Theoretical Prediction of the Efﬁciency of Hydrogen Production via Alkane Dehydrogenation in Catalytic Membrane Reactor

: The hydrogen economy is expected to dominate in the nearest future. Therefore, the most hydrogen-containing compounds are considered as potential pure hydrogen sources in order to achieve climate neutrality. On the other hand, alkanes are widely used to produce industrially important monomers via various routes, including dehydrogenation processes. Hydrogen is being produced as a by-product of these processes, so the application of efﬁcient separation of hydrogen from the reaction mixture can give double beneﬁts. Implementation of the dehydrogenation processes in the catalytic membrane reactor is that case. Since the use of dense metal membranes, which possess the highest perm-selectivity towards hydrogen, is complicated in practice, the present research is aimed at the optimization of the porous membrane characteristics. By means of a mathematical modeling approach, the effects of pore diameter on the hydrogen productivity and purity for the cases of ethane and propane dehydrogenation processes were analyzed. The pore size value of 0.45 nm was found to be crucial as far as the diffusion of both the alkane and alkene molecules through the membrane takes place. dehydrogenation, are able to permeate the membrane, and the hydrogen purity starts decreasing. Since ethane and ethylene molecules are bigger, hydrogen produced via the ethane dehydrogenation remains to be pure. A sharp fall of the hydrogen purity for both processes is observed when the pore size becomes 0.5 nm. The situation is more crucial for propane since the side reactions taking place during the dehydrogenation process are also being accelerated [15]. Methane formed as a by-product also penetrates the membrane and decreases the hydrogen purity already at the pore size of 0.45 nm.


Introduction
During the last decade, hydrogen has become the most attractive energy source playing a key role in multi-sectorial decarbonization processes [1][2][3][4]. Therefore, the longtimeknown processes for clean hydrogen production from hydrocarbons, including catalytic decomposition of methane, have gained a special actuality [5][6][7][8][9][10][11][12]. In most cases, hydrocarbons are being decomposed into carbon and hydrogen that give the highest efficiency towards hydrogen as a target product. The carbon formed as a by-product is often considered as a potential value-added product if its characteristics are good enough for the wide practical application [13,14].
On the other hand, the decomposition of hydrocarbon molecules can be incomplete, as in the case of dehydrogenation reactions when hydrogen is being co-produced along with unsaturated hydrocarbon (alkene) [15]. Since alkenes are highly demanded as monomers for the large-scale chemical industry, such an approach is very favorable from the economic point of view. The alkane dehydrogenation processes can be, therefore, considered as an alternative for the existing hydrogen production methods, including steam reforming of methane and natural gas [16,17]. The efficiency of the dehydrogenation processes in terms of hydrogen productivity can be noticeably improved by involving membrane technologies. Thus, the realization of the process in a catalytic membrane reactor (CMR) allows the elimination of produced hydrogen from the reaction zone through the permselective membrane along with an increase in conversion of the hydrocarbon substrate. In this case, the hydrogen-containing and alkene-containing gaseous flows are spatially separated. Another advantage of using the CMR is that due to a shift of thermodynamic Hydrogen 2021, 2 363 equilibrium, the dehydrogenation process can be realized at lower temperatures. Such a decrease in the process temperature positively affects the selectivity towards the target products (hydrogen and alkene) and allows minimizing the coke formation.
The main problem restricting the application of CMR in the industry is connected with the absence of inorganic membranes, which possess appropriate perm-selectivity and provide long-term stability at elevated temperatures. In general, a large number of research works reported in the literature deal with both dense metal membranes and porous membranes of different nature [18][19][20]. Among these materials, the dense metal membranes are characterized with the highest perm-selectivity towards hydrogen. However, such membranes are required to be almost ideal since any imperfectness or inconstancy of the structure (for instance, the formation of defects of cracks on the surface) causes an uncontrollable flux of all the reaction mixture components to the permeate side of the reactor. Usually, the embrittlement of the metal membranes takes place at high temperatures and can be facilitated by the action of the reaction medium. Porous ceramic membranes are significantly less perm-selective than the dense ones but exceed them in thermal stability and flux capability.
The present work is aimed at the optimization of the porous membrane's characteristics providing the maximal conversion of alkane along with the highest productivity of permeated hydrogen. The question of hydrogen purity is considered as well. The optimization procedures were performed by simulating the processes of ethane and propane dehydrogenation in the CMR. Mathematical modeling is widely used for the analysis of various processes in the reactors of different types, including membrane reactors, as well as for the optimization of the parameters of the membrane, reactor, and overall process [21,22]. Recently, a two-dimensional non-isothermal mathematical model of the CMR was developed and verified for the alkane dehydrogenation processes [23][24][25][26]. Here, the process efficiency (alkane conversion, hydrogen productivity) was analyzed as a function of the pore diameter of the ceramic membrane of a constant thickness. The prospective of hydrogen production via the catalytic dehydrogenation of alkanes is elucidated.

