˙ nAld,f − n˙ nAld,0

(A.33)

n˙ tAlk,0 − n˙ tAlk,f

Appendix B. Model equations of level 2 All additionally required equations to level 1 are summarized in this appendix. Energy balance The energy balance is formulated in terms of temperature assuming no technical work, and negligible influence of the pressure change on the temperature (refer to Eq. (B.1)). The total heat flux qtot in Eq. (B.1) accounts for the sensitive heat flux times the exchange area and for the enthalpy flux caused by the dosed components. In this energy balance, the contribution of the dissolved H2 and CO is neglected.







n˙ i,liq · cp,i



dT =− dt

qtot + VIL

i  COML







hi i  COM



i,j · rj

(B.1)

j ∈ NR

The heat capacity and enthalpy of each component is calculated according to Eqs. (B.2) and (B.3), respectively. The parameters were fitted in the investigated temperature range to data given in the literature [41,43] and are presented in Table 6. cp,i = Acp,i + Bcp,i · T + Ccp,i · T 2 , hi = Acp,i · T +

Bcp,i 2

· T2 +

Ccp,i 3

i  COM

· T 3 + Fi ,

(B.2) i  COM

(B.3)

Heat and mass transport The transfer of the gas phase components into the liquid phase is described by Eq. (B.4). Since the flux

Fig. 11. Balance envelope for each distributor.

(ji ) is the total flux into the fluid element (refer to Eq. (A.1)), it has to be scaled by the liquid volume (Vliq ). ∗ ji = (kL a)liq · (Ci,org − Ci,org ) · Vliq ,

i  COMG

(B.4)

Vliq = VIL + Vorg

(B.5)

In accordance to the equilibrium assumption of the film model, the surface concentration (Ci∗ ) depends on the partial pressure of the gas and its Henry coefficient in the organic phase according to Eq. (B.6). −1 Ci∗ = Hi,org · Ctot,org · pi ,

i  COMG

(B.6)

The heat transport coefficient between the fluid and the cooling media is assumed to be constant. Here, a mean value for a similar fluid system (organic solvent – water, ˛liq = 0.5 kW/(m2 · K)) is taken from the literature [45]. In case heat transfer problems occur, the exchange area is a degree of freedom to investigate whether reactors with high heat exchange areas will improve the reactor performance [18,24]. q = aliq Vliq ˛liq · (T − Tc )

(B.7)

Appendix C. Model equations of level 3 On this level, a reactor with static mixer elements and distributors for additional gas and alkene feed is modeled. The distributors are designed such that gas and liquid is added over the entire cross sectional area in order to avoid radial concentration gradients due to discrete dosing. In accordance to the literature, the flow characteristics are plug flow like when using static mixers [35]. Furthermore, for the investigated static mixers, no slip velocity between gas and liquid phase was observed [38]. Component balances for each segment The component balances in each segment (set of segments SEG) are given by Eq. (C.1) for the gas phase, for the liquid phase by Eq. (C.2), and by Eq. (C.4) for the total mass balance. dn˙ i,g dz

dn˙ i,liq dz

= −(kL a) · (Ci∗ − Ci,org ) ·  · R2 ,

i  COMG

⎧  ⎪ (k a) · (Ci∗ − Ci,org ) ·  · R2 + AIL · i,j · rj , ⎪ ⎨ L j ∈ NR  = ⎪ i,j · rj , i = else A · ⎪ ⎩ IL

(C.1)

i  COMG

(C.2)

j ∈ NR

AIL =

V˙ IL ui

(C.3)

n˙ i,liq = n˙ i,IL + n˙ i,org ,

i  COM

(C.4)

Mass balance for each distributor For each gas and liquid distributor (h  DIS with set of distributors DIS), Eq. (C.5) need to be solved for each component, respectively. Fig. 11 illustrates the balance envelope for a distributor.

⎧ ⎨

n˙ i,h,k,out =

⎩

h = 1 : k = g : i  COMG

n˙ i,h,k,in + n˙ i,h,k,add n˙ i,h,k,in

h = 2 : (k = g : i  COMG, k = liq : i = tAlk) h = 3 : k = g : i  COMG

(C.5)

else

In case of synthesis gas with a composition of H2 :CO=1 is used, Eq. (C.6) must be fulfilled for each distributor. n˙ H2 ,add = n˙ CO,add

(C.6)

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141

139

Table 6 Heat capacity parameters. Component

Acp,i [kJ/(kmol K)]

Decane (l) 1-Octene (l) 2-Octene (l) n-Nonanal (l) iso-Nonanal (l) n-Nonanol (l) iso-Nonanol (l) [Bmim][PF6 ] (l) [42] H2 (g) CO (g)

