# Modeling and Design Optimization of Multifunctional Membrane Reactors for Direct Methane Aromatization

^{*}

## Abstract

**:**

^{2}·atm

^{1/4}) through the hydrogen-permeable membrane. This modeling and design approach sets the stage for guiding further development of multifunctional membrane reactor models and designs for natural gas utilization and other chemical reaction systems.

## 1. Introduction

_{2}. However, some issues with this method are the overall low conversion of methane and the rapid catalyst deactivation by coking. A general benzene production trend for the DMA process is shown over time in Figure 2 to highlight some of these issues. There have been many attempts to mitigate these issues, with solutions involving membranes, catalysis, and selective oxidation.

## 2. Literature Review

_{2}or water suppresses coke formation through the use of steam reforming [12,13]. Also, oxidative coupling of methane by way of small additions of oxygen to the reactor have produced greater catalyst stability with an integrated recycle system [14]. Comprehensive kinetic models based on elementary steps have been formulated for DMA [15,16]. However, none of the reported research has shown the oxidative reaction mechanisms and kinetics.

## 3. Background

#### 3.1. DMA Membrane Reactor Model

_{4}) to hydrogen (H

_{2}) and benzene (C

_{6}H

_{6}). It is typically done over a molybdenum (Mo) catalyst on some zeolite support (typically HZSM-5, MWW or MFI). Equations (1)–(4) represent a two-step reaction mechanism, followed by the respective rate laws developed by Li et al. [26,37].

_{i}is the concentration in the gas phase of each i species, r

_{1}and r

_{2}represent the reaction rates of each reaction, k

_{1}and k

_{2}correspond to the respective reaction rate constants, and K

_{1}and K

_{2}are the equilibrium constants, determined by thermodynamic data from Yaws [38]. In the mathematical model, for a given reaction, if C

_{i}= 0 for a reactant, then the corresponding reaction rate is set to zero. This conditional is implemented in MATLAB

^{®}to avoid singularities in the rate denominators.

_{t,i}and F

_{s,i}are the molar flow rates for each species i inside the tube and the shell, respectively. The variables z, d

_{t}and A

_{t}are the differential reactor length, diameter and cross-sectional area, respectively. The membrane flux expression is considered to be proportional to the membrane partial pressure gradient and has a ¼ order dependence associated with an ion-transport membrane. Q is the hydrogen permeance through the membrane, and α

_{H2/i}is the selectivity between hydrogen and species i. p

_{t,i}and p

_{s,i}are the partial pressures of components i in the tube and shell sides, respectively. This model assumes plug flow and that the streams within each zone are well mixed. The reaction rate constant units from the original source [37] have been converted to a reactor volume basis by using the catalyst bed density in the packed bed reactor. The catalyst is packed along the length of the reaction zone with an assumed void fraction of 0.5 and catalyst effectiveness factor of 0.9. These two fractions are incorporated into the reaction rates and the cross-sectional area in the model. Also, the diameter of the catalyst particle considered between 180–425 µm is assumed small enough to neglect intraparticle transport limitations [20].

#### 3.2. Role of Selective Oxidation

_{x}products, as opposed to benzene.

## 4. Proposed Approach

#### 4.1. Reaction Modeling

_{10}H

_{8}) is formed from ethylene and benzene, as shown in Equation (14). Naphthalene is treated as an undesired product for the DMA system.

_{x}products into the system, as well as a water-gas shift reaction step.

_{2}H

_{2}], pyrene [C

_{16}H

_{10}], and elemental carbon [C]) added from this reaction scheme are considered as coking products in this model.

#### 4.2. Membrane Modeling

_{r,i}is the flow rate of species i in the reaction zone, z is the discrete reactor length, r

_{j,i}is the species reaction rate corresponding to reaction step j, A

_{r}is the cross-sectional area of the reaction zone, J

_{i,1}and J

_{i,2}are the molar fluxes across the membrane walls of M1 and M2, respectively, and d

_{1}and d

_{2}are the diameters of the tubes for M1 and M2, respectively. For elemental carbon in the reactor, J

_{i,1}= J

_{i,2}= 0 is assumed due to its solid phase. The sign of the molar fluxes depends on the partial pressure differences, shown below in Equations (23) and (24). The sign for the species reaction rate is positive for products and negative for reactants.