Description of the Reactor and the Mathematical Model
The scheme of the CMR used for the modeling is presented in Figure 1. A tubular reactor consisting of two tubes, inner and outer, is considered. The inner tube, in its turn, consists of the macroporous ceramic support covered with a layer of microporous oxide membrane. Therefore, the inner tube is permeable. The tube side (inside the inner tube) is filled with the dehydrogenation catalyst. The gaseous reaction mixture containing nitrogen with ethane or propane feeds the tube side of the CMR. The shell side (between the inner permeable and outer impermeable tubes) is purged with a sweep gas to output the hydrogen and other components fluxed through the inner tube.
Hydrogen 2021, 2, 2 gaseous flows are spatially separated. Another advantage of using the CMR is that due to a shift of thermodynamic equilibrium, the dehydrogenation process can be realized at lower temperatures. Such a decrease in the process temperature positively affects the selectivity towards the target products (hydrogen and alkene) and allows minimizing the coke formation. The main problem restricting the application of CMR in the industry is connected with the absence of inorganic membranes, which possess appropriate perm-selectivity and provide long-term stability at elevated temperatures. In general, a large number of research works reported in the literature deal with both dense metal membranes and porous membranes of different nature [18][19][20]. Among these materials, the dense metal membranes are characterized with the highest perm-selectivity towards hydrogen. However, such membranes are required to be almost ideal since any imperfectness or inconstancy of the structure (for instance, the formation of defects of cracks on the surface) causes an uncontrollable flux of all the reaction mixture components to the permeate side of the reactor. Usually, the embrittlement of the metal membranes takes place at high temperatures and can be facilitated by the action of the reaction medium. Porous ceramic membranes are significantly less perm-selective than the dense ones but exceed them in thermal stability and flux capability.
The present work is aimed at the optimization of the porous membrane's characteristics providing the maximal conversion of alkane along with the highest productivity of permeated hydrogen. The question of hydrogen purity is considered as well. The optimization procedures were performed by simulating the processes of ethane and propane dehydrogenation in the CMR. Mathematical modeling is widely used for the analysis of various processes in the reactors of different types, including membrane reactors, as well as for the optimization of the parameters of the membrane, reactor, and overall process [21,22]. Recently, a two-dimensional non-isothermal mathematical model of the CMR was developed and verified for the alkane dehydrogenation processes [23][24][25][26]. Here, the process efficiency (alkane conversion, hydrogen productivity) was analyzed as a function of the pore diameter of the ceramic membrane of a constant thickness. The prospective of hydrogen production via the catalytic dehydrogenation of alkanes is elucidated.

Description of the Reactor and the Mathematical Model
The scheme of the CMR used for the modeling is presented in Figure 1. A tubular reactor consisting of two tubes, inner and outer, is considered. The inner tube, in its turn, consists of the macroporous ceramic support covered with a layer of microporous oxide membrane. Therefore, the inner tube is permeable. The tube side (inside the inner tube) is filled with the dehydrogenation catalyst. The gaseous reaction mixture containing nitrogen with ethane or propane feeds the tube side of the CMR. The shell side (between the inner permeable and outer impermeable tubes) is purged with a sweep gas to output the hydrogen and other components fluxed through the inner tube. The two-dimensional non-isothermal stationary mathematical model includes the mass and energy balance equations with the appropriate boundary conditions for tube and shell sides, for the ceramic support and porous membrane. The modeling of the heat and mass transfer processes was carried out by means of a one-dimensional plug-flow model for the shell side.
A few simplifying assumptions were applied: 1.
The steady-state conditions are considered; 2.
The convective radial transfer is negligible; 3.
The internal mass and energy transport limitations inside the catalyst pellets, as well as external mass and heat transfer resistances at the surface of the pellets, are negligible.
The equations of the mass and energy balances are given in Appendix A. Appendix B presents the list of all symbols used in the equations.

