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This work shows the application of a validated mathematical model for gas permeation at high temperatures focusing on demonstrated scale-up design for H_{2} processing. The model considered the driving force variation with spatial coordinates and the mass transfer across the molecular sieve cobalt oxide silica membrane to predict the separation performance. The model was used to study the process of H_{2} separation at 500 °C in single and multi-tube membrane modules. Parameters of interest included the H_{2} purity in the permeate stream, H_{2} recovery and H_{2} yield as a function of the membrane length, number of tubes in a membrane module, space velocity and H_{2} feed molar fraction. For a single tubular membrane, increasing the length of a membrane tube led to higher H_{2} yield and H_{2 }recovery, owing to the increase of the membrane area. However, the H_{2} purity decreased as H_{2 }fraction was depleted, thus reducing the driving force for H_{2} permeation. By keeping the membrane length constant in a multi-tube arrangement, the H_{2} yield and H_{2 }recovery increase was attributed to the higher membrane area, but the H_{2} purity was again compromised. Increasing the space velocity avoided the reduction of H_{2} purity and still delivered higher H_{2} yield and H_{2} recovery than in a single membrane arrangement. Essentially, if the membrane surface is too large, the driving force becomes lower at the expense of H_{2} purity. In this case, the membrane module is over designed. Hence, maintaining a driving force is of utmost importance to deliver the functionality of process separation.

_{2}molar fraction

Global climate change is closely associated with energy production, particularly CO_{2} emissions from power generation and transportation using fossil fuels. One of the options to address this problem is the utilization of hydrogen, a clean energy carrier. In combustion or chemical processes to generate energy, hydrogen has the unique property of reacting with oxygen and producing water. Low temperature fuel cells are a clear example, where hydrogen disassociates into protons and electrons and, subsequently, recombines with oxygen from air to generate water. The major advantages of using hydrogen in fuel cell systems, such as polymer electrolyte fuel cells, include high efficiencies of up to 64% [

The most viable process to produce hydrogen is via natural gas reforming or coal gasification [_{2}. However, as hydrogen can be generated by a single plant, this facilitates CO_{2} capture storage for a single point source, a major advantage to tackle greenhouse gases in non-diffuse sources. In these processes, there is a need to separate hydrogen from CO_{2}. Conventional industrial processes for gas separation include amine absorption strippers and pressure swing adsorption. These processes are energy intensive, because the gases produced at high temperatures (>800 °C) needed to be cooled down to meet the temperature requirements for these technologies in order to operate effectively at lower temperatures (<50 °C) [

Metal- and silica-based inorganic membranes have been extensively investigated for hydrogen separation. Metal membranes are generally derived from palladium (Pd) and Pd alloy, where hydrogen is solubilised in the metal matrix, and its transport via the membrane follows the Sievert’s law, where the driving force is proportional to the square root of the partial pressure of hydrogen in the feed and the permeate streams. On the other hand, silica-derived membranes follow a molecular sieving transport, where the pore size allows for a very fast diffusion of hydrogen and, generally, hindering the diffusion of CO_{2}. In this case, the driving force is proportional to the partial pressure difference of hydrogen in the feed and the permeate streams. The molecular sieving transport is temperature-dependent, and generally, the flux of hydrogen increases with temperature, whilst the flux of CO_{2} reduces. This is generally the case for silica membranes prepared with the silica precursors, tetraethoxy silane [_{2} flux. Furthermore, the driving forces for gas permeation in silica-derived membranes are more significant, as any small increase in the partial pressure in the feed stream will increase the driving force instead of the square root law for the metal membranes.

The best silica membranes are those prepared with metal oxides, particularly cobalt oxide. These membranes have been shown to be hydrostable [_{2}/CO_{2} selectivities at high temperatures of 500 °C. However, inorganic membrane research has been mainly limited to laboratory scales, with the only exception to date being a multi-tube membrane module operating for 2000 h recently reported by Yacou and co-workers [_{2}, reducing its partial pressure in the feed domain and affecting the driving force. In principle, the flux of a gas is proportional to the driving force, which is essentially the partial pressure difference of the gas species of interest. Hence, as gases permeate though a membrane, the driving force reduces along the length of a membrane tube. This tends to affect the membrane performance in terms of H_{2} production.

