Polymeric Membrane Reactors

In: Simulation of Membrane Reactors ISBN: 978-1-60692-425-9 Editor: Angelo Basile and Fausto Gallucci © 2008 Nova Science Publishers, Inc. Chapter 6 POLYMERIC MEMBRANE REACTORS José M. Sousa1,2, Luís M. Madeira2, João C. Santos2 and Adélio Mendes2 ABSTRACT The aim of this chapter is the study of membrane reactors with polymeric membranes, particularly catalytic polymeric membranes. After an introduction where the main advantages and disadvantages of the use of polymeric membranes are summarised, a review of the main areas where they have been applied, integrated in chemical reactors, is presented. This excludes the field of bio-membranes processes, which is analysed in a specific chapter of this book. Particular attention is then given to modelling works in the fields of polymeric catalytic membrane reactors, where the membrane is catalytic itself. The various models that have been presented in the literature, as well as the numerical details regarding the respective mathematical equations, are shown and discussed. At the end, 3 different examples are presented and solved with the software package "MADONNA". INTRODUCTION To some extent, a membrane reactor (MR) is a device where a combination of a membrane and a chemical reactor must integrally couple in such a way that a synergy is created between the two systems. Particularly interesting is the possibility of overcoming equilibrium in reversible reactions, where MRs is one technology, among several others, to accomplish that. Improved selectivity is also of concern, as described below. A more comprehensive introduction on membrane reactors is present in chapter 1 of this book. 1 Chemistry Department, University of Trás-os-Montes e Alto Douro, Vila-Real, Portugal 2 LEPAE – Department of Chemical Engineering, Faculty of Engineering – University of Porto, Porto, Portugal José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 194 ADVANTAGES AND LIMITATIONS OF THE POLYMERIC MEMBRANE REACTORS In addition to the general advantages of combining reaction and membranes, the use of polymeric catalytic membranes in MRs can lead to some new important possibilities, as recently reviewed by Ozdemir et al. [1]. For instance, when a heterogeneous catalyst (nanosized metal cluster, zeolite or activated carbon, for example) is incorporated into a polymeric matrix, the selective sorption of reagents and products can be adjusted with a wellchosen polymeric environment, which may result in a potential beneficial effect on the catalytic performance. On the other hand, a much wider choice of polymeric membranes is available to select the most appropriate for membrane-assisted processes, as compared with metallic or ceramic membranes. Moreover, the technology to manufacture polymeric membranes is generally better developed. For these reasons, polymeric membranes have found a wide range of industrial applications in gas separation, and in some cases such processes have become a standard [2]. However, their use associated with reactors, particularly in the biotechnology field, is also of concern [1, 3]. Because polymeric membranes are less resistant to high temperatures, aggressive solvents/chemicals and oxidative conditions than their inorganic or metallic counterparts, polymeric membrane reactors (PMRs) may only be used in processes conducted at mild conditions. But yet, these limitations are relative: Nafion® and polydimethylsiloxane (PDMS), the two most widely used polymers to make catalytic polymeric membranes [4, 5], proved already to remain stable even under rather harsh conditions, showing an excellent thermal, mechanical and chemical resistance [5]. Moreover, the recent development of thermally resistant polymeric membranes [6, 7] provides promise for the more widespread use of such materials in MRs applications. On the other hand, polymeric membranes show some important advantages over the inorganic counterparts in their potential use as catalytic devices: their thickness can be controlled easily, a large scale preparation is not difficult, they are easier to be prepared crack-free and in different shapes (hollow, spiral wound, flat sheet, etc.), they are cheaper [1, 8] and they show versatile diffusivities and sorption capacities [9]. Besides, they may exhibit higher selectivities. Conversely, membrane permeability is often limiting. Another potential interesting feature of the polymeric catalytic membranes is that they may actively take part in the reaction by different ways. Firstly, the influence on the sorption capacity of reagents and products must be considered, because the concentration of the reaction species near the active sites depends directly on the sorption capacity. This sorption capacity depends strongly on the degree of polymer cross-linking, which is determined, in first instance, by the polymer composition, but the presence of a solid phase catalyst incorporated into the polymer may cause additional physical or even chemical cross-linking [10]. Secondly, the diffusivity of reagents and products is a very important parameter to have also into account, because it is influenced by the global sorption capacity: a high sorption of a component means a strong swelling of the polymer and thus an increased mobility of the diffusing species. Polymeric Membrane Reactors In general, almost only elastomeric polymers are utilized to incorporate homogeneous or heterogeneous catalysts. In fact, as the chains of glassy polymers are much less flexible, the incorporation of fillers disturbs its packing and leads to the occurrence of stresses, which may result in cracks. PDMS has been the most utilized polymer for making polymeric catalytic membranes, because it is cheap, chemically resistant, mechanically and thermally stable (up to 250 °C), easy to prepare and its flexible siloxane chains allow a fast mass transfer through the membrane. This mass transfer resistance could be important in the case of the diffusional controlled regime. APPLICATION FIELDS OF THE POLYMERIC MEMBRANE REACTORS In the following, some examples of chemical reactions that have been studied in PMRs, and particularly in polymeric catalytic membrane reactors (PCMRs), will be described, grouped according to the type of reaction. Such analysis excludes the applications in bioreactors, which have a specific chapter in this book devoted to its analysis (chapter 8), and also in fuel cells, a topic which has deserved the special attention of many reviews and textbooks. DEHYDROGENATION REACTIONS Dehydrogenations are endothermic and equilibrium-limited reactions, which should be performed at relatively high temperatures to proceed at reasonable rates and to shift the conversion to levels of practical significance. For such reasons, polymeric membrane reactors have been hardly ever used for conducting these reactions. Among the several tens of references on this subject, which are extensively discussed through this book in the chapter devoted to the inorganic membranes, only Frisch et al. [11] reported a study on dehydrogenation reactions conducted in PCMRs. These authors studied the dehydrogenation of cyclohexane to benzene using catalytic polymeric membranes based on polyethylacrylate and a 13X zeolite, which contained a dehydrogenation catalyst (Ti or Ni based). These catalytic membranes showed to be active for the reaction studied at temperatures as low as 50-90 ºC. Rezac et al. [12] also described a membrane-assisted dehydrogenation of n-butane, consisting of two plug flow reactors in series with an interstage hydrogen-removal polyimideceramic composite membrane. However, this is not a typical PCMR. HYDROGENATION REACTIONS Hydrogenation reactions conducted in MRs have been studied for long time, e.g. Gryaznov et al. [13]. In the early times, palladium membranes were utilized. However, PMRs and in particular PCMRs have also been considerably used for conducting such reactions in the most varied situations, namely gas, liquid and gas/liquid phases. Partial hydrogenation of multiple-unsaturated hydrocarbons is an important process in the petrochemical industry, used for both purification of alkene feed streams and production of José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 196 commodity chemicals from alkynes, dienes and aromatics. In choosing a catalyst for a partial hydrogenation process, selectivity for the intermediate product (e.g. alkene) is more important than catalytic activity. Thus, processes are often run at low temperatures and low hydrogen partial pressures, making promising the use of catalytic polymeric membranes. Table 1 summarises some of the works that have been carried out in this field using such membranes. Table 1. Some examples of hydrogenation reactions using catalytic polymeric membranes Main reagent(s) Catalyst(s) Membrane/ Support * Reacting Phases Operation mode and conditions Ref. Cyclopentadiene and isoprene Pd, Co, Cu, Ni PPO and PSF Liquid/gas Batch reactor, 40 ºC [14] Cyclopentadiene, isoprene and butadiene Pd PVP, EC and AR Gas Membrane contactor, 40 ºC, 0.1 MPa [15] 1-Octene Pd PES-C Liquid/gas Batch reactor, 40 ºC [16] Propadiene and propyne Pd PVP Gas Membrane contactor, 40 ºC [17] Butadiene Pd, Pd-Co PVP, EC and AR Gas Membrane contactor, 40 ºC [18] Cyclopentadiene Pd, Pd-Co PVP, EC and AR Gas Membrane contactor, 40 ºC [19] Ethylene, propylene and 1,3-butadiene Pd diblock copolymers (polyMTD) Gas Batch reactor, room temperature/120 °C [20, 21] Propylene Pd PDMS Gas Membrane contactor, 30 ºC [22] Propylene and propyne Pd PAI Gas Membrane contactor, 30 ºC [23] Propylene and propyne Pd PDMS Gas Membrane contactor, 30 ºC [24, 25] 1,5-Cyclooctadiene, 1octyne, phenyl acetylene and geraniol Pd PAA Liquid/gas Pore-through-flow CMR, 323 K, 40 bar H2 pressure [26] Sunflower oil Pd or Pt PES and PAI Gas/liquid Flow-through contactor, 100 ºC, 4 bar H2 [27] Sunflower oil Pd Nylon-6 Gas/liquid Flow-through reactor [28] Methylenecyclohexane Pd PVDF Gas/liquid Total flow-through reactor, 25-50 ºC [29] Cyclohexene Pt Nafion® and Tosflex®Gas/liquid Batch reactor [30] 4-Chlorophenol Pd/Rh PEBA Liquid Catalytic pervaporation (PV) reactor; ~25-60 ºC [31, 32] Acetophenone Pd PDMS and PEBA Gas/liquid Catalytic PV reactor; 3070 ºC, 1-4 bar H2 [33] * PPO – poly(phenylene oxide); PSF – polysulfone; PVP – polyvinylpyrrolidone; EC – ethylcellulose; AR – melamine-formaldehyde; PES-C – phenophtalein polyethersulfone; MTD – methyltetracyclododecene; PDMS – polydimethylsiloxane; PAI – polyamideimide; PAA – polyacrylic acid; PES – polyethersulfone; PVDF – polyvinylidene fluoride; PEBA – poly(ether-bamide); Polymeric Membrane Reactors Gao and co-workers [14], for example, prepared several metal-containing polymeric materials by incorporating transition metal chlorides (Pd, Co, Cu, Ni) into modified or unmodified poly(phenylene oxide) and polysulfone (cf. Table 1). The catalytic activities and selectivities of these materials were investigated, under mild conditions (40 °C), for the hydrogenation of cyclopentadiene in a liquid/gas phase batch reactor using ethanol as solvent and for the hydrogenation of cyclopentadiene and isoprene in a gas phase batch reactor. In these studies, the membrane acted only as a support of the catalyst. The authors found that both the activity and selectivity of the catalysts in the hydrogenation of cyclopentadiene in the gas phase were lower than those in liquid phase. The reason for the higher activity in liquid phase could be ascribed to the much higher concentration of