Parameters of the Mathematical Model
The effective coefficient of radial diffusion for the ceramic support layer and membrane is described by the equation: D c,m π M i /10 3 . The effective coefficient of molecular diffusion is determined by the Wilkes formula: For modeling the mass transfer processes in the microporous membrane, the correction coefficient b is taken into consideration. The coefficient b reflects the diffusion limitation, which appears when the pore diameter and the molecule size are of comparable values. Other dependences used for the determination of the mass and heat transfer coefficients are as reported elsewhere [15,23].
The mathematical model parameters used for the simulation of the CMR are summarized in Table 1.

Ethane Dehydrogenation
The ethane dehydrogenation reaction is considered in the tube side of the membrane reactor in accordance with the following equation: For the numerical study of the process, the observed kinetic parameters, reported for the Pd/Al 2 O 3 catalyst with a granule diameter of 3.35 mm and a height of 3.63 mm, are used [27]. The reaction rate of ethane dehydrogenation is described as follows:

Propane Dehydrogenation
The target and side reactions considered in the tube side of the membrane reactor during the propane dehydrogenation process are the following: The kinetic parameters for the mentioned reactions were determined for the Pt-Sn-K/Al 2 O 3 catalyst with a particle diameter of 0.16-0.25 mm by Lobera et al. [28]. Table 2 summarizes these parameters. Table 2. The kinetic parameters for the propane dehydrogenation process used for the modeling.

Reaction Rate and Rate Constant Equations
Reaction Rate Constant at T 0 , k 03 = 0.000236 E a2 = 154.54

Numerical Solution of the Model Equations
The mathematical model of the catalytic membrane reactor consists of the partial differential equations (Appendix A). For the obtained systems of nonlinear partial derivatives equations with the variable coefficients, an analytical solution does not exist. Therefore, the search for a solution was carried out using numerical methods.
The system was transformed into a non-linear set of the ordinary differential equations (ODE) to calculate the temperatures and the concentrations of all components. This transformation was performed using the methods and approaches reported elsewhere [29]. The obtained ODEs were solved using a semi-implicit Rosenbrock-type method of second-order accuracy with an automatic choice of the integration step [30].