Traditional membrane mass transfer models treat the feed-interface boundary and permeate-interface boundary as constant. However, it is questionable that this constant condition cannot be considered for large-scale modules, due to driving force variation. Hence, a gas transport model must be developed and validated to predict gas separation performance in the process industry using appropriate scales. In this study, a mass transfer model is investigated by incorporating both driving force changes in the gas flow and the mass transfer across a membrane. The simulation is validated against a multi-tube membrane module and, thus, predicting the hydrogen gas separation. The model is therefore applied to membrane modules by taking into consideration important process engineering parameters, such as H_{2} recovery, yield and purity in terms of membrane tube length and the number of membrane tubes per module.

A membrane module is depicted in

The structure of the membrane separation module.

There are two important gas diffusion mechanisms, namely: gas-through-gas diffusion and gas diffusion through the membrane. Gas-through-gas diffusion is severe at high temperature, and gases are constantly mixed to maintain the chemical equilibrium. The gas phase diffusivity is about four orders of magnitude of diffusivity across the silica membrane [

The basic mass balance in the gas phase can be described by the continuity equation:

The component mass balance of H_{2} is governed by the following solution conservation equation:
_{1} is the molar fraction of H_{2}, _{2} in the other gas, which can be estimated from the Fuller equation [_{1} is the source term for H_{2} permeation, which will be further discussed in the following.

It is important to observe that Equation (2) contains both the advection term,

Pressure is an important parameter in determining the driving force for permeation. The correlation between permeate pressure and flow rate is governed by the Hagen-Poiseuille equation [

The source terms in Equations (1) and (2) represent the mass transfer between the feed side and the permeate side and are derived from the following formulas [_{H2} are the total permeate flow rate and the hydrogen permeate flow rate, respectively, _{H2} is the hydrogen permeate flux, d

The major resistance of mass transfer occurs across the membrane; thus, this is a very important issue for consideration in modelling gas flux in membrane systems in the process industry. The membrane-mass-transfer mechanisms are always associated with the intrinsic properties of the membrane material. The widely used Fick’s law is proven to be less accurate than the Maxwell-Stefan model [_{i}_{ji}_{ij}_{i}

Equation (6) is usually cast into matrix form [

For an H_{2}/Ar binary gas system, the elements of [_{2} and 2 is for Ar):
_{1} is the Maxwell-Stefan single gas diffusivity of H_{2} and _{2} is that of Ar. _{12} is the Maxwell-Stefan interchange coefficient inside the membrane.

The permeate flux can be derived from Equation (7) as:

If the matrix [∆] is defined as [^{−1}

As both [∆] and [∇_{2}, fraction _{1}, it is necessary to solve the fraction _{1} profile across the membrane thickness in advance by flux conservation.
_{1} distribution across the membrane is reported in detail elsewhere [

The experimental data for this work were obtained from Yacou

The operating conditions for mixture gas separation test.

Temperature | Feed flow rate (mL min^{−1}) |
H_{2} Feed fraction |
Permeate flow rate in experiment (mL min^{−1}) |
Permeate flow rate in model (mL min^{−1}) |
H_{2} Permeate fraction in experiment |
H_{2} Permeate fraction in model |
Relative error for permeate flow | Relative error for permeate fraction |
---|---|---|---|---|---|---|---|---|

500 °C | 253.9 | 99% | 249.2 | 251.7 | 100% | 99% | 0.01 | 0.01 |

49.6 | 82% | 44.2 | 44.7 | 90% | 88% | 0.01 | 0.02 | |

35.1 | 76% | 30.7 | 31.3 | 85% | 82% | 0.02 | 0.03 | |

13.4 | 18% | 8.9 | 7.9 | 24% | 26% | 0.11 | 0.05 | |

400 °C | 142.7 | 98% | 137.7 | 140.8 | 100% | 98% | 0.02 | 0.02 |

44.6 | 84% | 39.8 | 40.8 | 91% | 89% | 0.03 | 0.03 | |

32.4 | 71% | 27.7 | 25.8 | 79% | 81% | 0.07 | 0.03 | |

16.8 | 41% | 12.2 | 10.8 | 51% | 53% | 0.11 | 0.05 |

The finite-difference method was used to solve the gas phase governing equations. The iteration stopped right after the calculation process reached a steady state, when the H_{2} mass balance converged to differences smaller than 1e^{−5}. This value is sufficiently small to attain accurate simulations in this work. The boundary conditions were set as follows: (i) retentate pressure is 6 atm; (ii) permeate outlet pressure is 1 atm; (iii) constant feed flow rate and (iv) constant gas composition at the feed inlet, in each case.

Different grid sizes were run by this model. A grid independence study in _{2} fraction profile along axial positions with different grid sizes. When the grid size, ∆_{2} fraction. Therefore, Δ

Grid independence simulation.

