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Ind. Eng. Chem. Res. 2009, 48, 1134–1139
Three-Dimensional Model of a Membrane Reactor Configuration with Cooling Tubes A. Ramı´rez Serrano,†,‡ F. Tiscaren˜o Lechuga,*,‡ and J. A. Ochoa Tapia§ Facultad de Quı´mica, UniVersidad Auto´noma del Estado de Me´xico, Paseo Colon esquina Paseo Tollocan S/N, Cipres, 50120 Toluca, Estado de México, Mexico, Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´gico de Celaya, AV. Tecnolo´gico S/N, Col. FoVisste, C.P. 38010, Celaya, Guanajuato, Mexico, and Area de Ingenieria Quimica, UniVersidad Autonoma Metropolitana, Unidad Iztapalapa, AV. San Rafael Atlixco 186, Col. Vicentina, C.P. 09340, Mexico City, D.F., Mexico
Membrane reactors are a promising alternative to carry out reactions in which the selectivity can be enhanced by the controlled addition of reactants. In addition, it also is very important to keep the temperature profile within a favorable range. In this work a new configuration is proposed for a membrane reactor with internal heat transfer sources. The configuration is similar to that of a shell and tube heat exchanger with the differences that the shell side is packed with catalyst and some of the tubes are membranes. Although the model assumes plug flow, it considers changes in the superficial velocities due to temperature profiles and variations of the molar flow rates. This geometry requires three-dimensional considerations; however, the solution took advantage of a pseudo-three-dimensional model. Controlled addition of reactant through the membrane not only reduces the effect of the side reactions but also alleviates the hot spots since the rates of reactions are decreased. Introduction Membrane reactors have been proposed as an alternative to improve the yield of reactions limited by equilibrium conversion.1-3 Although the technical feasibility has been proven for several dehydrogenation reactions with both ceramic and metallic membranes, this technology is not yet in industrial use.4-8 New membranes were tested with dehydrogenation reactions proving that conversions above the equilibrium conversion can be achieved due to the selective removal of the produced hydrogen. However, for some cases with permselective membranes, the major contribution to the increase in conversion was the dilution of the reacting mixture rather the removal of hydrogen.9 Saucedo et at.10 showed that the endothermic nature of this reaction leads to operating conditions in which the presence of the membrane does not present a real advantage. Later, Cordova11 studied a modified Cosden-Badger process for the dehydrogenation of ethylbenzene that considered heated membrane reactors. Even assuming ideal membranes and low-priced membrane reactors, economic analysis revealed that this process is unfeasible. If membrane reactors are not as attractive for dehydrogenation reactions as initially believed more than 20 years ago, there is another major field of opportunity for membrane reactors. Membrane reactors have been widely studied for partial oxidation reactions in which selectivity can be enhanced by the controlled addition of reactants.12-16 Membrane reactors promise to dose oxygen in a safe way, preventing the formation of undesired products with low commercial value (CO2 and H2O). Several research groups have reported that better conversions and yields were obtained using membrane reactors than using conventional fixed bed reactors.17-20 Production of defect-free membranes with adequate and consistent properties is still required at reasonable costs. At present, no evidence has been found of any process on the industrial scale that is applying this new technology. * To whom correspondence should be addressed. Tel.: (461) 6117575, ext 146. Fax: (461) 611-7744. E-mail:
[email protected]. † Universidad Auto´noma del Estado de Me´xico. ‡ Instituto Tecnolo´gico de Celaya. § Universidad Auto´noma Metropolotana.
Usually, isothermal conditions are pursued for laboratory- and pilot-scale studies; therefore, membrane reactors are placed inside controlled temperature environments such as furnaces or fluidized sand baths. However, once suitable membranes are available and aiming for industrial-scale operation, it will be relevant to evaluate the return of placing cooling elements within membrane reactors for partial oxidation applications. Our premise is that, along with dosing the oxygen, cooling the reacting chamber not only maintains the temperature profiles within favorable ranges but also alleviates explosion hazards. The proposed reactor configuration is similar to that of a heat exchanger, where the catalyst is packed in the shell side and some of the pipes are used for heat transfer while the rest are for mass transfer (membranes). Mathematical statements for membrane reactors with multiple internal pipes may require onedimensional, two-dimensional (2D), or three-dimensional (3D) modeling depending on the specific geometry. However, since the catalyst is packed in the shell side, three-dimensional models are necessary even in the absence of membranes. This approach is intended to be used in future optimization studies of the internal configuration of cooled membrane reactors for partial oxidations. Theory The model is based on mass and energy balances in terms of average concentration and temperature with effective transport coefficients.21 The reacting chamber is packed with catalyst, but it is described as a pseudohomogeneous system with explicit catalytic rate expressions.22,23 The reactor configuration has a packed bed chamber with some imbedded tubular membranes and some imbedded heat exchanger tubes. Figure 1 depicts a schematic representation of the proposed configuration for the membrane reactors. As shown in the top left corner of this figure, this particular arrangement has 17 heat exchanger tubes and 16 tubular membranes corresponding to the white and black circles, respectively; however, it is more efficient to solve only a symmetry unit. The complete cross section can be formed with 16 of these symmetry units; each unit is adjacent to its mirror
10.1021/ie800641h CCC: $40.75 2009 American Chemical Society Published on Web 11/05/2008
Ind. Eng. Chem. Res., Vol. 48, No. 3, 2009 1135
Figure 1. Cross section of a symmetry unit of a membrane reactor with internal heat transfer tubes.