Results and Discussion
As mentioned in the previous section, hydrogen permeated through the membrane from the reaction zone (tube side) to the shell side was outputted by a flow of the sweep gas. Such a regime complicates the illustration of the hydrogen concentration profile for the shell side. Therefore, the values of the hydrogen concentration in each point of the reactor length to image the concentration profile were obtained as the subtraction of the hydrogen amount left in the tube side from the amount of produced hydrogen during the dehydrogenation process. Figure 2 demonstrates the resulting hydrogen concentration profiles along the reactor length for the shell side of the CMR, for the cases of ethane and propane dehydrogenation. The dependence of the alkane conversion values at the reactor outlet on the pore diameter is plotted in Figure 3. As seen, in the case of ethane dehydrogenation, the hydrogen concentration for pore diameter (d p ) values less than or equal to 0.45 nm increases as along the reactor length as with the growth of d p (Figure 2a). At d p values above 0.45 nm, the profiles exhibit a falling tendency. The outlet conversion values tend to be near 39% (Figure 3).
hydrogen amount left in the tube side from the amount of produced hydrogen during the dehydrogenation process. Figure 2 demonstrates the resulting hydrogen concentration profiles along the reactor length for the shell side of the CMR, for the cases of ethane and propane dehydrogenation. The dependence of the alkane conversion values at the reactor outlet on the pore diameter is plotted in Figure 3. As seen, in the case of ethane dehydrogenation, the hydrogen concentration for pore diameter (dp) values less than or equal to 0.45 nm increases as along the reactor length as with the growth of dp (Figure 2a). At dp values above 0.45 nm, the profiles exhibit a falling tendency. The outlet conversion values tend to be near 39% (Figure 3). In the case of propane (Figure 2b), the shape of the hydrogen concentration profiles has also changed with the pore diameter increase but not so crucial. The breaking dp value here is 0.5 nm. Propane conversion at such a value of the pore diameter reaches 86%, and the corresponding outlet hydrogen concentration in the shell side is equal to 0.08 m.f.
The observed decrease of the hydrogen concentration in the shell side of the CMR with the growth of the pore diameter is connected with the size comparability of membrane pores and hydrocarbon molecules. The membrane becomes permeable for the other components of the reaction volume (initial alkane, reaction products). It should be mentioned that the diffusion rates for these components are noticeably lower if compared with the hydrogen permeation rate. Nevertheless, these diffusion processes negatively contribute to most of the dehydrogenation process parameters (hydrogen purity, alkane conversion, etc.).  Another important aspect of the hydrogen production in the CMR, which sh taken into consideration, is the distribution of hydrogen between the shell and tu of the reactor at the reactor outlet (named as a hydrogen S/T ratio). Figure 4 sh hydrogen profiles along the reactor length for the tube and shell sides and t amount of the produced hydrogen. Additionally, the curve for a traditional tub actor (TR, inner tube is not permeable for any gases) is presented for compariso curve represents the thermodynamic limitations of the corresponding dehydrog processes.  In the case of propane (Figure 2b), the shape of the hydrogen concentration profiles has also changed with the pore diameter increase but not so crucial. The breaking d p value here is 0.5 nm. Propane conversion at such a value of the pore diameter reaches 86%, and the corresponding outlet hydrogen concentration in the shell side is equal to 0.08 m.f.
The observed decrease of the hydrogen concentration in the shell side of the CMR with the growth of the pore diameter is connected with the size comparability of membrane pores and hydrocarbon molecules. The membrane becomes permeable for the other components of the reaction volume (initial alkane, reaction products). It should be mentioned that the diffusion rates for these components are noticeably lower if compared with the hydrogen permeation rate. Nevertheless, these diffusion processes negatively contribute to most of the dehydrogenation process parameters (hydrogen purity, alkane conversion, etc.).
Another important aspect of the hydrogen production in the CMR, which should be taken into consideration, is the distribution of hydrogen between the shell and tube sides of the reactor at the reactor outlet (named as a hydrogen S/T ratio). Figure 4 shows the hydrogen profiles along the reactor length for the tube and shell sides and the total amount of the produced hydrogen. Additionally, the curve for a traditional tubular reactor (TR, inner tube is not permeable for any gases) is presented for comparison. This curve represents the thermodynamic limitations of the corresponding dehydrogenation processes. Another important aspect of the hydrogen production in the CMR, which should be taken into consideration, is the distribution of hydrogen between the shell and tube sides of the reactor at the reactor outlet (named as a hydrogen S/T ratio). Figure 4 shows the hydrogen profiles along the reactor length for the tube and shell sides and the total amount of the produced hydrogen. Additionally, the curve for a traditional tubular reactor (TR, inner tube is not permeable for any gases) is presented