_{M1,i}and F

_{M2,i}are the flow rates in the outer shell and inner tube, respectively. The resulting model can be solved using an ODE initial-value problem solver. MATLAB subroutine “ode15s” was used due to the nature of the governing rate equations that produce a stiff ODE problem. The model for M1, derived from Li et al. [41], is based on a SrCe

_{0.7}Zr

_{0.2}Eu

_{0.1}O

_{3−δ}(SCE) ion-transport membrane. The flux through M1 is assumed to have a ¼ order dependence on partial pressure, as shown in Equation (23), in accordance with Equations (5)–(12):

_{1}is the permeance of hydrogen through M1, α

_{i,1}is the selectivity of species i to hydrogen for M1, and p

_{r,i}and p

_{M1,i}are the partial pressures of each species in the reaction zone and M1, respectively.

_{2}NiO

_{4}

_{+δ}ion-transport membrane. The flux through M2 is also assumed to have a ¼ order dependence on partial pressure, as well as a temperature dependence, as shown in Equation (24):

_{2}is the permeance of oxygen through M2, α

_{i,2}is the selectivity of species i to oxygen for M2, p

_{M2,i}is the partial pressure of each species in M2, B is an effective activation energy for M2, and T is the system temperature.

#### 4.3. Simulation and Optimization Setup

_{2}and B are taken from Mancini and Mitsos [42]. This base case design is used to simulate and validate the use of the multifunctional membrane reactor. Coking is assumed to have no effect on the membrane transport. The oxygen flux along the length of the reactor is ~10

^{−2}–10

^{−3}mmol/cm

^{2}·h leading to CH

_{4}/O

_{2}ratios of ~100–1000:1.

_{1}and α

_{i,1}) are also changed to initially produce two-dimensional input analyses. The operability mapping is performed to independently analyze the effects of reactor dimensions and M1 design parameters, respectively, on the performance variables. Then all four of the above design parameters are varied to determine the best design for a DMA system via this model.

_{4}conversion (X

_{CH4}):

_{6}H

_{6}production rate (F

_{C6H6}):

_{C}):

_{h}is the hydraulic diameter for an annulus, defined in Equation (29) below:

## 5. Results and Discussion

#### 5.1. Base Case Performance Studies

_{x}products. The benzene production loss could be an issue with this design, but a cost analysis would need to be done to compare the tradeoffs with the reduction in coking effects.

#### 5.2. Sensitivity Studies

_{i,1}, is considered to be a design variable.

_{1}, Q

_{1}, and α

_{i,1}significantly affect all three performance criteria, while M2 permeance (Q

_{2}) and diameter (d

_{2}) primarily affect the coke products. As only d

_{2}significantly affects the coking products, shown in Figure 6, Q

_{2}is assumed to be 1.3 × 10

^{−3}mol/s·m

^{2}·atm

^{1/4}as in the base case. As d

_{2}increases, which causes a larger oxygen molar flux through the membrane, the coking effects decrease, which confirms that the more oxygen that is in the system, the less coke products will be produced. Due to low effects of this variable on the other performance criteria, the low, fixed value of 0.25 cm is chosen for d

_{2}for the rest of the studies. This low d

_{2}value is chosen to allow for d

_{1}to change according to the range in Table 3.

_{1}) affects the methane conversion the most when compared to other parameters, as depicted in Figure 8, as expected, because the more hydrogen flows out of the reaction zone, the more the equilibrium shifts toward the products. However, too high of a permeance allows for benzene permeation out of the reaction zone.

#### 5.3. Optimization and Operability Mapping

_{1}= 0.01 mol/s·m

^{2}·atm

^{1/4}and α

_{i,1}= 1000, chosen as typical estimates of membrane parameters, as shown in Figure 9. In particular, from points “A” to “B,” as the length increases, the coking effects also increase and the benzene production decreases due to pyrolysis and benzene permeation as reported above. From points “B” to “C” the diameter is increased, so more cross-sectional area for the DMA reaction, along with all the other reactions, is present to take place in. Thus, the coking effects dramatically increase with the larger area and benzene is consumed in the process. The best design based on the optimization for these cases is determined to be a small reactor (d

_{1}= 1.65 cm, L = 5 cm) with an L/D ≈ 3. This L/D is far too small for the plug flow assumption of L/D > 15 [43], and, therefore, not a realistic solution. Whenever this constraint is applied, a new optimal design is obtained, denoted by the star point in Figure 9 (d

_{1}= 1.5 cm, L = 20 cm). These results are interesting because they demonstrate the flexibility of the formulated optimization problem and the variability of the optimal result depending on the incorporated process constraints. Thus, this means that the best design based on the objective function in Equation (30) calls for a short reactor with a total volume just large enough to allow for the DMA reactions to reach equilibrium.