image. It should be pointed out that this configuration has not been optimized, but it contains all the mathematical elements of a general model. For purposes of demonstration, the model assumes that only one of the reactants, oxygen, permeates through the membranes. However, it is not difficult to consider that other components may also pass through the membrane. Heat transfer through the membrane walls is neglected compared to that of the heat exchanger tubes; as a consequence, zero heat flux boundary condition was imposed in this location. The reaction occurs in the shell side, which is packed with catalyst; the model uses a pseudohomogenous representation. Therefore, all the transport parameters are effective coefficients. The explicit rate expressions employed already include the effects of internal and external mass and heat resistances. The mathematical model was developed assuming the following: (a) Steady state was assumed, but 3D implications result by actually solving a modified 2D transient state model. (b) Axial dispersion and conductivities are negligible compared to the order of magnitude of the corresponding convection terms:
(
∂Ci ∂ ∂ (u C ) . D ∂z s i ∂z e ∂z
)
and ∂ ∂T ∂ (u Fc T) . k ∂z s p ∂z e ∂z (c) Effective diffusivities and conductivities in the x and y directions are constant. (d) Heat transfer coefficients for the internal wall of each heat transfer tube, hc, are equal and constant. (e) The membrane is made of a dense permselective layer, and only oxygen has a significant permeability. (f) The membrane support behaves as a thermal insulator. (g) Heat transfer associated with the sensible heat of the oxygen flux that permeates the membrane is neglected (oxidation of ethylene was selected as the case of study). (h) Superficial velocity may change along the axial position. However, it is independent of the x or y coordinates; that is, the axial velocity by z is recalculated locally with the plug flow assumption. (i) All flow rates are concurrent to keep simple “initial” conditions for the parabolic pseudo-3D governing equations. Boundary Conditions. Based on the nomenclature given in Figure 1, boundaries I-VIII are simply symmetry conditions
(
)
for both heat and mass transfer. Boundary XIV assumes that the external reactor wall is insulated and impermeable to oxygen; however, it could be easily modified to consider an external heated jacket. Boundaries IX, X, and XI correspond to cooling tubes.24-26 Each tube side is assumed to behave as plug flow; therefore, the internal temperature is imposed equally for a given axial position. It is assumed that the temperature of the tube wall is that of the packed bed side and the only boundary layer that exists is in the tube side; on the packed bed or shell side the radial heat transfer is described by a unique radial effective heat transfer coefficient. Since the temperature on the shell side changes with position, it is necessary to locally solve the following equation to calculate the axial changes on the heat exchanger tube side: dTc,j ) dz
∫
aj
0
hc,j(Tc,j - T) daj j c,jjcp,j F
(1)
where aj is the perimeter of the heat exchanger tube and the subscript j refers to each of the three tubes in this given configuration. Boundaries XII and XIII are associated with the membranes. In this case, it is assumed there are two-dimensional composition profiles and the internal axial velocity with the membrane tube behaves as a plug flow. Mass transfer through the membrane’s wall is given by6,27,28 n·
|
De Pm ∇Ni wallNi ) [(PO2)shell - (PO2)membrane] us δ
(2)
All the walls including those associated with the membranes are assumed to be impermeable to the rest of the components of the reacting mixture: n · ∇Ni ) 0. Mass and Energy Balances within the Reacting Chamber. The model assumes plug flow to avoid the need for solving the momentum balance and to still be able to employ the pseudo3D model. If this assumption were not imposed, it would have been necessary to solve the Navier-Stokes equation considering the geometry shown in Figure 1. Even if plug flow is assumed, it must be recalculated at each axial position based on the cross section temperature profiles and the variation of the total molar flow rate and pressure.29 The mass and energy balances are
(
)
∂Ni De ∂2Ni ∂2Ni ) + 2 - FBrp,i ∂z us ∂x2 ∂y
(
)
ke ∂2T ∂2T ∂T ) + ∂z usFcp ∂x2 ∂y2
∑
FB(-∆Hr)rp,r usFcp
(3) (4)
1136 Ind. Eng. Chem. Res., Vol. 48, No. 3, 2009
where Ni ) usCi is the molar flux of component i, subscript r refers to a given independent reaction, and rp is the catalytic reaction rate that includes internal and external resistances. It is necessary to solve eq 3 at least for each one of the stoichiometrically independent chemical species. Since there are two independent reactions in the case of study, the two key components are oxygen and ethylene. In addition, due to the assumption that within the membrane tube the concentration profiles depend on the position, it is necessary to include the mass balances for that region, but in this case without the reaction term:
(
( )
)
Dm ∂2Gi ∂2Gi ∂Gi ) + 2 (5) ∂z k um ∂x2 ∂y Here Gi is the molar flux of oxygen and k is the membrane number within the symmetry unity. All the concentrations or partial pressures can be easily calculated from the fluxes, temperatures, and pressures; then, the reaction rates are calculated with the explicit catalytic or global rate expressions. All the variables and parameters on the right-hand sides of these equations are known or calculated as the integration in the z direction is carried out. If the superficial velocity us is kept constant, the model can be easily solved with initial conditions for this pseudo-3D model. However, if this is not the case, the recalculation of the local superficial velocity requires a few local calculations. First, it is necessary to obtain the molar flow rate for the independent components with the local fluxes: Fi )
∫ N dS
where S is the cross section area of the reacting chamber. With the flow rate of these two independent components and stoichiometry, the rest of the molar flow rates can be easily obtained as long as their transport equations have the same effective transport coefficients. Then, the total molar flow rate, which includes any inert gas, is known: num of species
FT )
∑
Fi
(7)
i)1
The temperature is obtained as an average temperature for each local cross section: Tavg )
1 T
∫ T dS S
(8)
Finally, the superficial velocity can be corrected using the ideal gas law for an open system: us ) TavgFT
k1
2C2H4 + O2 98 2C2H4O k2 2 2 1 C2H4 + O2 98 CO2 + H2O 3 3 3
(6)
i
S
balances were handled with coupling variables that had axial profiles but did not correspond to any specific x-y position within the cross section. For the Comsol environment, in fact, it would be easier to solve for internal cross section profiles because a flat profile within either tube also requires solving for local integrations such as that in eq 1 at each given axial position. We chose these assumptions because they represent a more interesting and general solution approach; however, the model can be readily modified so that both tube types could consider cross section two-dimensional profiles or flat profiles. Case Study: Oxidation of Ethylene. One of the most attractive reactions reported in the literature that has been considered for the implementation of membrane reactors is the oxidation of ethylene.17,18,20 Its selectivity can be significantly improved by the controlled addition of reactants; however, it is critical to avoid temperature increases not only because the yield drops but also because of serious safety issues. Industrial production of ethylene oxide in the gas phase is carried out using multitubular packed bed reactors with Ag/Al2O3 as catalyst. However, it should be clear that those tubes are packed with catalyst whereas in our configuration the catalyst is placed in the shell side. For our case of study, the following reaction system was chosen:
( ) Rg SPT
(9)
According to this procedure, even if plug flow assumption is imposed, the superficial velocity is affected as result of significant changes in total molar flow rates, temperature, or pressure. The differential equations of the pseudo-3D model have a parabolic nature, and therefore only initial conditions are required for the axial position. The model was solved with the commercial software Comsol,30 which employs finite element techniques.31,32 For the configuration presented in Figure 1, three subdomains were employed: one subdomain for the reacting chamber with two independent mass balances and the energy balance; two subdomains for each membrane tube with the corresponding mass balance for oxygen. Since uniform temperatures were assumed across each heat exchanger tube, temperatures and energy
k3 2 C2H4O + O2 98 2CO2 + 2H2O 5
It should be noted that only two of these reactions are independent. At a large excess of ethylene as applied in our base case, literature reports agree that the rate equations simplify to first-order kinetics in the oxygen concentration.33-35 Westerterp and Ptasinski33 presented expressions for oxygen consumption and corresponding rate constants for industrial operating conditions: r1 ) k1CO2 and r2 ) k2CO2, where k1 ) 70.4 exp (-7200/T) and k2 ) 4.94 × 104 exp (-10800/T). Inspection of this reaction system reveals that the contribution of the secondary reactions can be diminished if the concentration of oxygen is kept low. Moreover, since the activation energies of the secondary reactions are larger than that of the main reaction, a temperature increase will also decrease the selectivity. Therefore, cooling the reaction chamber should increase the yield besides avoiding operating conditions near explosion risks. In the numerical simulations the internal diameter and length of the reactor were set equal to 10 cm and 1.125 m, respectively. The internal diameter of the membrane and the heat exchanger tubes were both 1.6 cm. Other specifications were as follows: (a) Reacting chamber: total feed flow rate, 120 mol/min at 220 °C and 10 bar; ethylene molar fraction, 0.9; the rest was an inert. Although it was not considered at this point because it should be also optimized, some air or oxygen could also be fed along with the ethylene. (b) Membrane tubes inlet: 61 mol/min air at 220 °C and 1.5 bar. (c) Heat exchange tubes inlet: 168 cm3/min oil at 220 °C and 1.013 bar. Transport parameters and physical properties were taken from the literature.17,18,35,36 Effective transport properties for the reacting chamber were a diffusivity of 8.03 cm2/s and a