for comparison. This curve represents the thermodynamic limitations of the corresponding dehydrogenation processes. As seen, the total outlet hydrogen concentration can be significantly improved by performing the dehydrogenation process in CMR. In the case of ethane, this value for CMR exceeds that for TR by a factor of 3.3 (Figure 4a). Dehydrogenation of propane in CMR gives a 2.3 times higher concentration of hydrogen at the reactor outlet if compared with TR. Due to the hydrogen removal from the reaction zone, the equilibrium of the dehydrogenation reaction shifts towards the product side, thus improving the yields of hydrogen and alkene. By comparing the hydrogen profiles for tube and shell sides of the CMR, the hydrogen redistribution along the reactor length can be inferred. Initially, at the beginning of the reaction, hydrogen starts to form according to the dehydrogenation reaction, and its concentration in the tube side exceeds that in the shell side. Then, the  As seen, the total outlet hydrogen concentration can be significantly improved by performing the dehydrogenation process in CMR. In the case of ethane, this value for CMR exceeds that for TR by a factor of 3.3 (Figure 4a). Dehydrogenation of propane in CMR gives a 2.3 times higher concentration of hydrogen at the reactor outlet if compared with TR. Due to the hydrogen removal from the reaction zone, the equilibrium of the dehydrogenation reaction shifts towards the product side, thus improving the yields of hydrogen and alkene. By comparing the hydrogen profiles for tube and shell sides of the CMR, the hydrogen redistribution along the reactor length can be inferred. Initially, at the beginning of the reaction, hydrogen starts to form according to the dehydrogenation reaction, and its concentration in the tube side exceeds that in the shell side. Then, the curves intersect at 0.018 and 0.025 m in the cases of ethane and propane, correspondingly.  Figure 5 demonstrates how the hydrogen S/T ratio depends on the pore diameter. Both curves show a fast-growing slope until the pore diameter reaches 0.45 nm. Then, in the case of ethane, a sharp decrease followed by a relative plateau is observed. For the propane, the hydrogen S/T ratio curve continues to grow until 0.5 nm and then smoothly declines, reaching the value of 9.7 at the pore diameter of 0.7 nm. on the pore diameter. Both curves show a fast-growing slope until the por reaches 0.45 nm. Then, in the case of ethane, a sharp decrease followed by plateau is observed. For the propane, the hydrogen S/T ratio curve continu until 0.5 nm and then smoothly declines, reaching the value of 9.7 at the pore d 0.7 nm. The obtained hydrogen concentrations allow estimating the hydrogen p for both processes. The resulting dependences of this parameter on the pore d shown in Figure 6. The maximal values correspond to the pore diameters (ethane) and 0.5 nm (propane). Further increase in the pore diameter decrea drogen productivity 1.6 and 1.3 times, accordingly. It should also be men propane dehydrogenation seems to be more efficient and preferable in terms o productivity if compared with ethane dehydrogenation. Since the mathematical modeling allows following the distribution of gents along with the target and side products of the dehydrogenation proces The obtained hydrogen concentrations allow estimating the hydrogen productivity for both processes. The resulting dependences of this parameter on the pore diameter are shown in Figure 6. The maximal values correspond to the pore diameters of 0.45 nm (ethane) and 0.5 nm (propane). Further increase in the pore diameter decreases the hydrogen productivity 1.6 and 1.3 times, accordingly. It should also be mentioned that propane dehydrogenation seems to be more efficient and preferable in terms of hydrogen productivity if compared with ethane dehydrogenation. until 0.5 nm and then smoothly declines, reaching the value of 9.7 at the pore d 0.7 nm. The obtained hydrogen concentrations allow estimating the hydrogen p for both processes. The resulting dependences of this parameter on the pore d shown in Figure 6. The maximal values correspond to the pore diameters (ethane) and 0.5 nm (propane). Further increase in the pore diameter decrea drogen productivity 1.6 and 1.3 times, accordingly. It should also be men propane dehydrogenation seems to be more efficient and preferable in terms o productivity if compared with ethane dehydrogenation. Since the mathematical modeling allows following the distribution of gents along with the target and side products of the dehydrogenation process Since the mathematical modeling allows following the distribution of all the reagents along with the target and side products of the dehydrogenation processes, in both the tube and shell sides, parameters such as hydrogen purity can be estimated as well. As evident, the hydrogen purity parameter depends on the amount of other reaction mixture components diffused from the tube side to the shell side along with hydrogen. As shown in Figure 7, at pore sizes less than or equal to 0.4 nm, hydrogen dominantly transfers through the porous membrane, and its purity is about 100%. In order to simulate the transfer process via diffusion, the critical dimensions of the molecules were used (see Figure 8). If the pore size reaches 0.45 nm, methane molecules, as by-product of propane dehydrogenation, are able to permeate the membrane, and the hydrogen purity starts decreasing. Since ethane and ethylene molecules are bigger, hydrogen produced via the ethane dehydrogenation remains to be pure. A sharp fall of the hydrogen purity for both processes is observed when the pore size becomes 0.5 nm. The situation is more crucial for propane since the side reactions taking place during the dehydrogenation process are also being accelerated [15]. Methane formed as a by-product also penetrates the membrane and decreases the hydrogen purity already at the pore size of 0.45 nm.