cyclopentadiene. For the higher selectivity, the very low solubility of hydrogen in ethanol could be pointed out as a reason, which, associated with the higher concentration of cyclopentadiene, inhibited the further hydrogenation of the monoene. The same group [15] has also studied the selective hydrogenation of cyclopentadiene, isoprene and butadiene in CMRs made of a polymer-anchored palladium catalyst. They used different polymers (cf. Table 1) to anchor the Pd catalyst, which was deposited on the inner wall of hollow fibres. The authors found that all the MRs were active for the selective hydrogenations, though with a strongly dependent efficiency on both the hollow fibre support (cellulose acetate, polysulfone or polyacrylonitrile) and the polymer anchored palladium complex. Besides, they concluded also that the segregated feed of reactants (hydrogen fed to the shell side of the hollow fibre, cyclopentadiene fed to the bore side) was better than the premixed feed (fed to the bore side). As in the previous study [14], the selectivity showed to be strongly dependent on the hydrogen concentration at the reaction site. Slowly feeding hydrogen along the reactor length improves the selectivity by keeping its partial pressure low throughout the reactor, allowing to achieve values as high as 93.4% at a cyclopentadiene conversion of 99.0% [15]. Yu et al. [16], on its turn, studied the hydrogenation of 1-octene using a high temperature withstanding polymer, the phenophtalein polyethersulfone. This Pd-containing polymeric material exhibited a large activity for this reaction, as well as the membranes made from this material showed a relatively high permeability. Based on their results, the authors stated that such material may be promising to build a catalytic polymeric membrane reactor for gas phase hydrogenation and dehydrogenation at relatively high temperatures. Liu et al. [17] have also prepared catalytic membranes of poly(N-vinyl-2-pyrrolidone) (PVP)-Pd/cellulose acetate alike the ones referred above [15] for the selective hydrogenation of propadiene and propyne to propene (cf. Table 1). The content of propadiene and propyne reduced from 1.2% and 1.3% to less than 10 and 5 ppm, respectively (the requisites for propene use in polypropylene production), and a high selectivity in the conversion of such components to propene (97.8%) showed that this MR was very effective to conduct these reactions, provided the appropriated hollow fibres and operation conditions had been chosen. The same group has also studied other dehydrogenation reactions, as mentioned in Table 1. For instance, it is worth noting the selective hydrogenation of butadiene to 1-butene [18], where a content of butadiene less than 10 ppm and a maximum 1-butene loss of about 2% could be achieved, while simultaneously inhibiting 1-butene isomerization to 2-butene. Another example is the selective conversion of cyclopentadiene into cyclopentene [19]. In this case, and to further improve the catalytic performance, the reaction was carried out in the bimetallic (Pd-Co) polymeric hollow fibre reactors, with conversion of cyclopentadiene and José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 198 selectivity to cyclopentene attaining values as high as 97.5% and 98.4%, respectively. Besides, the activity of the catalytic hollow fibres kept stable during the selective hydrogenations [19], indicating that the polymer-anchored catalysts were firmly retained in the microporous structure of the fibres. Some groups have used catalytic polymeric membranes with incorporated metallic nanoclusters, an arrangement that offers the possibility to combine the catalytic activity (higher surface area per unit of catalyst volume) with the capacity to selectively remove components from the reaction medium. With this in mind, Ciebien et al. [20, 21] prepared catalytic polymeric films by synthesizing palladium nanoclusters within diblock copolymers, with a palladium content of 14 wt. %. These materials showed to be active catalysts for the hydrogenation of ethylene and propylene [20] and 1,3-butadiene [21], even though the clusters were completely surrounded by a bulk polymer matrix. The polymer was able to stabilize the clusters against gross aggregation, but could no prevent some systematic increase in clusters size [20, 21]. The authors found that the hydrogenations were a complex function of the sorption and diffusion coefficients of the reactants in the polymeric matrix, sorption of reactants on the palladium clusters surface, molecular size of reactants, size of clusters, temperature, etc. In another group, Theis et al. [22] studied the catalytic activity of a polydimethylsiloxane (PDMS) membrane loaded with palladium nanoclusters (0-15 wt. % metal content), selecting the hydrogenation of propene to propane as test reaction (cf. Table 1). Later on, they studied the same reaction in a polyamideimide ultrafiltration membrane loaded with palladium nanoclusters [23]. The membrane surface was impregnated with an inorganic titanium dioxide layer (up to 40 wt. %), further activated by finely dispersed nanosized palladium clusters. In this way, the palladium catalyst was decoupled from the polymeric surface. For the most active membranes, high conversion of the alkene (up to 100%) and high flux were achieved. In the selective hydrogenation of a stream containing 5% of propyne in propene using the same membranes, a selectivity of 99% to propene at a propyne conversion of 100% was achieved. Besides, these membranes proved to be stable up to temperatures of 200 °C and for operation times up to 50 h. More recently, these hydrogenations were studied using PDMS-incorporated nanosized Pd, aiming the development and evaluation of mathematical modelling of catalytic membrane reactors [24, 25] as it will be discussed at the end of section “Modelling of Polymeric Catalytic Membrane Reactors”. The main goal of this hydrogenation was the selective hydrogenation of propyne into propylene, avoiding the subsequent conversion into the alkane. Table 1 summarizes other relevant works in this field, some of them consisting of reactions conducted in three-phase reactors (gas/liquid/solid-catalyst), where mass transfer limitations often constitute a serious problem in achieving high activities and selectivities and that can be partially overcome with the use of CMRs [28-29]. One field that deserves special attention is the edible oil hydrogenation, particularly selective conversion of sunflower oil [27, 28], in which trans-isomerization is avoided and the expensive noble metal catalysts employed are easily recovered by using MRs. Finally, one should mention the works by Bengtson et al. [31-33], who studied the simultaneous enrichment of organic compounds by pervaporation and the concerning reaction within the same catalytic membrane reactor. Firstly, they used poly(ether-b-amide) (PEBA) membranes, an elastomer known to effectively concentrate slightly polar chemicals from water [31]. Due to the low affinity of chlorophenol towards palladium and also the very low volume fraction of catalyst (∼1% v/v), a considerable amount of this reactant crosses the membrane without meeting any reaction site. Polymeric Membrane Reactors In an attempt to reduce such losses, the experiments were repeated with a catalytic membrane filled with nano-sized silica particles [32]. This membrane showed to be much more active than the one previously used [31], as a result of the considerably lower cluster size and the elongation of the diffusion path through the membrane, increasing thus the contact rates with the catalyst. More recently, acetophenone hydrogenation was studied by a pervaporative catalytic membrane reactor coupled to a mass spectrometer [33], cf. Table 1. WATER AND WASTEWATER TREATMENT Although the use of polymeric membrane reactors for water treatment (through hydrogenation processes) could be included in the previous section, we consider that this application is sufficiently relevant to be discussed separately. In this concern, one should first mention the elimination of nitrates from drinking water, a serious problem in many agricultural areas in Europe and in the US. Indeed, limits for nitrate concentrations in drinking water have been imposed, because it has been linked to a number of health hazards. Conventional techniques for nitrates removal (e.g. chemical precipitation of complex salts, distillation, reverse osmosis, electrodialysis, ion exchange and biological treatment) have a number of disadvantages; the physicochemical processes create a waste of highly concentrated brines that must be disposed-off, whilst the biological treatment requires the use of a co-metabolite that, for drinking water applications, also raises other safety concerns. An alternative procedure involving the hydrogenation of nitrates to N2 in a CMR has been proposed recently, offering several advantages over the established technologies for denitrification. Beyond the CMRs based on inorganic membranes which can be found in many studies, also polymeric membranes have been considered for conducting such processes, as shown in Table 2. Table 2. Some examples of water and wastewater treatment applications using catalytic polymeric membranes Pollutant Catalyst(s) Membrane/Support * Phases Operation mode and conditions Ref. Nitrates Pd-Cu PEI Liquid/gas Membrane contactor [34] Nitrates Pd-Cu PA Liquid/gas Membrane contactor, 25 ºC [35] Chlorobenzene Pd PDMS/PAN Liquid/gas Membrane contactor, 25 ºC [36] Chloroaliphatics TiO2 PHOTOPERMTM Liquid/gas Batch mode with total recycling in a flow-through configuration, pilot -scale** [37, 38] Phenol TiO2 PHOTOPERMTM Liquid idem [39] Chlorophenols TiO2 PHOTOPERMTM Liquid idem [40] n-Alkanoic acids TiO2 PHOTOPERMTM Liquid/gas idem but at lab-scale [41] Ethylene glycol TiO2 microporous PTFE Batch mode with total recycling in a flow-through configuration [42] Several TiO2 Several nanofiltration Liquid/gas Batch photoreactor and [43] José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 200 pharmaceuticals membranes (NTR 7410, PAN GKSS HV3/T, N30 F, NF PES 10) photocatalytic MR and with total recirculation Azo dye Orange II Fe3+ Nafion® Liquid Batch photoreactor with recycling [44] Phenol H3PW12O40 (W12) PVDF Liquid/gas Continuous flat-sheet MR with recirculation, 30 ºC [45] * PEI – polyetherimide; PA – polyamide; PDMS – polydimethylsiloxane; PAN – poly(acrylonitrile); PTFE – polytetrafluoroethylene; PVDF – polyvinylidene fluoride; ** Membrane module PHOTOPERM® CPP/313 in a coaxial configuration. Very briefly, Lüdtke et al. [34] studied the reduction of nitrate using catalytic microporous polyetherimide membranes. The catalyst incorporated into the membrane matrix consisted of bimetallic microsized clusters containing 4.45 wt. % of palladium and 0.95 wt. % of copper coated on aluminium oxide. Hydrogen (pressure slightly above atmospheric) was permeating from the outer side of the hollow fibre membrane into the nitrate-containing water flowing in the bore side. It was found that the activity decreased by increasing the pH of the solution and increased with the temperature of the solution, though the selectivity to the final product N2 remained constant in both cases. The authors concluded that the formation of ammonia could be minimized by reducing the catalyst contact time. The reaction within the membrane matrix was dominated by mass transfer. Later on, Ilinitch et al. [35] studied the same reduction reaction using catalytic