2.0029 × 10 1.2122 × 102 1.3358 × 102 2.2532 × 102 2.2532 × 102 2.7779 × 102 2.3221 × 102 3.8801 × 102 2.7884 × 101 2.7956 × 101 2

Bcp,i [kJ/(kmol K2 )] −1

3.7248 × 10 3.6872 × 10−1 3.5997 × 10−1 2.8945 × 10−1 2.8945 × 10−1 3.4317 × 10−1 3.7669 × 10−1 −4.0162 × 10−1 3.1192 × 10−3 3.6324 × 10−3

Energy balance for each segment With respect to temperature a pseudo-homogeneous gas liquid system is assumed and the energy balance for each segment is given by Eq. (C.7). Here, the contribution of the dissolved H2 and CO in the liquid phase is neglected while the general contribution of the gas phase to the heat capacity flowrate is considered.





n˙ i,liq · cp,i +

i  COML

⎛





n˙ i,g · cp,i

i  COMG



= − ⎝2 ·  · R · q + AIL ·

i  COM

hi ·

dT dz



i,j · rj ⎠

R·T ·

(C.7)

n˙ i  COMG g,i

(C.16)

 n˙ liq,i · Mi i  COML

(C.17)

i

V˙ g A

(C.8)

(C.9)



p

ug,s =

˛ = liq · ˛liq + g · ˛g

(C.15)

Constitutive equations In order to calculate the interstitial velocity and the superficial velocities of gas and liquid Eqs. (C.16)–(C.21) are required.

V˙ liq =

The overall heat transport coefficient ˛ is modelled assuming a linear contribution of the gas and liquid phase. Here, ˛g is obtained from a mean value for a comparable system (water – compressed gas, ˛g = 0.17 kW/(m2 K)) [45] and ˛l as in level 2. The contribution of the solid phase is neglected due to its small volume fraction (for the chosen mixer s = 0.13 [38]).

– −2.173344 × 105 −2.206307 × 105 −4.677970 × 105 −4.677970 × 105 −5.659289 × 105 −5.538329 × 105 – −8.4480 × 103 −1.190221 × 105

(kL a) = (kL a)liq · liq

j ∈ NR

q = ˛ · (T − Tc )

Fi [kJ/kmol]

– – – – – – – 1.56763 × 10−3 – –

volume on level 2. The values can be converted according to Eq. (C.15).

V˙ g =

⎞

Ccp,i [kJ/(kmol K3 )]

uliq,s =

(C.18)

V˙ liq

(C.19)

A

The interstitial velocity (ui ) is calculated assuming no slip velocity between the phases and considering the solid fraction (s ) of the investigated mixers and the cross sectional area of the tube (A). ui =

V˙ liq + V˙ g

(C.20)

A · s

A =  · R2

(C.21)

g V˙ g = liq V˙ liq

(C.10)

Assuming ideal gas behavior the partial pressure of each gas is calculated using Eqs. (C.22) and (C.23).

1 = liq + g + s

(C.11)

pi = p · yi ,

Energy balance for each distributor In order to calculate the mixing temperature at each distributor outlet, an energy balance (Eq. (C.12)) has to be solved at each distributor. The temperature of the dosed alkene (373 K) and synthesis gas (298 K) is fixed.



i  COM

n˙ i,out · hi,out =



i  COM

n˙ i,in · hi,in +



n˙ i,add · hi,add

(C.12)

yi =



i  COMG

n˙ g,i

n˙ i  COMG g,i

2 · uK3 (kL a) = K1 · uKliq,s g,s

(C.13)

dp 5 · uK6 = −K4 · uKliq,s g,s dz

(C.14)

The validity range of the correlations is 0 ≤ ug,s ≤ 0.5 m/s and 0.1 ≤ uliq,s ≤ 2 m/s [38], and these ranges are used as bounds in the optimization. The superficial gas and liquid velocities are calculated according to Eqs. (C.16)–(C.19). On level 3, the (kL a)-value and exchange area are based on the total reactor volume, while these values are based on the liquid

i  COMG

(C.23)

System bounds The total alkene stream is calculated from the inlet alkene stream and the added alkene at each liquid distributor according to Eq. (C.24).

i  COM

G/L mass transfer and momentum balance For the calculation of the (kL a)-value and the pressure drop, correlations from the literature [38] are used (K1 = 3.45, K2 = 1.5, K3 = 0.64, K4 = 0.346, K5 = 1.71, K6 = 0.070).

,

(C.22)

n˙ tAlk,tot = n˙ tAlk,0 +



n˙ tAlk,add,d

(C.24)

d  DIS

In accordance to the previous levels, the solvent stream is connected to the reactant stream by Eq. (C.25) and the volume fractions of organic and IL phase are restricted by Eq. (C.26).