_{1}and α

_{i,1}and using L = 20 cm and d

_{1}= 1.5 cm, chosen from the previous study’s optimal point, as depicted in Figure 10. In this figure, from points “A” to “B”, the membrane selectivity is increased, which leads to a sharp increase in benzene production and a small increase in coking effects. This is due to less benzene leaving the reaction zone, possibly allowing for more benzene and other reaction products to be converted to coke. In segment “B” to “C”, the coking effects decrease possibly because more hydrogen is left in the reaction zone due to low permeance. The benzene production decreases because there is less conversion when permeance is low, again due to more hydrogen in the reaction zone. In the “C” to “D” segment in the AIS, selectivity is decreased, but interestingly, the changes in the AOS are negligible, highlighting that at low permeance, selectivity has little effect on the performance criteria. From point “D” back to “A” it is observed that the benzene production first increases sharply, then it decreases while the permeance increases. This makes sense, since at a low selectivity, increasing permeance allows for more hydrogen to pass through the membrane, thus leading to higher conversion; however, the permeance eventually gets too large for benzene and its reactants to flow through the membrane, thus losing production overall. Hence, there is a sensitive balance at work with the permeance at low selectivity. The optimal design can thus be determined to have a membrane to allow for a mid-range to high permeance with a high selectivity, denoted by the star point (Q

_{1}= 5 × 10

^{−4}mol/s·m

^{2}·atm

^{1/4}, α

_{i,1}= 2 × 10

^{6}). As it may be difficult to achieve a very high selectivity in laboratory conditions, it is suggested that the best design has a lower selectivity as an alternative for a mid-range permeance, as the optimal results in this case would change by <10%, seen at the red circle point (Q

_{1}=5 × 10

^{−4}mol/s·m

^{2}·atm

^{1/4}, α

_{i,1}= 10

^{3}). The performed operability mapping thus also allows for determining alternative optimal designs for the membrane reactor system.

_{i,1}, is not considered in the figure for graphical purposes and due to the fact that the optimal designs always had the maximum selectivity. Table 4 shows the optimal designs obtained within the range simulated, as well as the optimal performance criteria for different cases considering higher weights for specific performance criteria. Case 1 uses the objective function φ shown in Equation (30), while Cases 2–4 use 100 times weight toward benzene production, methane conversion, and coke production, respectively. The considered ranges are similar to the ones used in the mapping studies with two inputs. The maximum/minimum range values assumed for some of the inputs are also presented in Table 4 to account for membrane properties that would be more feasible in a laboratory setting. As noted before, high permeance and high selectivity allows for the most hydrogen to leave the system, allowing for increased conversion of methane, while also allowing for the desired benzene to stay in the reaction zone. A small reactor is shown to give the lowest coke formation, but the reactor must be large enough to convert the feed and still be considered a plug flow reactor. Given these considerations, zone “a” in Figure 11 shows the optimal region: high permeance and selectivity with small reactor dimensions. This zone contains the best designs according to the objective function for Cases 1 and 2, denoted by the large star and the circle, respectively. Cases 1 and 2 have very similar results, showing that high benzene production coincides with low coke production and high conversion. The zone “b” shows the region with small reactor dimensions with low permeance, not allowing for the equilibrium of the DMA reactions to be shifted toward the products through hydrogen removal. The optimal point for Case 4 is contained in zone “b”, denoted by the diamond point, and since permeance is low, the reactor acts as a tubular reactor with no M1, only M2, such as in the base case simulations above. The criteria are at very low values due to low reaction volume, so minimal coking also coincides with little feed reacting. Region “c” is where larger reactor dimensions are used with high permeance, highlighted by the large amount of coke production that occurs. The best design for Case 3 is in this zone, denoted by the square point, showing that maximal conversion can also lead to high coke production and low benzene production. This is due to the large reactor dimensions allowing for pyrolysis and benzene permeation as explained above. The optimization studies performed thus demonstrate that this multifunctional membrane reactor for a DMA system should have a small volume to inhibit the coking effects and a high permeance through M1 for conversion enhancement.

## 6. Conclusions

^{2}·atm

^{1/4}) and a high selectivity (greater than 10

^{5}) to hydrogen. If a high selectivity cannot be achieved, a high permeance is best to allow for hydrogen to be removed and shift the equilibrium toward the products. The optimization and operability mapping performed allowed the determining of feasible ranges of expectations for outputs when further developing this system.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Equilibrium conversion of methane for direct conversion methods under non-oxidative conditions [6]. Copyright 2003 Elsevier.

**Figure 2.**Benzene formation in typical DMA processes (I. Activation. II. Pseudo-steady state. III. Deactivation [7]). Copyright 2015 Wiley.