macroporous polyamide membranes (Table 2). The main goal of the authors was to explain why the coupling of the palladium and copper catalysts in the same membrane gave rise to a multifold increase in the catalytic activity of the aqueous nitrate reduction by hydrogen, comparatively to the values obtained with similar membranes impregnated with each of the catalysts alone. They concluded that it was most likely the hydrogen spillover the main responsible for such behaviour, which consisted in providing reducing agent (hydrogen species) for the reductive regeneration of copper sites. Beyond nitrates, ground water has been also contaminated with halogenated hydrocarbons, mainly in some industrial areas, despite their low solubility. To study the possibility of treatment of such waters in CMRs, Fritsch et al. [36] selected the hydrodechlorination of chlorobenzene as a representative test reaction; the halogen-free hydrocarbons are then more readily degraded by microorganisms in subsequent biological treatment units. The experiments were conducted in a three-phase CMR operating in interfacial contact mode (cf. Table 2). With such an arrangement, the supplying of hydrogen from the gas phase to the catalyst through the PDMS membrane is decoupled from its limited solubility in water. PDMS is a hydrophobic elastomer that, besides preventing direct access of all potential contaminants in the aqueous phase (e.g. heavy metals or some sulphur compounds) to the catalyst (allowing nevertheless that organic reactants cross it), shows a high permeability towards hydrogen. Though the catalytic activity attained in the MR proved to be enough for the purpose of the target application, the authors found a continuous decrease of activity, suspecting that this might be due to a possible partial poisoning of the catalyst by adsorbed chlorine. Apart from water treatment, wastewater management is nowadays another issue of concern because conventional methods are often not convenient (e.g. incineration for diluted aqueous wastes or biodegradation for toxic compounds). Therefore, attention has recently Polymeric Membrane Reactors been given to alternative and innovative technologies for elimination or detoxification of hazardous organic wastes, frequently referred to as advanced oxidation processes (AOPs). Briefly, the AOPs generally include the addition of oxidizing agents (hydrogen peroxide, ozone, or molecular oxygen itself) in the presence of a catalyst, ultraviolet radiation (UV) or both, therefore providing the formation of highly oxidative radicals, like the hydroxyl ones. Among several others AOPs, the photocatalytic processes are an important class, which has been proposed as a viable alternative for the decontamination of either wastewater or drinking water for human use, namely for the degradation of different toxic organic compounds − particularly chlorinated ones. In these processes, the activation of a finely divided semiconductor by UV radiation, usually titanium dioxide, in intimate contact with an aqueous solution of the pollutants, develops a redox environment capable of oxidizing them into nontoxic substances. Slurry-type reactors making use of TiO2 suspensions seem to be more efficient than those based on immobilized catalyst, but, for engineering applications, they suffer from an intrinsic drawback: the need of a post-treatment for catalyst recycling and for the ultimate goal of obtaining clean and powder-free water [46]. For this reason, anchoring the TiO2 to a suitable support or impregnating it in a polymer is desirable. In this last case, a UV transparent membrane matrix with a good adsorption capacity for the organic compounds is required. With these goals in mind, many studies have been carried out focusing in the degradation of several compounds in potable waters and wastewater, being of particular relevance those of Bellobono and co-workers, a few of which are summarised in Table 2. Outstanding results obtained by the photocatalytic membrane technology used in these works, patented by Chimia Prodotti e Processi, Milan, Italy [47] and developed up to pre-industrial scale, have showed that these membranes are very useful and promising for degrading several compounds. Comparing with the suspended semiconductor reactors, the CMRs (even without promoting photocatalysts) showed more than twofold gain in rate [37]. Besides photomineralization, the PHOTOPERM® process showed also to be suitable for the pre-treatment of wastewater containing a broad variety of non-biodegradable contaminants, specifically toxic compounds, preceding a biological treatment [37]. Ingested pharmaceuticals, used for medical and veterinary purposes, are usually present in wastewaters, because they are not completely metabolized. Because they are often difficult to (bio)degrade or remove using conventional treatments, they appear in the effluents in concentrations of up to several mg/L. This way, the photooxidation of different pharmaceuticals with oxygen by using a hybrid photocatalytic MR was performed recently [43], cf. Table 2, but further investigation seems to be still in progress. In particular, the membrane rejection towards the pollutants was not very satisfactory. Another interesting example is the use of the photo-Fenton process to abate a nonbiodegradable azo dye (Orange II) using a CMR, as described by Lopez and Kiwi [44]. In their work, the authors impregnated a Nafion® membrane with Fe3+ ions through a ionexchange process. The membrane was floating freely in the solution and was used to eliminate the need of removal of the free iron ions in wastewater after pollutant degradation, as occurs in a homogeneous process. The degradation was based on the enhanced generation of mainly OH radicals from H2O2 in the presence of the Fe3+ species by UV radiation. Finally, it was reported recently a study in which another photocatalyst has been used – Keggin type phosphotungstic acid, H3PW12O40 (W12) –, one of the most widely used photocatalytic José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 208 phase PCMRs. We will focus specifically in membrane reactors with polymeric catalytic membranes. The examples that deal with membrane reactors with polymeric non-catalytic membranes are usually related with different subjects, namely bio-reactors, photocatalysis and others (discussed in different chapters of the present book). In the case of packed-bed membrane reactors with polymeric non-catalytic membranes, where the specific function of the membrane is only the separation, the models are similar to the ones devoted to inert inorganic membranes, which are widely covered and discussed in other chapters of the present book. The only difference between both types of membranes is the transport equation for the reaction species. LIQUID/VAPOUR PHASE MEMBRANE REACTORS Yawalkar et al. [94] developed a one-dimension mathematical model to describe theoretically the steady-state liquid phase epoxidation of alkenes to epoxides in a CMR. The model’s emphasis was put on the effect that some variables and parameters had on the membrane reactor performance, namely the peroxide and alkene concentrations in the liquid phase, the peroxide and alkene sorption coefficients and the catalyst particles loading and size. The reactor was operated in interfacial contact mode (segregated feed of the reactants to each side of the membrane), with homogeneous concentration in both bulk organic and aqueous phases (well-stirred chambers) and in conditions such that there was no concentration polarization. The main assumptions of the model considered by the authors were the following: • The catalytic membrane consisted of a homogeneous polymeric phase with cubic zeolite catalyst particles, all of the same size, built-in and distributed uniformly throughout, i. e., the distance between successive particles was the same. • Membrane partition coefficients for the peroxide and organic phases were assumed to be independent of each other. • The authors considered two simultaneous reactions to describe the alkene epoxidation: the epoxidation reaction itself, between the peroxide and the alkene to give epoxide and by-products, and the side undesirable peroxide decomposition. • Due to the organophillic nature of the membrane, the concentration of alkene inside the membrane was assumed to be much higher than that of the peroxide. In this way, the epoxidation reaction was considered to be of pseudo first order concerning the peroxide concentration. • The proposed model considered fickian diffusion of the reactant peroxide in the polymeric phase and in the catalyst particles. • Membrane partition coefficients of peroxide between the bulk liquid phase and the membrane surface, as well as between the polymer phase and the catalyst surface, were assumed to be described by the Henry's law. Polymeric Membrane Reactors According to these simplifications, the model equations consist of the fickian mass transport of peroxide across the polymeric phase as well as the fickian mass transport and reaction across the catalyst particles. The boundary conditions include the sorption equilibrium for the interfaces gas/polymer and polymer/catalyst (described by Henry’s law) and equality of fluxes at the interfaces polymer/catalyst. The model predictions showed how the organophillic membrane phase, along with the zeolite particles, lowered the excess of peroxide at the catalyst active sites, thus reducing drastically its decomposition and the catalyst deactivation. By proper selection of the various parameters, high peroxide efficiency and significant rate of epoxide generation could be obtained. The Yawalkar’s model was extended very recently by Nagy [95]. The author used a simple physical model for the distribution of the catalyst particles inside the membrane layer, as well as for the mass transport throughout it, and studied the influence of some membrane properties, such as the size and geometric distribution of the catalyst particles, the catalyst phase hold-up, the distance of the first catalyst particle from the membrane surface and the membrane thickness, as well as the diffusion coefficients in the membrane phase and in the catalyst particles. A chemical irreversible first-order reaction occurring inside the catalyst particles was considered, although the developed methodology could also be extended for higher-order chemical reactions. Depending on the particle size, the author considered two different models: a pseudo-homogeneous one, suitable to be applied when the size of the catalyst particles was very low (submicrometer-sized), and a heterogeneous model, recommended for larger particles (several micrometers in size). Specifically for the case of first order irreversible reactions, the author developed analytical mathematical equations for the prediction of the mass-transfer rates as a function of every physical and chemical parameters that characterizes the catalytic membrane layer, namely the diffusion and solubility coefficients, reaction rate constant, catalyst particle size, particle hold-up and membrane thickness. One of the main conclusions from that study was that the mass-transfer rate could be significantly improved by decreasing the size of the catalyst particles and the distance between the first particle and the membrane surface. On the other hand, the low diffusion coefficient through the catalytic particles, which is very often the case, could strongly lower the global mass-transfer rate, as well as the effect of the