components diffused from the tube side to the shell side along with hydrogen in Figure 7, at pore sizes less than or equal to 0.4 nm, hydrogen dominantl through the porous membrane, and its purity is about 100%. In order to si transfer process via diffusion, the critical dimensions of the molecules were Figure 8). If the pore size reaches 0.45 nm, methane molecules, as by-product dehydrogenation, are able to permeate the membrane, and the hydrogen pu decreasing. Since ethane and ethylene molecules are bigger, hydrogen produ ethane dehydrogenation remains to be pure. A sharp fall of the hydrogen puri processes is observed when the pore size becomes 0.5 nm. The situation is m for propane since the side reactions taking place during the dehydrogenation p also being accelerated [15]. Methane formed as a by-product also penetrates brane and decreases the hydrogen purity already at the pore size of 0.45 nm.  The diffusion of hydrocarbons through the porous membrane worsens process efficiency. As shown in Figure 9a,b, at a pore diameter of 0.5 nm and alkene concentration profiles go through the maximum and exhibit a declinin at the reactor outlet. Figure 9c,d demonstrates the cumulative amount of hyd that appeared in the shell side. In both cases, this amount increases with the in Figure 7, at pore sizes less than or equal to 0.4 nm, hydrogen dominantl through the porous membrane, and its purity is about 100%. In order to si transfer process via diffusion, the critical dimensions of the molecules were Figure 8). If the pore size reaches 0.45 nm, methane molecules, as by-product dehydrogenation, are able to permeate the membrane, and the hydrogen p decreasing. Since ethane and ethylene molecules are bigger, hydrogen produ ethane dehydrogenation remains to be pure. A sharp fall of the hydrogen pur processes is observed when the pore size becomes 0.5 nm. The situation is m for propane since the side reactions taking place during the dehydrogenation also being accelerated [15]. Methane formed as a by-product also penetrates brane and decreases the hydrogen purity already at the pore size of 0.45 nm.  The diffusion of hydrocarbons through the porous membrane worsens process efficiency. As shown in Figure 9a,b, at a pore diameter of 0.5 nm and alkene concentration profiles go through the maximum and exhibit a declinin at the reactor outlet. Figure 9c,d demonstrates the cumulative amount of hy that appeared in the shell side. In both cases, this amount increases with th The diffusion of hydrocarbons through the porous membrane worsens the overall process efficiency. As shown in Figure 9a,b, at a pore diameter of 0.5 nm and above, the alkene concentration profiles go through the maximum and exhibit a declining tendency at the reactor outlet. Figure 9c,d demonstrates the cumulative amount of hydrocarbons that appeared in the shell side. In both cases, this amount increases with the pore size Hydrogen 2021, 2 370 increase, and the shape of the concentration profile becomes curved. This indicates that the higher hydrocarbon diffusion rates give a higher contribution to the concentration of hydrocarbons in the shell side at the initial part of the reactor. from 0.35 to 0.45 nm, hydrogen preferably diffuses through the membrane that positively affects productivity. An increase in the pore size within this range raises the hydrogen diffusion rates at the same 100% purity of permeated hydrogen. The further increase in the pore diameter to 0.5 nm and higher sharply leads to a decrease in the olefins productivity due to the accelerated elimination of both initial alkanes and target alkenes from the reaction zone (the tube side). Therefore, the process of hydrogen and olefin co-production in the CMR can be efficient and reasonable when the ceramic membrane with an average pore size not exceeding 0.45 nm is applied. It is worth noting that the obtained results correlate well with the literature data on gas separation using porous membranes [31]. The productivity of the considered processes towards the target alkenes is presented in Figure 10. When the pore size is 0.3 nm or below, hydrogen molecules do not permeate through the membrane, and the olefins productivity corresponds to thermodynamic values, like in the traditional tubular reactor (see Figures 2 and 6). Within the pore size range from 0.35 to 0.45 nm, hydrogen preferably diffuses through the membrane that positively affects productivity. An increase in the pore size within this range raises the hydrogen diffusion rates at the same 100% purity of permeated hydrogen. The further increase in the pore diameter to 0.5 nm and higher sharply leads to a decrease in the olefins productivity due to the accelerated elimination of both initial alkanes and target alkenes from the reaction zone (the tube side). Therefore, the process of hydrogen and olefin co-production in the CMR can be efficient and reasonable when the ceramic membrane with an average pore size not exceeding 0.45 nm is applied. It is worth noting that the obtained results correlate well with the literature data on gas separation using porous membranes [31].