Ind. Eng. Chem. Res., Vol. 48, No. 3, 2009 1137
Figure 2. Cross section temperature profile within the reacting chamber at the exit of the reactor.
Figure 3. Variation of the cross section average temperature with the axial position.
conductivity of 0.000 014 17 W/cm K. Both of these transport properties correspond to any direction parallel to the cross section. If cylindrical coordinates were employed, these parameters would be referred to as radial effective properties; however, Comsol employs Cartesian coordinates. The heat transfer coefficient for the inner wall of the heat exchange tubes was 0.029 W/cm2 K. The permeability of the membrane was 4.56 × 10-8 cm3/cm2 s bar. The packed bed density was 0.850 g/cm3. Results In Figure 2 are shown the temperature profiles of a cross section at the exit of the reactor. The letters “T” and “M” indicate if the neighboring area is a heat exchanger tube or a membrane tube, respectively. Because of the exothermic nature of the reacting system, higher temperatures are found close to the membrane where one of the reactants is fed and far from the cooling tubes. Inspection of these profiles makes clear the need to consider 2D cross section profiles giving rise to a 3D model. The temperature difference between the hottest and the coldest spots at the exit was only about 20 °C, which could be further decreased if required. However, the final optimization should be based on the economic analysis of the complete plant and not just based on the reactor performance or flattering the profiles. In Figure 3 is presented the longitudinal profile for the cross section average temperature of the reacting chamber. The average temperature only increased by less than 6 °C between the inlet and the outlet. In Figure 4, temperature radial profiles at the exit of the reactor for three specific angular positions are presented. This
Figure 4. Radial temperature profiles for three angular positions at the reactor exit.
figure was included to complement information that is harder to visualize from the color scale in Figure 2. As indicated in the top left corner of Figure 4, red curves in this figure refer to 0°, while the green and blue curves correspond to 11.25° and 22.5°, respectively. The angles 0° and 22.5° correspond to the boundaries of our symmetry unit. The profiles are pertinent only to the reacting chamber, and the discontinuities are due to the presence of cooling or membranes tubes. Inspection of Figure 4 reveals that in general the temperature decreases as the radius increases except for the profile for 0° which exhibits a minor increase. These variations suggest the need to optimize the relative position and number between heat transfer and membrane tubes. Figure 5 depicts the oxygen concentration two-dimensional profile at the exit of the reactor. The concentration exhibits a significant variation from 0.0264 to 0.0512 mol/m3. The center of the reactor has higher oxygen concentrations because this region has a higher ratio of membrane transfer area to weight of catalyst for this configuration. Concentration profiles for ethylene are not shown because variations were minimal since it was fed in excess. Figure 6 shows axial concentration profiles for the limiting reactant and the desired product. Axial profiles are given for some specific x-y positions that are declared in Figure 5. Continuous curves relate to oxygen, whereas dotted curves relate to ethylene oxide. The product concentration increased nearly linearly through the reactor excluding the entrance, where it shows a lag. This lag can be diminished if oxygen is also fed at the inlet instead of just being fed through the membranes. The oxygen average concentration reached a maximum around
1138 Ind. Eng. Chem. Res., Vol. 48, No. 3, 2009
Figure 5. Cross section concentration profile within the reacting chamber at the exit of the reactor.
tubes had significant effects on the reactor performance. If a membrane reactor without cooling were operated with the same conditions, the temperature would increase rapidly leading to explosion limits and resulting in numerical overflows while attempting to solve the mathematical modeling. On the other hand, if the inlet temperatures were lowered so that the model without cooling could also be solved, then the advantages of providing additional cooling would be veiled. Conclusions
Figure 6. Variation of oxygen and ethylene oxide concentrations with reactor length.