chemical reaction on it. Kaliaguine and co-workers [56] studied and modelled a similar catalytic membrane reactor used in the oxyfunctionalization of n-hexane with hydrogen peroxide into a mixture of hexanols and hexanones, as mentioned above (Table 3). The authors used a solvent-free biphasic liquid-vapour catalytic polymeric membrane reactor operating in interfacial contact mode, where the catalytic membrane was a composite PDMS polymer built-in with titanium silicalite zeolite TS-1. The authors developed a simple steady-state diffusion-reaction model for describing the reactants concentration profiles and oxygenates formation rates inside the membrane based on the following main assumptions: • Isothermal and isobaric reaction conditions. • Mass transport of the reactants truly unidirectional, that is, occurring only through the membrane thickness (this is generally the case, as the membrane thickness is usually much smaller than its surface area), and described by the Fick’s law. José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 210 • Homogeneous and isotropic membrane, with the reactants concentration in the liquid aqueous (H2O2) and vapour organic (n-hexane) phases in equilibrium with the ones at the respective membrane surface. • Diffusion coefficients independent of the concentration, that is, constant throughout the membrane thickness. • Independence of the reaction rate relatively to the internal mass transfer resistance (diffusion) in the catalyst particle, that is, particle effectiveness factor close to unity. • No boundary layer on the membrane surfaces, that is, no mass transfer resistance for the reactants and reaction products at the interface membrane surface/bulk phase solution. • Effective (experimental) values for the partition coefficients of peroxide (interface membrane/aqueous phase) and n-hexane (interface membrane/organic phase). The solubility of each of these components in the other phase was assumed to be negligible. • Negligible concentration of the reaction products in both membrane surfaces, that is, as these species reach the membrane surface, they are immediately separated from it, due to their immiscibility in the bulk liquid/vapour phases. According to these simplifications, the model equations consisted of the mass transport (fickian) and reaction balances for all the reaction species. The respective simulations showed to describe accurately the average formation rates of the oxygenate species, which were found to fit quite well with the ones obtained from the experiments. They demonstrated also that the hydrogen peroxide diffusivity in the catalytic polymeric membrane had a strong effect on the rates of oxygenates formation (the reaction was conducted in an excess of n-hexane). The authors still evidenced the feasibility of the principle of using a catalytic membrane as interphase contactor in a biphasic reaction. Specifically for this system, the catalytic membrane was effective not only in the production of hexanols and hexanones, but also in the separation of these products from the organic feed. Despite the authors discussed the influence of the catalyst loading and the membrane modifications on the catalytic performance, the presented results showed only the influence of the hydrogen peroxide diffusivity. A simple numerical example based in this model and solved using the software MADDONA will be presented later in the section “Appendix: Examples of Application Using Madonna”. Vital and co-workers [75] also developed a simple model to describe the hydration of α-pinene into α-terpineol and a series of other products, being the reaction carried out in a batch catalytic polymeric membrane reactor. The catalytic membrane was a composite PDMS polymer built-in with zeolite USY 750 as catalyst. The reactant solution used was a mixture of water and α-pinene in acetone (as solvent). The authors developed a simple transient model to describe the reactants and reaction products concentration as a function of time in the reactor chamber, which was developed based on the following main assumptions: • Isothermal and isobaric reaction conditions. • Pseudo steady-state conditions for diffusion and reaction inside the membrane. Polymeric Membrane Reactors • Mass transport of the reactants inside the membrane truly unidirectional and described by the Fick’s law, as also assumed in the Kaliaguine’s model [56]. • Homogeneous and isotropic membrane macrostructure, as also assumed by Kaliaguine et al. [56]. • Equilibrium concentration of the reactants in the liquid phase with the one in the membrane surface, described by a linear relationship (Henry’s law). • No mass transport resistance for the reactants from the solution bulk phase to the membrane surfaces (no concentration polarization effect). • The diffusivity of α-pinene was assumed to be independent of its own concentration, but considered to depend on the concentration of α-terpineol. • Independence of the observed reaction rate relatively to the internal diffusion in the catalyst particle (zeolite). • Consumption of the α-pinene reactant (species A) according to a parallel reaction network. The reaction product α-terpineol (species B) was considered to not react subsequently. A series of the others components were lumped into a generic component (species C). • Since the sorption of water in a PDMS membrane is low and there is a large excess of this reactant relatively to α-terpineol, the authors assumed a pseudo first order elementary reaction rate for the two parallel reactions. According to these simplifications, the model equations considered the mass balance equation for component A inside the membrane at pseudo steady-state conditions (fickian transport and reaction rate) and the mass balances in the reactor chamber for components A, B and C at transient conditions. The reactor was operated in total recycle mode. The results obtained with this model showed to describe quite well the average concentration history of reactants and reaction products in the liquid phase, as well as the selectivity to α-terpineol. The model predicted also the increase of the reactants permeability as a function of the catalyst loading. In a later work by this group [72], the authors applied a similar model to the same reaction system, but now conducted over molybdophosphoric acid immobilized in hydrophobic polyvinylalcohol membranes modified with acetic anhydride. In that study, the authors considered a slightly more complex model to describe the parallel chemical reaction network: a second order elementary reaction rate for the reaction concerning the production of α-terpineol and a first order elementary reaction rate for the secondary (parallel) reaction. Like in the previous work [75], the model proved to describe reasonably the obtained experimental results. Frisch et al. studied the dehydrogenation of cyclohexane to benzene [11] and the cis to trans isomerization of piperylene [77], both carried out in a batch catalytic polymeric membrane reactor at low temperature. The catalytic membrane was a composite polyethylacrylate polymer built-in with a zeolite 13X containing a catalyst based on Ni or Ti [11] and on Co [77]. The feed to the membrane consisted of pure reactants in vapour phase. In order to calculate the reaction rate constants and the activation energies in both studies, the authors developed and solved a simple transient diffusion-reaction model, considering pseudo first order reaction relatively to the reactants. The diffusion coefficient of the reactants in the membranes, necessary to solve their model, was obtained by the time-lag method using José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 212 control membranes, that is, membranes containing the same zeolite 13X, but free of catalyst. The calculated values were consistent with kinetics controlled by the surface diffusion. GAS PHASE MEMBRANE REACTORS Modelling and simulation of polymeric catalytic membrane reactors for carrying out gas phase reactions is a subject that has been deserved some attention mainly by our research group, conducting either theoretical [96-104] or experimental studies [24, 25]. Various studies are now going to be presented, following the relevant publications. First of all, it will be described an isothermal reactor for conducting equilibrium-limited gas phase reactions with perfectly-mixed flow pattern. Then, a different flow pattern, the plug flow, will be introduced and discussed, keeping, though, the same reaction type. Finally, a nonisothermal reactor applied for conducting a consecutive-parallel gas phase reaction system, also with perfectly-mixed flow pattern, will be introduced and discussed. Isothermal/perfectly-mixed flow pattern models. Concerning the theoretical aspects, the first model developed for gas phase catalytic membrane reactors considered an equilibrium-limited reaction of the type ⎯⎯→ ++ ←⎯⎯ d i k k aA bB cC dD occurring in a polymeric catalytic nonporous membrane reactor (PCMR) with nanosized catalyst distributed homogeneously across the membrane, as sketched in Figure 1 [96, 97]. Figure 1. Schematic diagram of the polymeric catalytic membrane reactor (PCMR). Adapted from [96]. This model was developed based on some assumptions, which will be described along the text when the presentation of the respective mathematical equations. These equations comprise the steady-state differential mass balance for the membrane in planar coordinates and the respective boundary conditions, as well as the algebraic mass balances for the upstream and downstream chambers. Polymeric Membrane Reactors MASS BALANCE AND BOUNDARY CONDITIONS FOR THE MEMBRANE 2 20 + ν= i iidi dc D kf(c) dz (1) The first term describes the diffusive transport through the membrane and the second one describes the chemical reaction. The main assumptions considered in this model are: steadystate and isothermal conditions, transport of the reaction species throughout the membrane described by the sorption-diffusion model, fickian diffusion, and constant sorption and diffusion coefficients throughout the membrane and for the whole range of concentrations. Symbol i refers to the ith component, D is the diffusion coefficient, c is the concentration inside the membrane, z is the spatial coordinate perpendicular to the membrane surface, ν is the stoichiometric coefficient (taken positive for products and negative for reactants), kd is the direct reaction rate constant and f is the local reaction rate function, defined as follows: () ()() () () () ()() () () ()() () () 1 / ⎛⎞ ⎜⎟ =−=−= ⎜⎟ ⎝⎠ ⎛⎞ ⎜⎟ − ⎜⎟ ⎝⎠ cd c ab d abCD idABiCD AB ddi cd abCD AB k cc fc k c c kc c c c kkk cc cc R (2) It was considered an elementary reaction rate law and that such reaction occurred only at the surface of the nanosized catalyst particles. It was still assumed that the concentration of the reaction species (at the catalyst surface) was equal to the local concentration on the polymer phase. In principle, any relationship could be considered for this partition coefficient, but this one simplifies the original problem without compromising the main conclusions. Symbol ki is the reverse reaction rate constant and Rk is the ratio between the direct and reverse reaction rate constants, that is, an equilibrium constant. Sousa et al. [96, 97] considered this parameter equivalent to the thermodynamic equilibrium constant based on the gas phase feed conditions, Ke. However, this assumption is only strictly valid when activity coefficients