Conclusions
The hydrogen production via the dehydrogenation of alkanes (ethane, p the catalytic membrane reactor was shown to be prospective if the membrane istics are well optimized. The hydrogen molecules start to permeate the mem ticeably when the pore diameter exceeds 0.3 nm. An increase of the pore size nm improves the process parameters (hydrogen and alkene productivity) an hydrogen purity at a 100% level. However, it should be mentioned that in propane dehydrogenation, a pore size approaching the value of 0.45 nm inte side reactions, including the methane formation. A further increase of the pore size affects them more dramatically. Molecules, such as ethane and prop reagents) and ethylene and propylene (target products), became able to diffu the pores from the inner tube side of the reactor to the shell side, where only h expected to be. As a result, all the process parameters, as well as the hydro worsen significantly. Finally, it can be concluded that there is a narrow win optimal pore sizes near 0.4-0.45 nm when the membrane technology can provi production of industrially important monomers along with pure hydrogen.

Conflicts of Interest:
The authors declare no conflicts of interest.

Appendix A. Model Equations
Mass Balances: Tube Side:

Conclusions
The hydrogen production via the dehydrogenation of alkanes (ethane, propane) in the catalytic membrane reactor was shown to be prospective if the membrane characteristics are well optimized. The hydrogen molecules start to permeate the membrane noticeably when the pore diameter exceeds 0.3 nm. An increase of the pore size up to 0.45 nm improves the process parameters (hydrogen and alkene productivity) and provides hydrogen purity at a 100% level. However, it should be mentioned that in the case of propane dehydrogenation, a pore size approaching the value of 0.45 nm intensifies the side reactions, including the methane formation. A further increase of the membrane pore size affects them more dramatically. Molecules, such as ethane and propane (initial reagents) and ethylene and propylene (target products), became able to diffuse through the pores from the inner tube side of the reactor to the shell side, where only hydrogen is expected to be. As a result, all the process parameters, as well as the hydrogen purity, worsen significantly. Finally, it can be concluded that there is a narrow window of the optimal pore sizes near 0.4-0.45 nm when the membrane technology can provide efficient production of industrially important monomers along with pure hydrogen.
The calculations of the hydrocarbon conversion are made under consideration of the gas-mixture volume change: X = y C n H 2n+2 , in − (u t l C tot ) L 2 (u t l C tot ) in R 2 1 R 1 0 (y C n H 2n+2 ) L r 1 dr 1 + c -pore diameter of ceramic support, m d p -membrane pore diameter, nm G t,s -gas flow rate, ml min −1 −∆H j -heat effect of reaction j, kJ mol −1 l-length of reactor, m M i -molecular weight of ith compound, g mol −1 N R -number of reactions within the tube side of reactor N t,s -number of components within the tube side/shell side of reactor perm-permeability P 0 -pressure at normal conditions, atm P i t,s,c,m -partial pressure of components, atm r 1,2,3 -radial coordinate into the tube side, in the ceramic support, in membrane, m r cap -capillary radius, m R r -reactor radius, m R-universal gas constant, J mol −1 K −1 S sp1,sp2 -specific surface area, m −1 T 0 -temperature at normal conditions, K T t,s,c,m,w -temperature, K T av -average temperature, K u i -average thermal velocity of molecule, cm s −1 u l t,s -axial velocity, m s −1 w j -rate of reaction, kmol kg −1 s −1 X-conversion, % y i -mole fraction of ith component Greek letters α 1,2 -heat-transfer coefficient between the membrane and fixed bed catalyst (shell); between the exterior wall of reactor and fixed bed catalyst (shell), kJ m −2 s −1 K −1 β-mass transfer coefficient from membrane surface to shell side, m s −1 δ c,m -ceramic support and membrane thickness, m ε t,c,m -porosity of catalyst layer (tube side); ceramic support and membrane γ ij -stoichiometric coefficient for ith component into jth reaction λ t ef -effective coefficient of radial thermal conductivity, J m −1 s −1 K −1 λ c,m -thermal conductivity of the ceramic support, membrane, J m −1 s −1 K −1 µ-dynamic viscosity of a gas mix, kg m −1 s −1 ρ G t,s -gas density, kg m −3 ρ k t -density of catalyst, kg m −3 Indexes c-ceramic support in-inlet m-membrane s-shell side t-tube side W-wall of reactor m.f.-mole fraction