CtAlk =

n˙ tAlk,tot n˙ tAlk,tot · MtAlk /tAlk (373 K) + n˙ Dec · MDec /Dec (373 K)

= 2 kmol/m3

min ≤ Vfrac

V˙ Dec max ≤ Vfrac V˙ IL

(C.25)

(C.26)

140

A. Peschel et al. / Chemical Engineering Journal 188 (2012) 126–141

Table 7 Reference values used for scaling. Figure

Reference variable

Fig. 6(c) Fig. 6(d) Fig. 6(e) Fig. 6(f)

Fig. 7(b)

Fig. 7(b), Fig. 10(c) Fig. 10(b)

rref rref jref ,H2 , jref,CO qref jref,tAlk jref ,H2 jref,CO qref min (Tc ) Tref jref,tAlk pref n˙ ref ,tAlk , n˙ ref ,syngas

Appendix E. Numerical solution approach Reference value −2

2.19 × 10 3.14 × 10−2 6.1 × 10−3 6.62 × 10−1 2.48 × 10−2 8.3 × 10−3 7.9 × 10−3 8.99 × 10−1 354.18 10.35 1.6 × 10−2 60 2.4 × 10−3

Unit [kmol/(m3IL [kmol/(m3IL

s)] s)]

[kmol/s] [kW] [kmol/s] [kmol/s] [kmol/s] [kW] [K] [K] [kmol/s] [bar] [kmol/s]

Recycle condition Due to the recycling of the terminal alkene and the solvent, Eq. (C.27) is enforced. n˙ tAlk,0 ≥ n˙ tAlk,f

(C.27)

STY and X The space time yield of the technical reactor based on the liquid phase (otherwise not comparable to level 1 and 2) is given by Eq. (C.28). The conversion for the continuously operated reactor is fixed according to Eq. (C.29). n˙ nAld,f − n˙ nAld,0

STY =

X=



L

 · R2 dz z=0 liq

n˙ tAlk,tot − n˙ tAlk,f n˙ tAlk,tot

≥ 5.09 × 10−3 kmol/(m3liq s)

= 0.8

(C.28)

(C.29)

Energy balance for the coolant The energy balance for the coolant in each reactor segment is formulated in temperature form. After simplifications the change in the coolant temperature is proportional to the heat flux (according to [18]). For the coolant constant Kc , a value below zero corresponds to counter-current cooling, above zero to co-current cooling, and Kc = 0 to cooling with constant cooling temperature, respectively. dTc = 2RKc q dz

(C.30)

In principle, different methods can be used to solve the arising dynamic optimization problems such as the Hamilton-JacobiBellmann equation, the Pontryagin-Minimum-Principle, or NLP based methods. For all calculations, the dynamic optimization problems are transferred to large scale NLP problems using the simultaneous approach [46]. Here, orthogonal collocation on finite elements is used as discretization method [47]. This approach offers high numerical accuracy, fast solution times, and easy integration of algebraic constraints in the dynamic optimization problem. On level 1 and 2 a fixed number of finite elements is used, while on level 3 the number of finite element in each segment is chosen such that they are approximately equidistant. For further information on methods to solve dynamic optimization problems and on the used numerical solution approach, please refer to the literature [18,46]. The optimization problems were implemented in the optimization environment 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. Due to the use of local NLP solvers only local optimality can be guaranteed. However, each problem was solved several times using different starting values and only the reported solutions were observed. Alternatively, global optimization solvers such as BARON can be used to obtain global optimal solutions or criteria such as defined by Mangasarian [48] and Arrow [49] can be applied to check for global optimality. Similar to the sufficient KKT conditions for NLP problems, the dynamic optimization problem needs to be convex in the objective functions and the system equations on the whole feasible solution space spanned by the the control functions and states. However, the applicability of these criteria is very limited due to the complexity of the models. In order to obtain reliable solutions within reasonable computing time, good starting values are required. One advantage of the used discretization method is that the starting values can be interpreted physically and hence are relatively easy to guess. In addition, the followed three level optimization approach offers the possibility to transfer the solution of a subjacent level to a level above, thereby simplifying the choice of starting values.

Appendix D. Definition of normalized variables The definition of all normalized variables are given in Eqs. (D.1)–(D.7). The according reference values are summarized in Table 7. rj

rrel,j =

rref

(D.1)

jrel,i =

ji jref ,i

(D.2)

qrel =

qtot qref

(D.3)

Trel,c = prel =

Tc − min (Tc ) Tref

p pref

prel,i =

pi pref

n˙ rel,i,h =

n˙ i,h,k,add nref ,i

(D.4) (D.5) (D.6)

(D.7)

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