are unitary, which was assumed by the authors [96, 97]. More generally, the equilibrium constant obtained from the concentration ratio between products and reactants is different when considering the gas and membrane phases. In this way, the sorption-based enhancement of the equilibrium conversion reported by Sousa et al. [96, 97] is not generally valid. The second order differential equation (1) needs two boundary conditions, in this case an equilibrium condition at the interfaces membrane/gas: At 0=z (retentate side), = R iii cSp (3) At =δz (permeate side), = P iii cSp (4) José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 214 It was assumed that such equilibrium condition is described by the Henry's law and also that there was no external mass transport limitations in the interface (no occurrence of concentration polarization). Symbol p represents the partial pressure in the bulk gas phase and the superscripts R and P refer to the retentate and permeate stream conditions, respectively. δ represents the membrane thickness and S represents the membrane partition coefficient between the bulk gas phase and the membrane surfaces. PARTIAL AND TOTAL MASS BALANCES FOR THE RETENTATE CHAMBER It was assumed perfectly-mixed flow pattern with negligible drop in the total pressure and ideal gas behaviour in the development of the mass balances for the retentate chamber. The corresponding mathematical equations are: 0 0 = −+ = ℜℜ FF RR m ii i i FR z Qp Qp dc AD TT dz (5) 0 0 = −+ = ℜℜ ∑ FF RR mi i FRiz dc QP QP AD TT dz (6) The first and second terms of these equations describe the molar flux that comes into and out of the reactor on the retentate side, respectively. The third term describes the molar flux exchanged between the membrane and the retentate chamber. Symbol Q is the volumetric flow rate, P represents the total pressure, Am is the membrane surface area, ℜ is the universal gas constant and T is the absolute temperature. The superscript F refers to the feed stream conditions. PARTIAL AND TOTAL MASS BALANCES FOR THE PERMEATE CHAMBER In the development of these mass balances, it was also assumed perfectly-mixed flow pattern with negligible drop in the total pressure and ideal gas behaviour. The corresponding equations are now: 0 =δ += ℜ PP m ii i Pz Qp dc AD Tdz (7) 0 =δ += ℜ∑ PP mi i Piz dc QP AD Tdz (8) By analogy with the retentate chamber, the first term describes the molar flux that comes out of the reactor by the permeate side and the second term describes the molar flux exchanged between the membrane and the permeate chamber. Polymeric Membrane Reactors Equations (1)-(8) were made dimensionless, becoming as follows: 2 2 2 d() 0 d + νΦ = ζ * ** i iii c Dfc (9) () () ()() ()() Δ ⎡ ⎤ =− ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ n ref **a*b*c*d iABCD e C fc c c c c K (10) 0 ζ= = **R* iii ,c Sp (11) 1 ζ= = **P* iii ,c Sp (12) 0 d 0 dζ= − +Γ = ζ * F* F* R* R* * i iii c Qp Qp D (13) 0 d 0 dζ= − +Γ = ζ ∑* F* F* R* R* * i i i c QP QP D (14) 1 d 0 dζ= + Γ= ζ * P* P* * i ii c Qp D (15) 1 d 0 dζ= + Γ= ζ ∑* P* P* * i i i c QP D (16) where /= * iiref ccC , /= *ref P PP , /= * iiref ppP , /= * iiref D DD , /= *ref QQQ , /= * iiref SSS , / ζ =δz, Δ =+−−ncdab () 1/2 1+− ⎛⎞ ⎜⎟ Φ=δ⎜⎟ ⎝⎠ ab dref ref kC D, ℜδ Γ= = δ ℜ ref m mref ref ref ref ref ref ref ref P AD S AD C T QP QP T The total feed pressure and total feed volumetric flow rate were taken as reference conditions to define the dimensionless pressure and volumetric flow rate, respectively. Species A was taken as the reference component to define the dimensionless diffusion and sorption coefficients and Cref was defined as the reference concentration (Cref = SrefPref). The set of dimensionless equations contains two dimensionless groups: the Thiele modulus, Φ, which represents the ratio between the characteristic intramembrane diffusion time for the reference component and the characteristic direct reaction time, and a dimensionless contact time, Γ, which represents the ratio between the maximum possible flux through the membrane for the reference component, that is, permeation of pure species against null permeate pressure, and the total molar feed flow rate (see the respective mathematical definitions above). This last parameter is somehow related with the stage-cut, that is, the feed fraction that permeates through the membrane. For the case of 100 % stage-cut, the reactor operates in what is called “total flow-through configuration” or in “dead-end flow” [105]. José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 216 Due to the characteristics of the membrane reactors, it is expected that, in some way, they perform better than a conventional reactor. In these specific studies [96, 97], the main objective of the authors was to compare the performance of the catalytic polymeric membrane reactor with the more conventional counterpart catalytic reactor exclusively in terms of the achieved conversion for an equilibrium-limited gas phase reaction. More specifically, they studied the role played by the different diffusivity and sorption selectivities for the reaction species on the performance of a catalytic polymeric membrane reactor. “Conventional”, in the context of their work, meant a catalytic reactor operating in similar conditions as the ones of the catalytic membrane reactor, that is: the same flow pattern (perfectly-mixed), fed with a gaseous mixture in the same conditions of composition and pressure, containing the same catalyst and with the same chemical reaction taking place at the catalyst surface. Additionally, the chemical reaction was considered to be described by equivalent kinetic equations in both reactors and with the same ratio between the direct and reverse reaction rate constants, that is, with the same equilibrium constant. At last, it was assumed that the maximum conversion achieved in the conventional catalytic reactor was the thermodynamic equilibrium value, being then its equilibrium constant value the thermodynamic equilibrium one. Through this assumption, the conversion achieved in the catalytic membrane reactor was compared with the thermodynamic equilibrium value. To perform such an analysis, it was considered the variable “relative conversion”, ΨA, defined as the ratio between the conversion of the reactant A attained in the membrane reactor, XA, and the thermodynamic equilibrium conversion based on the gas phase feed conditions, E A X . The conversion reached in the membrane reactor was calculated by the following equation: 1 ⎛⎞ −+ ⎜⎟ ℜℜℜ + ⎝⎠ ==− ℜ FF RR PP AAA R *R* P*P* AA AFF F*F* AA Qp Qp Qp TTTQp Qp XQp Q p T (17) The numerical details concerning the resolution of the model equations are described below, in the section “Numerical Methods for Membrane Reactors”. RESULTS AND DISCUSSION In the studies by Sousa et al., [96, 97], several simulations were performed in order to understand how the relative conversion depended on the dimensionless contact time and Thiele modulus parameters for different dimensionless diffusion and sorption coefficients of the reaction components and for different reaction stoichiometries. From the set of the respective results, the authors concluded that it is possible to reach conversions higher than the thermodynamic equilibrium value, for a given set of conditions. This conversion enhancement could be obtained, for example, by selecting a membrane where the diffusion coefficients of the reactants are lower than the ones of the reaction products. These conclusions can be observed in Figure 2, where the relative conversion of a PCMR is presented as a function of the dimensionless contact time and for different Thiele modulus Polymeric Membrane Reactors values. The relevant variables are: 01= * B D ., 05= F* A p., 05= F* B p., 0= F* C p, 0= F* D p, 01= e K ., 1ν= i and 0 01= P* P . [96]. This figure show a continuous increase of the conversion with the contact time until the reactor operates in the total flow-through configuration, also named as “total permeation condition”, TPC (described by the line TPL in the figure). That is, the maximum conversion is attained when the reactor operates in conditions of no reactants loss on the retentate stream and where the reacting species have the longest possible contact with the catalyst inside the membrane. The maximum value of the contact time parameter (at the TPC) for each Thiele modulus depends on the pressure difference between both sides of the membrane, for a given set of sorption and diffusion coefficients values. Figure 2. Relative conversion of component A as a function of the dimensionless contact time for various Thiele modulus values. 0.1 * B D=, 1 * iB D≠ = and 1 * i S = . The other variables have the values referred in the text. Due to the higher average reaction products permeability than the one of the reactants, and according to the Le Chatelier principle, the local chemical reaction condition inside the membrane, that is, the ratio () ( ) ()() () Δ cd ** n CD ref ab ** AB cc C cc [97], defined as the chemical reaction coefficient, Θ (see equation (10)) goes beyond the thermodynamic equilibrium value in a fraction of the membrane thickness on the downstream side (see Figure 3). As a result, the chemical reaction is shifted favourably towards the reaction products and the chemical conversion increases beyond the thermodynamic equilibrium value (Figure 2). This displacement of the local chemical equilibrium condition arises from the net balance between the separation effect (which shifts the reaction to the right side) and the backward reaction effect (which tries to hinder such shifting) and depends directly on the Thiele José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 224 d d 20 dd = +π = ℜ∑P PP Pi i irr,x c PQ mr D Tx r (26) The corresponding boundary conditions for Eq.s (25)-(26) are as follows (considering tube side feed): For m=1 (co-current flow), QP=0 at x=0 (inlet side) (27) For m=-1 (counter-current flow), QP=0 at x=L (outlet side) (28) These equations states that there is no volumetric flow rate at the inlet of the permeate side (no sweeping gas). rR is the membrane radius for the permeate side (rP=rS for tube side feed and rP=rT for shell side feed) and L is the length of the tubular membrane. The boundary condition for the partial pressure on the permeate side at x=0 or x=L, according to the type of flow (co-current or counter-current, respectively), is not necessary. As Q P=0 for such boundary, the term P P i Qp is always zero, independently of the partial pressure value. Anyway, its value can be calculated from the fluxes of each component at the membrane surface (see eq. (29) below). For cross flow operation mode, the composition of a species on the permeate stream at each axial coordinate is considered to be the ratio between its flux and the total flux of all species that crosses the membrane surface [100], as follows: d d d d = = = ∑ P P i i P P rr,x i i i irr,x c Dr p(x) P c Dr (29) The volumetric flow rate for this arrangement, which is independent of the flow direction, is given by: d d 20 dd = +π = ℜ∑P PP Pi i irr,x c PQ rD Tx r (30) Equations (18)-(30) were made dimensionless, becoming as follows: () 2 2 2 10 / ⎛⎞ ++νΦ= ⎜⎟ ζζ+δζ ⎝⎠ ** ** ii iii t dc dc Dfc drd (31) () () Δ ⎛⎞ ⎜⎟ =− ⎜⎟ ⎝⎠ n ref *** iAB e C fc c c K (32) ( ) () () ζ =∀λ λ= λ **R* iii j,cSp (33) (1 ) ( ) () () ζ =− ∀λ λ= λ **P* iii j,cSp (34) Polymeric Membrane Reactors () 10 ζ= λ δ ⎛⎞ −+ Γ = ⎜⎟ λζ ⎝⎠ R* R* j* i*i i t j, dQ p dc D drd (35) 10 ζ= λ δ ⎛⎞ −+ Γ = ⎜⎟ λζ ⎝⎠ ∑ j* R* R* * i i tij, dc dQ PD dr d (36) 0 and 1λ= = = R* F* R* ii ,p p Q (37) () 1 1 10 − ζ= − λ δ ⎛⎞ ++ Γ = ⎜⎟ λζ ⎝⎠ P* P* j* i*i i t j, dQ p dc mD drd (38) 1 1 10 − ζ= − λ δ ⎛⎞ ++ Γ = ⎜⎟ λζ ⎝⎠ ∑ j* P* P* * i i tij, dc dQ Pm D dr d (39) 10 0=λ= = P* m: ,Q (40) 11 0=− λ= = P* m: ,Q (41) 1 1 ζ= − λ ζ= − λ ζ λ= ζ ∑ * *i i j, P *P* i* *i i ij, dc Dd p() P dc Dd (42) 1 1 10 − ζ= − λ δ ⎛⎞ ++ Γ = ⎜⎟ λζ ⎝⎠ ∑ j* P* P* * i i tij, dc dQ Pf D dr d (43) where ()/ ζ =− δ T rr , λ= x L, () 1/2 1− ⎛⎞ ⎜⎟ Φ=δ⎜⎟ ⎝⎠ a dref ref kC D, 2 π ℜ Γ= δ Tref ref ref ref rLD C T QP The symbol j defines the configuration feed: j=0 for tube side feed and j=1 for shell side feed. The parameters Φ and Γ were described in the previous model. The remaining symbols are described in the nomenclature section. The numerical details concerning the resolution of this model are described below, in section “Numerical Methods for Membrane Reactors”. RESULTS AND DISCUSSION In the work regarding this model [99, 100, 102], the authors performed several simulations in order to understand how the relative conversion (as defined in the previous model) depended on the different dimensionless parameters and variables. According to the results, the dependence of the reactor conversion with the relative diffusion coefficients, total pressure drop between the upstream and downstream sides and reaction stoichiometry, along the contact time and Thiele modulus parametric space, is basically the same that was described above for the perfectly mixed flow pattern model. They studied further the José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 226 influence of the flow configuration (co-current, counter-current and cross flow arrangements), as well as the influence of the feed configuration (through the tube side or through the shell side). Concerning the flow configuration, it was concluded that the co-current operation mode was the best one, while the counter-current was the worst, being the differences in the reactor performance relevant only for a medium/high dimensionless contact time and for an intermediate range of Thiele modulus values (Figure 7) [100]. For low dimensionless contact times, the net result depends almost exclusively on the retentate composition and flow rate, so the flow configuration has almost no influence. For Φ→∞, the chemical reaction tends to be in equilibrium throughout all the membrane thickness, resulting then a limit conversion equal to the thermodynamic equilibrium one. Figure 7. Relative conversion of component A as a function of the Thiele modulus for different flow patterns and dimensionless contact time values. 1 * A D = , 10 * B D = , 1 * i S = , 1 F* A p=, 01 P* P .=, /10 t rδ= , 0j= and 025 k R.=. Adapted from [100]. The higher efficiency of the counter-current operating mode when there is only gas separation is the result of a better exploitation of the pressure gradients between the tube and shell sides along the fibre length, as also happens in other processes in chemical engineering (e. g., heat transfer). However, for a catalytic membrane reactor like the one described in this study [100], the partial fluxes through the membrane are not constant. In fact, they depend on the Thiele modulus, on the relative sorption and diffusion coefficients and on the total pressure difference between the upstream and the downstream sides, for a given contact time value [100]. Anyway, the main differences in terms of composition along the reactor length for all flow configurations occurs essentially on the permeate side and for a region of low axial coordinates (inlet reactor region), as it can be realized from Figure 8. Polymeric Membrane Reactors Figure 8. Partial pressure of component B in the retentate (upper part) and permeate (lower part) streams as a function of the axial coordinate, at the TPC and for different Thiele modulus values and flow patterns. Same variable values as in Figure 7. Adapted from [100]. As a consequence, the performance of the reactor depends essentially on what happens in this part of the reactor. Either for co-current or cross flow operating modes, the pressure of component B in the permeate stream at this region is the minimum possible (the shell side inlet is closed), maximizing thus the reaction rate inside the membrane (maximum driving force and maximum reactant concentration). For counter-current flow, on the other hand, the concentration of the reaction product for x=0 (exit of the permeate chamber) is higher than for co-current flow, as a result of the production earlier on, that is, for higher axial coordinates. This higher concentration leads to a diffusion of component B back to the membrane, as it can be seen in Figure 8 from the slight decreasing of its partial pressure [100]. Relatively to the influence of the reactor feed configuration [100], the simulation results showed that the conversion for a shell side feed was always higher than the one obtained for a tube side feed, for a reaction product permeability higher than the one of the reactant. Figure 9 shows these results for the case of higher reaction product diffusion. José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 228 Figure 9. Relative conversion of component A as a function of the Thiele modulus and for different feed locations and / t rδ ratios. TPC Γ=Γ and 1m = . The other variables have the same values as in Figure 7. Adapted from [100]. The simulation results showed also that the reactor conversion depends on the ratio / T rδ, as a direct consequence of the cylindrical geometry of the membrane. For high /δ T r values (membrane approaching the flat shape), the diffusion crossing area is nearly constant as a function of the radial coordinate and the influence of the feed location (tube side or shell side) is negligible (see Figure 10). As the rT/δ value decreases (the hollow fibre wall becomes thicker for a given internal radius), the diffusion crossing area as a function of the radius changes more and more, leading to a favourable or unfavourable impact on the concentration of the reaction components. For a tube side feed, the reactant concentration decreases doubly, leading this way to a conversion penalization. This occurs either due to the chemical reaction (consumption of reactant) or due to the membrane cylindrical geometry, which diffusion crossing area increases with the radial coordinate. When the reactor is fed from the shell side, on the other hand, the decreasing of the diffusion crossing area as the reactants proceeds across the membrane thickness lessens the impact of the decreasing of the reactant concentration due to the chemical reaction, enhancing thus the conversion (Figures 9 and 10). Polymeric Membrane Reactors Figure 10. Dimensionless concentration of component A inside the membrane as a function of the radial coordinate for different feed locations and / t r δ ratios. TPC Γ =Γ , 3 Φ =, 1 λ = and 1m=. The other variables have the same values as in Figure 7. Adapted from [102]. For the case where the reaction product permeability is lower than the one of the reactant, the conclusions are different: the best feed location (with respect to the conversion) depends now on the Thiele modulus and contact time values (see Figure 11). For low Thiele modulus, the shell side feed is always better than the tube side feed, whatever is the dimensionless contact time. For medium/high Thiele modulus values, the shell side feed is better than the tube side feed only for low to medium dimensionless contact times. This behaviour is a consequence of two factors: the cylindrical geometry of the membrane, that is, the change of the diffusion crossing area as a function of the membrane radius, and the separation effect [100]. As in the previous case, the influence of the feed location on the conversion becomes irrelevant as the membrane tends to the flat shape. José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 230 Figure 11. Relative conversion of component A as a function of the contact time for different feed locations, Thiele modulus and / t rδ ratios. 1 * i D = , 1 * A S = , 01 * B S.=, 1 F* A p = , 01 P* P .=, 1m= and 025 k R.=. Adapted from [102]. Nonisothermal/Perfectly-Mixed Flow Pattern Models More recently, Sousa and Mendes extended the previously developed perfectly mixed flow pattern model to consider now nonisothermal and nonadiabatic conditions [103]. Moreover, the model reaction now considered was much more complex and completely different in nature: a consecutive-parallel reaction describing the hydrogenation →→ P ropyne Propene Propane . The main objective of such work was to analyze in which conditions a catalytic polymeric membrane could take advantage of its effective diffusivity and sorption selectivities to improve the performance of a PCMR over the one obtained in a conventional reactor for this reaction system. More specifically, the authors tried to answer the question “can the concentration of propyne in the outlet stream be lowered under the levels obtained using a conventional catalytic reactor in a more efficient way”? To answer it, they developed a proper model and performed a set of simulation results to illustrate some key points about the use of such membrane reactor. They performed an analysis of the propyne concentration on the permeate stream along the model parametric space, as well as the possibility of enhancing the selectivity and overall yield to the desirable intermediate product (propene) and the conversion of the main reactants propyne and hydrogen. Polymeric Membrane Reactors DEVELOPMENT OF THE MEMBRANE REACTOR MODEL The catalytic membrane reactor considered in this study is the same as the one depicted in Figure 1. The reaction studied, the hydrogenation of propyne, was of the consecutiveparallel type: Reaction 1: +→ A BC Reaction 2: + →BC D with A≡Propyne; B≡Hydrogen; C≡Propene; D≡Propane. The model proposed for this reactor is based on the same main assumptions as described above, excepting the nonisothermal nature. The corresponding steady-state mass and energy balance equations are described in the following: MASS BALANCE FOR THE MEMBRANE () () 22 2 1 0 = +ν = ∑ i iijjji j dc DkTfc dz (44) where i refers to the ith component, j refers to the jth reaction and ν is the stoichiometric coefficient, taken negative for reactants, positive for reaction products, and null for the components that do not take part in the reaction. k(T) is the reaction rate constant based on the conditions of the reaction medium (in terms of temperature). f is the local reaction rate function, which was given by the following rate expressions: () 1= iAB f ccc (45) () 2= iB f cc (46) The details about the choice of these reaction rate laws are described in [103]. Basically, such a choice was supported in two main points. Firstly, this formulation considerably simplifies the problem, without compromising the main conclusions, although any type of reaction rate expression could be inserted into the catalytic membrane reactor model. Anyway, some published works report that the power-law type rate equations represent the experimental results better than the Langmuir-Hinshelwood-Hougen-Watson models for the same hydrogenation reactions as the one considered in the study now discussed [103]. Secondly, they considered that the reaction rate defined in equation (45) depends on the concentration of reactants propyne and hydrogen, because the concerning concentrations were considered close to each other. For the reaction rate defined in equation (46), on the other hand, they considered zero order relatively to the hydrocarbon (propene), as its concentration was in very large excess relatively to that of hydrogen. So, this reaction was considered of pseudo first order relatively to hydrogen. Concerning the temperature dependence of the reaction rate constants, the authors assumed that they followed an Arrhenius’ dependence: José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 232 () () 011 11 1 11 exp exp ⎡ ⎤ ⎛⎞ ⎛⎞ =−= −− ⎢ ⎥ ⎜⎟ ⎜⎟ ⎜⎟ ℜℜ ⎝⎠ ⎢ ⎥ ⎝⎠ ⎣ ⎦ ref ref EE kT k kT TTT (47) () () 022 22 2 11 exp exp ⎡ ⎤ ⎛⎞ ⎛⎞ =−= −− ⎢ ⎥ ⎜⎟ ⎜⎟ ⎜⎟ ℜℜ ⎝⎠ ⎢ ⎥ ⎝⎠ ⎣ ⎦ ref ref EE kT k kT TTT (48) where 0 j k and Ej are the pre-exponential reaction rate constant and the activation energy for reaction j, respectively. ENERGY BALANCE FOR THE MEMBRANE () () () 242 2 11 () 0 == ⎛⎞ λ+ +−Δ = ⎜⎟ ⎝⎠ ∑∑ r i eP,ii jjji ij dc dT dT CTD HkTfc dz dz dz (49) The first term in this energy balance equation is related to the transport of energy by conduction, while the second term is related with the enthalpy carried by the reaction species. The last term is the energy generation term related to the heat of reaction. In eq. (49), λe is the effective thermal conductivity that depends on both thermal conductivities of the solid and sorbed species, Δr j H is the reaction enthalpy for the reaction j and P ,i C is the heat capacity of species i in the sorbed phase. Following the assumption of negligible external transport limitations at the membrane surface, the boundary conditions for the mass and energy balances are: At z=0 (retentate side), = R iii cSp and = R TT (50) At z=δ (permeate side), = P iii cSp and = P TT (51) PARTIAL AND TOTAL MASS BALANCES FOR THE RETENTATE/PERMEATE SIDES The partial and total mass balances for the retentate and permeate sides are the same as the ones concerning the model described above (equations 5, 6, 7 and 8), tacking into account, however, the non isothermal conditions. Polymeric Membrane Reactors ENERGY BALANCE FOR THE RETENTATE SIDE () () 44 4 11 1 0 0 4 10 0 == = = = == − ++λ+ ℜℜ −Δ − − = ∑∑ ∑ ∑ FF F RR R mm ii ii i ii e FR ii i z z m s t,R t,R R ext i ii iz QpH QpH dc dT AHD A TT dzdz dc AHD UATT dz (52) The first and second terms of equation (52) account for the enthalpy of the gas phase in the feed and retentate streams; the third term accounts for the enthalpy transported by the reaction species that cross the membrane boundaries; the fourth term accounts for the heat exchange by conduction between the bulk gas and the membrane surface; the fifth term accounts for the sorption enthalpy; the last term accounts for the heat transfer between the bulk gas and a heat exchanger where the coolant is at a fixed temperature, Text. H is the enthalpy for the gas phase and ΔHS is the sorption enthalpy (that is, the change between the enthalpy of a reaction species in the gas and membrane phases). Ut,R and At,R are the external overall coefficient and external area of heat transfer for the retentate side, respectively. ENERGY BALANCE FOR THE PERMEATE SIDE () () 44 11 4 1 0 == =δ =δ ==δ + +λ + ℜ −Δ + − = ∑∑ ∑ PP P mm ii i ii e P ii z z m s t,P t,P P ext i ii iz QpH dc dT AHD A Tdzdz dc AHD UATT dz (53) The meaning of the different terms in equation (53) is identical to the corresponding ones in equation (52), as well as the hypotheses considered. DIMENSIONLESS EQUATIONS The model variables were made dimensionless with respect to the feed conditions (QF, PF and TF), to component A (DA, SA and CP,A) and to the membrane thickness, δ. The external and feed temperatures were considered to be equal. The reference temperature was set to 298 K. In this study, the authors assumed as reasonable to consider a uniform temperature for the catalytic membrane reactor chambers (TR=TP), in view of the usually low membrane thickness (few hundred microns, normally) and the perfectly mixed flow pattern assumption for both chambers. According to this hypothesis, the energy balances for the retentate and permeate chambers, equations (52) and (53), respectively, were simplified to a single global energy balance, described by equation (66) below. José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 240 Despite the efficiency of this strategy in terms of achieving a solution with high accuracy, it demands a considerable computational effort. In a tentative to simplify the resolution of this kind of problems, an alternative strategy based on a transformation of the independent variable (spatial coordinate) was developed [101]. Basically, this different approach has the same goal as the dynamically adaptive grid described above [98], that is, only a necessary density of collocation points along the spatial domain is used. However, while this task is done dynamically in the wavelet-based algorithm, a constant grid of collocation points along all the integration time is used in this new strategy. Moreover, the discretization grid is defined automatically in the wavelet-based algorithm, while it is more or less empirical in this new strategy, as it implies a previous knowledge about the solution of the problem. This numerical scheme was applied to the solution of some of the developed models using orthogonal collocation for the discretization of the spatial derivative terms and the results obtained were compared with the ones obtained using the wavelet-based method [101], showing to be very efficient to get the solution with high accuracy and with low computational effort. APPENDIX: EXAMPLES OF APPLICATION USING MADONNA In this appendix, some examples based on the works presented along the text are solved by the software package MADONNA. This software package solves initial and boundary value problems, so it is not able to solve problems in two or more regions connected through variable boundary conditions. Typically, model equations describing membrane reactors are defined in more than one region. However, there are a few situations where MADONNA can be successful in solving such models. If the membrane model equations have analytical solution, the software can be used to solve the model equations for the retentate and permeate chambers, either as a function of time (for the perfectly-mixed flow pattern) or as a function of the spatial coordinate, that is, along the reactor length (for the plug flow pattern). If the membrane model equations do not have analytical solution, MADONNA can be successful in solving the problem only if the boundary conditions are constant. This situation is valid only for perfectly-mixed flow pattern. The conditions for applicability of MADONNA in this situation are even more restrictive, because the solution is possible only for isothermal regime. In the following, 3 examples are going to be presented in order to illustrate these particular situations. EXAMPLE 1: This example is based on the model presented by Kaliaguine and co-workers [56], described in section “Liquid/Vapour Phase Membrane Reactors”, and shows a situation where the boundary conditions are constant. We should call the reader attention for some small misprints in this reference, concerning the model equations. Anyway, the objective of the presented example is to show how to use the software to solve a specific problem and not to solve a specific model published by anyone. Contrarily to what happen in the original paper [56], the equations of the model presented here are in dimensionless form, because the Polymeric Membrane Reactors numerical errors are minimized, though the reference variables are correctly selected, and the equations become more compact (reduction of the independent variables number). Moreover, the arising of dimensionless groups with physical significance makes the overall analysis of the system easier to perform. As referred in section “Liquid/Vapour Phase Membrane Reactors”, this model describes the oxyfunctionalization of n-hexane with hydrogen peroxide into a mixture of hexanols and hexanones. The chemical equations considered in this model are the following: ( ) 614 614 22 1 614 614 614 614 2 1 614 22 614 2 1 1 CH CH HO k CH CH CH O CH O kK C C CH HO CH O HO r KC K C +⎯⎯→+ = ++ (67) ( ) 614 614 22 2 614 614 614 614 2 2 614 22 612 2 2 21 CH O CH O HO k CH CH CH O CH O kK C C CH O HO CH O HO r KC K C +⎯⎯→+ = ++ (68) () 3 22 2 22 2 2 3 3 1 2 kHO HO HO O r k C⎯⎯→+ = (69) Considering, for simplification, 614 ≡ A CH , 614 ≡BCHO and 22 ≡CHO, and considering the sorption-diffusion model for the transport through the membrane as assumed by the authors [56], the dimensionless mathematical model equations are as follows: () 2 2 2 20 1 ⎡⎤ ⎢⎥ −Φ = ⎢⎥ ζ++ ⎣⎦ ** *AC *A A** AAref BBref cc dc DdKccKcc (70) () () 22 2 22 2 1 0 11 ⎡⎤ ⎢⎥ −Φ − = ⎢⎥ ζ++++ ⎣⎦ ** ** *BC AC *BB B** ** A AAref BBref AAref BBref cc cc dc k K DdkKKccKccKccKcc (71) () () () 222 2 23 2 2 11 10 11 ⎡⎤ ⎢⎥ −Φ + + = ⎢⎥ ζ++ ++ ⎣⎦ ** ** * *AC BC C *CB C** ** A A ref B B ref A A A ref B B ref A ref cc cc c dc k kK DdKccKcckKKccKcckKc (72) The corresponding boundary conditions are as follows: At 0 ζ =, 0 = * A c, 0= * B c, 0 159= * C c. (73) At 1 ζ =, 100= * A c. , 0= * B c, 0 = * C c (74) These boundary conditions describe an interfacial reactor where each of the reactants (A and C) is fed to the different sides of the membrane and is immiscible in the other phase, as well as the immiscibility of the main reaction product (B) in both aqueous and organic phases. The dimensionless variables (referred with an “*” in superscript) are defined in the same way José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 242 as the equations presented in section “Gas Phase Membrane Reactors”. Species A (n-hexane) was chosen to be the reference component, hence 721 256 10 m h − − =× ref D. . The reference concentration was considered the one of the reference component at the membrane surface, hence 3 5 11 kmol m− = ref c. . ki is the kinetic constant for reaction i and Ki is the equilibrium constant of adsorption for component i. Φ is the Thiele modulus 1 ⎛⎞ ρ Φ=δ ⎜⎟ ⎜⎟ ⎝⎠ bA ref ref kK cD, where () -3 kg mρb is the density of the catalyst in the catalytic membrane. The other variables have the usual meaning and are defined in the nomenclature section. Table 4 contains the values considered for the different variables in the simulations according to the authors [56]. Table 4. Nondimensionless variables values used in the simulations Reaction Components C6H14 C6H14O H2O2 Surface concentration at z=0 (kmol m-3) 0 0 0.81 Surface concentration at z=δ (kmol m-3) 5.11 0 0 Equilibrium adsorption constants (m3 kmol-1) 19.3 0.21 −− Diffusion coefficients (m2 h-1) 2.56x10-7 1.33x10-7 1.12x10-8 Kinetic constants (m6 kmol-1 kg-1 h-1) k1=8.60x10-3 k 2=1.75x10-2 k 3=2.90x10-3 The authors [56] analysed the influence of the membrane thickness (δ) for the different values given in the paper, as well as the respective diffusivities. However, the membrane thickness is incorporated in the Thiele modulus parameter in the present example. So, the reader can analyse which is the influence of that variable in the respective concentration profiles of the reaction species inside the membrane by performing the simulations for different Thiele modulus values. Relatively to the density of the catalyst (ρb), the authors do not say anything about. However, despite this variable appears only in the Thiele modulus parameter, we should call the reader's attention that its influence in real systems embraces the kinetic parameter, as well as the diffusion coefficients of the reaction components. Figure 