0.5 m from the inlet. Near the reactor entrance, oxygen fed through the membranes was higher than its consumption by the reactions. As the average temperature increased, reaction rates were higher than mass transfer across the membranes. However, inspection of the oxygen profiles reveals that the oxygen concentration was maintained somewhat constant throughout the reactor length. Temperature profiles indicate that the cooling tubes prevented a significant temperature increase, whereas membranes helped to maintain a somewhat constant oxygen concentration of ca. 0.035 mol/m3. In general, the proposed reactor configuration is performing as desired. However, these results are shown for demonstration purposes and it should be noted that the internal configuration has not been optimized yet. Furthermore, the inlet conditions and flow rates do not necessarily need to be the same for each tube. Some of these issues are to be explored with the tool presented in this work. At initial stages, the configuration could be optimized based on simplified objective functions such as maximizing the yield or decreasing the temperature differences; however, eventually the optimization should consider costs for the complete plant including raw materials and sale prices. It is clear from the results presented up to this point that the general configuration proposed in this work needs threedimensional considerations. However, it may raise a doubt if our configuration presents advantages over membrane reactors without internal cooling. This computational tool was developed to answer that question. However, it should be explained why a comparison was not provided. Operating conditions were selected so that both heat transfer and mass transfer across the
Different configurations for this membrane reactor with internal heat exchange were explored, and it can be concluded that the insertion of cooling tubes enhances the performance of the membrane reactor for the partial oxidation of ethylene. Controlled addition of reactant through the membrane not only reduces the contribution of the side reactions but also alleviates the hot spots since the rate of reactions are decreased, leaving more time for the heat dispersion within the catalyst bed. However, partial oxidations are highly exothermic with side reactions that become more significant as the temperature increases; additional internal cooling has the advantage that the rate of addition of reactant can be increased without sacrificing selectivity and leading to more compact reactors. Inclusion of heat exchanger tubes led to flatter temperature profiles for each cross section, reducing the hot spots along the axial profile. Packing the shell side of the proposed reactor configuration requires a 3D model if several membrane and heat exchanger tubes are employed. However, the design equations and the energy balance were derived from parabolic differential equations of a transient 2D model. This artifice along with solving for only a symmetry unit reduced significantly the computer memory requirements and the computational time. The model and its solution method represent a useful tool that can be used in future studies to optimize reactor configurations compatible with the family of symmetry units presented. The effect of total number of membrane and cooling tubes and their relative positions on the conversion and yield must be studied. Later, a refined configuration can be used to assess advantages and determine if the industrial implementation of membrane reactors with internal cooling is economically feasible. Notation a ) perimeter (cm) Ci ) molar concentration of species i (mol/cm3) cp ) heat capacity (cal/mol K) De ) effective diffusivity in the shell (cm2/s)
Ind. Eng. Chem. Res., Vol. 48, No. 3, 2009 1139 Dm ) effective diffusivity in the membrane (cm /s) Fi ) molar flow (mol/s) Gi ) molar flux of species i in the membrane (mol/s cm2) hc ) film of heat transfer coefficients (cal/cm2 s K) Hi ) enthalpy of the species i (cal/mol) k ) reaction rate constant (cm3/g s) ke ) effective conductivity (cal/cm s K) Ni ) molar flux molar of the species i (mol/s cm2) n ) unit normal vector Pi ) partial pressure of the species i (bar) Pm ) permeability of the species i (cm3/cm2 s bar) PT ) total pressure (atm) Q ) heat flux (cal/cm2 s) Rg ) gas constant S ) cross-sectional area at planes (cm2) T ) absolute temperature (K) um ) superficial velocity in the membrane (cm/s) us ) superficial velocity (cm/s) x, y, z ) Cartesian coordinates (cm) ∇ ) nabla operator 2
Subscripts B ) bed catalytic j ) number of tube i ) species i r ) reaction number ref ) reference Greek Symbols δ ) thickness of the membrane (cm) F ) density (kg/cm3)
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ReceiVed for reView April 20, 2008 ReVised manuscript receiVed August 20, 2008 Accepted September 3, 2008 IE800641H