**Figure 3.**Membrane reactor co-current configuration [36].

**Figure 4.**Methane conversion over time for (

**a**) membrane reactor (MR) and fixed-bed reactor (FR) and (

**b**) Product selectivity [31]. Copyright 2013 Wiley.

**Figure 6.**Sensitivity studies of d

_{2}for model performance criteria: (

**a**) X

_{CH4}; (

**b**) C

_{C}; (

**c**) F

_{C6H6}.

**Figure 7.**Sensitivity studies of L for model performance criteria: (

**a**) X

_{CH4}; (

**b**) C

_{C}; (

**c**) F

_{C6H6}.

**Figure 8.**Sensitivity studies of Q

_{1}for model performance criteria: (

**a**) X

_{CH4}; (

**b**) C

_{C}; (

**c**) F

_{C6H6}.

Parameter (Unit) | Value | Parameter (Unit) | Value |
---|---|---|---|

Temperature (K) | 1050 | d_{2} (cm) | 0.5 |

Pressure (atm) | 1 | Q_{1} (mol/s·m^{2}·atm^{1/4}) | 0.01 |

F_{CH4,feed} (mmol/h) | 4.98 | Q_{2} (mol/s·m^{2}·atm^{1/4}) | 1.3 × 10^{−3} |

F_{air,feed} (mmol/h) | 23.8 | α_{i,1} (H_{2}/all) | 10^{6} |

F_{He,sweep} (mmol/h) | 6.24 | α_{i,2} (O_{2}/all) | 10^{6} |

L (cm) | 25 | B (K) | 10,240 |

d_{1} (cm) | 1.25 | – | – |

**Table 2.**Base case performance criteria results. Base uses no membrane, M1 uses H

_{2}-permeable membrane, M2 uses O

_{2}-permeable membrane, and multifunctional uses both membranes.

Output (Unit) | Base | M1 | M2 | Multifunctional |
---|---|---|---|---|

X_{CH4} (%) | 19.52 | 38.36 | 19.57 | 38.15 |

F_{C6H6} (mg/h) | 10.49 | 19.85 | 8.65 | 18.09 |

C_{C} (%) | 2.28 | 4.95 | 2.06 | 4.64 |

Input (Unit) | Range |
---|---|

L (cm) | 20–200 |

d_{1} (cm) | 0.5–3 |

d_{2} (cm) | 0.2–2 |

Q_{1} (mol/s·m^{2}·atm^{1/4}) | 10^{−6}–0.1 |

Q_{2} (mol/s·m^{2}·atm^{1/4}) | 10^{−7}–10^{−2} |

α_{i,1} (H_{2}/all) | 10^{2}–10^{7} |

**Table 4.**Optimal reactor designs and outputs with modified objective functions: Case 1 has equally weighted criteria, as seen in φ; Case 2 has 100 times weight on F

_{C6H6}; Case 3 has 100 times weight on X

_{CH4}; Case 4 has 100 times weight on C

_{C}.

Input/Output (Unit) | Case 1 | Case 2 | Case 3 | Case 4 |
---|---|---|---|---|

L (cm) | 25 | 13 | 37 | 9 |

d_{1} (cm) | 0.7 | 1.1 | 2.1 | 0.5 |

Q_{1} (mol/s·m^{2}·atm^{1/4}) | 0.01 * | 0.01 | 0.01 | 2.15 × 10^{−5 #} |

α_{i,1} (H_{2}/all) | 4.64 × 10^{5} * | 4.64 × 10^{5} | 1000 ^{#} | 4.64 × 10^{5} |

F_{C6H6} (mg/h) | 20.66 | 20.88 | 5.22 | 5.97 |

X_{CH4} (%) | 37.82 | 38.18 | 42.17 | 13.06 |

C_{C} (%) | 1.30 | 1.99 | 15.32 | 0.064 |

^{#}Maximum/minimum value in simulated range considered.

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**MDPI and ACS Style**

Fouty, N.J.; Carrasco, J.C.; Lima, F.V.
Modeling and Design Optimization of Multifunctional Membrane Reactors for Direct Methane Aromatization. *Membranes* **2017**, *7*, 48.
https://doi.org/10.3390/membranes7030048

**AMA Style**

Fouty NJ, Carrasco JC, Lima FV.
Modeling and Design Optimization of Multifunctional Membrane Reactors for Direct Methane Aromatization. *Membranes*. 2017; 7(3):48.
https://doi.org/10.3390/membranes7030048

**Chicago/Turabian Style**

Fouty, Nicholas J., Juan C. Carrasco, and Fernando V. Lima.
2017. "Modeling and Design Optimization of Multifunctional Membrane Reactors for Direct Methane Aromatization" *Membranes* 7, no. 3: 48.
https://doi.org/10.3390/membranes7030048