14 shows the dimensionless concentration profiles across the catalytic membrane thickness for Φ=25 and Table 5 contains the values from the simulations. These results were obtained using the software MADONNA with a step size of 0.001 and a tolerance of 1x10-8. The concerning code for example 1 is presented in the following: METHOD RK4 { Run this example through Model -> Modules -> Boundary Value ODE Set as Boundary Conditions: CA(1)=1 CB(1)=0 CC(1)=0 Set as Unknowns: Polymeric Membrane Reactors INIT CA'; Min=0.5; Max=1.5 INIT CB'; Min=-0.1; Max=0.1 INIT CC'; Min=-1; Max=0 Set Tolerance=1E-8 } STARTTIME = 0 STOPTIME= 1 DT = 0.001 DTOUT=0.1 RENAME TIME= r {Model parameters} thiele=25. CRef=5.11 DRef=2.56E-7 {Diffusion coefficients} DA=2.56E-7/DRef DB=1.33E-7/DRef DC=1.12E-8/DRef {Kinetic constants} k1=8.6E-3 k2=1.75E-2 k3=2.9E-3 {Equilibium constants} KA=19.3 KB=0.21 {Initial conditions} INIT CA = 0 INIT CA' = 1 INIT CB = 0 INIT CB' = 0.1 INIT CC = 0.81/CRef INIT CC' = -1 {Component A} FA=CA*CC^2/(1+KA*CA*CRef+KB*CB*CRef) CA'' = thiele^2*FA/DA {Component B} FB1=CA*CC^2/(1+KA*CA*CRef+KB*CB*CRef) FB2=CB*CC^2/(1+KA*CA*CRef+KB*CB*CRef) José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 244 FB=-FB1+k2/k1*KB/KA*FB2 CB'' = thiele^2*FB/DB {Component C} FC=FB1+k2/k1*KB/KA*FB2+k3/k1*1/KA*1/CRef*CC^2 CC'' = thiele^2*FC/DC Figure 14. Dimensionless concentration of reaction components inside the membrane as a function of the dimensionless spatial coordinate. The values of the variables used in the simulations are referred in the text. Table 5. Dimensionless concentration values of the reaction components along the membrane thickness (although the results were obtained using a step size of 0.001, the table shows only a few points for simplicity) ζ * A c * B c * C c 0.0 0 0 0.15851 0.1 0.09885 0.00221 0.10670 0.2 0.19837 0.00314 0.07585 0.3 0.29824 0.00339 0.05615 0.4 0.39831 0.00325 0.04258 0.5 0.49850 0.00289 0.03256 0.6 0.59875 0.00241 0.02460 0.7 0.69904 0.00184 0.01782 0.8 0.79936 0.00124 0.01168 0.9 0.89968 0.00062 0.00580 1.0 1 0 0 Polymeric Membrane Reactors EXAMPLE 2: This example is based on the model presented above in section “Isothermal/PerfectlyMixed Flow Pattern Models” [98]. It considers perfectly-mixed flow pattern and a chemical reaction of the type ⎯⎯→ ←⎯⎯ A B, for which there is an analytical solution for the diffusionreaction equations across the membrane thickness. The model equations to be solved by the package MADONNA comprehend the unsteady-state ODEs that result from the mass balances for the retentate and permeate chambers, as follows: 0 dd dd ζ = =−+Γ θζ R* * F* F* R* R* * ii iii pc Qp Qp D (75) 1 dd dd ζ= ⎡ ⎤ τ =+Γ ⎢ ⎥ θ ζ τ ⎢ ⎥ ⎣ ⎦ P* * R P* P* * ii ii P pc Qp D (76) These equations were obtained from equations (5) and (7) by including the transient term. θ=τ R t is the dimensionless time based on the retentate conditions. τ= R R ref V Q and τ= P P ref V Q are the residence times for the retentate and permeate chambers, respectively, where V means the chamber volume. The other symbols have already been described above and are referred in the nomenclature section. The reference variables and conditions are the same as the ones described in section “Isothermal/Perfectly-Mixed Flow Pattern Models”. We should call the reader’s attention for the fact that the membrane process was considered to be in pseudo steady state, that is, the residence time for the membrane is much less than the residence time for the retentate and permeate chambers. If this assumption is not considered, the mass balance for the membrane should be written in addition to equations (75) and (76), considering the necessary relation between all the residence times. The corresponding initial conditions, defined as pure reactant A in both retentate and permeate chambers at the respective dimensionless total pressures, are as follows: At 0θ= , 1= R* A p, 0= R* B p, 001= P* A p., 0 = P* B p (77) The dimensionless retentate and permeate volumetric flow rates are calculated from the global mass balance equations (eqs. (14) and (16)), described in the following (remember that the retentate and permeate total pressures are constant, so dd 0 dd = = θθ ∑∑ R* P* ii pp and 1== ∑∑ F* R* ii pp , equations (75) and (76) above): 0 d 1 d ζ = =+Γ ζ ∑* R* * i i i c QD (78) José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 246 1 d d ζ = Γζ =− ∑* *i i i P* P* c D QP (79) The derivative terms of the flux at the membrane surfaces are described by the following analytical equations [98], considering the reference variables and conditions described in section “Isothermal/Perfectly-Mixed Flow Pattern Models”: 0 d dζ= =+ψ−ψ ζ * A c J JJ 23 4 (80) 1 d() (-) dζ= =+ψ ψ−ψ ψ ζ * A cJJexp Jexp 23 4 (81) 0 d dζ= ψψ =−+ ζ * Be** B B J cJ JK D D 34 2 (82) 1 () d(-) dζ= ψψ ψ ψ =− + ζ * B** BB Jexp cJexp JK DD 34 2e (83) where 12 1 1 ⎛⎞ ψ=Φ + ⎜⎟ ⎝⎠ / * Be DK 12 ⎛⎞ Φ=δ⎜⎟ ⎜⎟ ⎝⎠ / d ref k D, 1 + =+ R ***R* ABBB * eB pDSp JKD 1 , ()() 1 +−+ =+ P ***P* R***R* A BBB A BBB * eB pDSp pDSp JKD 2 , () () () 3121 ⎡⎤ ψψ −−+ ⎣⎦ =+ψ− * R* * R* P* * P* BeABBeABB * eB D exp(- ) exp(- ) K p S p K p S p JKD exp(- ) , ( ) () () 4121 ⎡⎤ −−−ψ− ⎣⎦ =+ψ− *R**R* P**P* BeABB eABB * eB DKpSpexp(-)KpSp JKD exp(- ) At this point, we should call the reader’s attention for the following: after a threshold Thiele modulus value, the term exp(ψ) in equations (81) and (83) takes values too high and an overflow error appears during the calculations. However, the term J3.exp(ψ) can be simplified adequately (exp(ψ)exp(-ψ)=1), avoiding such a possible overflow error. The values of the variables and parameters considered in the simulations are presented in Table 6. Figure 15 shows the dimensionless partial pressure of the reaction components as a function of time (retentate and permeate chambers) and Table 7 contains the respective values. These results were obtained with MADONNA using a step size of 1x10-5. The concerning code for example 2 is presented in the following: Polymeric Membrane Reactors METHOD RK4 STARTTIME = 0 STOPTIME= 0.22 DT = 0.00001 DTOUT=0.02 {Parameters} THIELE=4 TAU=0.50538 TAUR=1 TAUP=1 {Feed composition} PAF=1 PBF=0 {Dimensionless total pressures} PP=0.01 PR=1 {Dimensionless diffusion coefficients} DA=1 DB=10 {Dimensionless sorption coefficients} SA=1 SB=1 {Equilibrium constant} Ke=0.25 {Initial conditions} INIT PAR = 1.0 INIT PBR = 0.0 INIT PAP = 0.01 INIT PBP = 0.0 PSI=THIELE*(1+1/(DB*Ke))^0.5 J1=(PAR+DB*SB*PBR)/(1+Ke*DB) J2=((PAP+DB*SB*PBP)-(PAR+DB*SB*PBR))/(1+Ke*DB) J3=DB*EXP(-PSI)*(EXP(-PSI)*(Ke*PAR-SB*PBR)- Ke*PAP+SB*PBP)/((1+Ke*DB)*(EXP(-2*PSI)-1)) J4=-DB*(Ke*PAR-SB*PBR-EXP(-PSI)*(Ke*PAP-SB*PBP))/((1+Ke*DB)*(EXP(- 2*PSI)-1)) José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 248 DCADR0=J2+J3*PSI-J4*PSI DCADR1=J2+J3*PSI*EXP(PSI)-J4*PSI*EXP(-PSI) DCBDR0=J2*Ke/SB-J3*PSI/(DB*SB)+J4*PSI/(DB*SB) DCBDR1=J2*Ke/SB-J3*PSI*EXP(PSI)/(DB*SB)+J4*PSI*EXP(-PSI)/(DB*SB) QR=(1+TAU*(DA*SA*DCADR0+DB*SB*DCBDR0))/(PR) QP=-TAU*(DA*SA*DCADR1+DB*SB*DCBDR1)/(PP) {Component A in retentate chamber} PAR' = PAF-QR*PAR+TAU*DA*DCADR0 {Component B in retentate chamber} PBR' = PBF-QR*PBR+TAU*DB*DCBDR0 {Component A in permeate chamber} PAP' = (-QP*PAP-TAU*DA*DCADR1)*TAUR/TAUP {Component B in permeate chamber} PBP' = (-QP*PBP-TAU*DB*DCBDR1)*TAUR/TAUP Table 6. Dimensionless variables and parameters values used in the simulations Component A Component B * i D 1 10 * i S 1 1 F * i p 1 0 Φ 4 Γ 0.50538* e K 0.25 R * P 1 P * P 0.01 /ττ R P 1 * This value for the contact time parameter corresponds to the total permeation condition. Polymeric Membrane Reactors Figure 15. Dimensionless retentate and permeate pressure of reaction components along the dimensionless time. The values of the variables used in the simulations are referred in the text. Table 7. Dimensionless pressure values of the reaction components along the dimensionless time θ R * A p R * B p P * A p P * B p 0.000000 1.000000 0.000000 0.010000 0.000000 0.020004 0.975867 0.024133 0.005681 0.004319 0.040009 0.956770 0.043230 0.004249 0.005751 0.060013 0.941681 0.058319 0.003800 0.006200 0.080017 0.929748 0.070252 0.003640 0.006360 0.100021 0.920296 0.079704 0.003563 0.006437 0.120026 0.912798 0.087202 0.003516 0.006484 0.140030 0.906843 0.093157 0.003484 0.006516 0.160034 0.902108 0.097892 0.003460 0.006540 0.180038 0.898340 0.101660 0.003442 0.006558 0.200043 0.895339 0.104661 0.003428 0.006572 0.220047 0.892949 0.107051 0.003417 0.006583 EXAMPLE 3: This example is based on the model presented and discussed above in section “Isothermal/Plug Flow Pattern Models” [99, 100]. It considers plug flow pattern with constant pressure on the retentate and permeate chambers and a chemical reaction of the type ⎯⎯→ ←⎯⎯ A B, for which there is an analytical solution for the diffusion-reaction equations through the membrane, as referred in the previous example. The model equations to be solved José M. Sousa, Luís M. Madeira, João C. Santos and Adélio Mendes 256 E Activation energy. [J mol-1] H Enthalpy. [J mol-1] d k Direct reaction rate constant (reversible reactions). [(mol m-3)1-a-b s-1] e K Thermodynamic equilibrium constant based on feed conditions. [(mol m-3)c+d-a-b] 0 j k Pre-Exponential reaction rate constant for the reaction j (irreversible reactions). [(m3 mol-1)2-j s-1] j k Reaction rate constant for the reaction j (irreversible reactions). [(m3 mol-1)2-j s-1] H P e Modified heat Peclet number. [-] P Total pressure. [Pa] p Partial pressure. [Pa] Q Volumetric flow rate. [m3 s-1] r Spatial radial coordinate perpendicular to the membrane surface. [m] ℜ Universal gas constant. [J mol-1 K-1)] E R Ratio of the activation energies. [-] H R Ratio of the heat of reaction. [-] k R Ratio between the direct an reverse reaction rate constants. [(mol m-3)c+d-a-b] r R Ratio of the reaction rate constants. [-] S Henry’s sorption coefficient. [mol m-3 Pa-1)] St Stanton number. [-] T Absolute temperature. [K] U Heat transfer coefficient. [J m-2 s-1 K-1] x Tube/Shell spatial coordinate. [m] X Conversion. [-] y Molar fraction. [-] C Y Overall yield to species C. [-] z Membrane spatial coordinate. [m] Greek symbols β Prater number based on reaction 1. [-] γ Arrhenius’ number based on reaction 1. [-] Γ Dimensionless contact time based on species A. [-] δ Membrane thickness. [m] ζ Dimensionless membrane spatial coordinate. [-] θ Dimensionless time. [-] Θ Relative reaction coefficient. [-] κ Dimensionless reaction rate constant. [-] λ Dimensionless tube/shell spatial coordinate. [-] λe Effective thermal conductivity. [J m-1 s-1 K-1] ν Stoichiometric coefficient. [-] Polymeric Membrane Reactors ( 11ν=− A, , 11ν=− B, , 11ν= C, , 10 ν = D, , 20 ν = A, , 21 ν =− B, , 21 ν =− C, , 21ν= D, ). [-] σC Selectivity to species C. [-] τ Residence time. [s] ϕ Ratio of the reactants composition in the feed stream (based on reaction 1). [-] Φ Thiele modulus. [-] ΨA Relative conversion (ratio between the conversion of reactant A , A X , and the thermodynamic equilibrium one based on the feed conditions, E A X ) [-] Ω Dimensionless heat generation parameter. [-] Subscripts i Component i. [-] j Reaction j. [-] ref Reference conditions or component. [-] Superscripts * Dimensionless variable. [-] F Relative to the feed stream conditions. [-] G Relative to the entire reactor (Stanton number definition). [-] m Relative to the membrane. [-] O Relative to the exit conditions (CSTR model equations). [-] P Relative to the permeate chamber conditions. [-] r Relative to reaction. 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