The process parameters of interest to be investigated in this work are: H_{2} purity, H_{2} yield and H_{2} recovery, as follows:

H_{2} purity is defined as the H_{2} permeate molar fraction at the permeate outlet.

H_{2} yield is the permeate flow rate at the permeate outlet multiplied by H_{2} purity.

H_{2} recovery is the H_{2} yield divided by the H_{2} feed flow rate.

The parameters of interest were simulated by solving the derivative equations by the finite difference method. The domain (e.g., feed or permeate side) was divided into numerous grids from first (entry) to ^{th} (exit) along the axis. Upon conversion of the simulated results at steady state conditions, the permeate flow rate and permeate fraction were determined from the ^{th} grid.

The properties of the membrane and the constant operating conditions were also sought from the work of Yacou _{2} and Ar separation. The use of Ar instead of CO_{2} was to avoid the reverse of the water gas shift reaction, which would result in the production of CO and water. Hence, Ar was used as a subrogated molecule to maintain a binary gas mixture of H_{2}/Ar instead of a multiple transient gas mixture. In addition, Ar (d_{k} = 3.42Å) and CO_{2} (d_{k} = 3.3Å) have similar kinetic diameters (d_{k}) and show similar trends and fluxes and negative apparent energy of activation in high quality silica membranes. These trends are contrary to the smaller kinetic diameter of H_{2} (d_{k} = 2.89Å), showing a positive apparent energy of activation. The simulation in this work investigates these process parameters, which are affected primarily by changing the surface area of the membranes inside a module. This can be done by altering the design specification, either by increasing the length of the membrane tubes or by adding extra tubes of the same length to a membrane module in a parallel configuration.

Membrane properties and operating conditions [

Operating conditions | Value |
---|---|

Temperature | 500 (°C) |

H_{2} permeance |
5.80 × 10^{−}^{8} (mol s^{−1} m^{−2} Pa^{−1}) |

Ar permeance | 5.67 × 10^{−}^{10} (mol s^{−1} m^{−2} Pa^{−1}) |

Radius of module | 0.05 (m) |

Membrane radius | 0.007 (m) |

Retentate pressure | 6 (atm) |

Permeate pressure | 1 (atm) |

^{−1}, feed H_{2} fraction = 0.5). The simulation shows that by increasing the membrane length, it benefited H_{2} yield and H_{2} recovery, though it was detrimental to H_{2} purity. In order to explain the decline of the H_{2} purity, it is important to understand the H_{2} fraction distribution in the feed side of the membrane in _{2} feed fraction reduced from 0.5 to 0.35 for the 1 m length tube. By increasing the membrane length to 5 and 10 m, the hydrogen molar fraction reduced even further to 0.15 and 0.10, respectively. Hence, this caused a significant reduction in the driving force for the hydrogen permeation. Consequently, this allows for the flux of the less permeable gas to increase. As a result, the H_{2} purity decreases as a function of the membrane length.

H_{2} processing performance as a function of membrane length: (_{2} purity; (_{2} yield; (_{2} recovery.

H_{2} molar fraction distribution along the module for different membrane lengths.

In industrial process design, multiple membrane tubes can be fixed in a module. If the membranes are connected in series, then the membranes would be equivalent to a single long membrane with the same total membrane length. Therefore, this section focuses on the separation performance of a multi-tube membrane in parallel, as displayed in

The structure of quintuple membranes in parallel.

_{2} purity, higher H_{2} yield and H_{2} recovery. The reduction of H_{2} purity as a function of the number of membrane tubes in parallel is associated with a reduction of the H_{2} molar feed fraction in the feed side, as shown in

H_{2} processing performance as a function of the number of membranes in a module: (_{2} purity; (_{2} yield; (_{2} recovery.

H_{2} molar fraction distribution along the 1 m membrane of different membrane numbers.

However, in terms of process design, these results strongly suggest that full recovery of H_{2} is conflicting with H_{2} purity. Hence, excessive membrane area (five tubes of five metres in length) causes the membrane to deliver H_{2} purity almost similar to the H_{2} molar ratio in the feed stream. In this case, the membrane module has been over-designed, and its function to separate gases is no longer attainable. Hence, there is a need to maintain a driving force for H_{2} permeation through the membrane. If H_{2} is depleted because the surface area is too large, then this creates the conditions for other gas/gases to start permeating through the membrane and, likewise, reducing H_{2} purity in the permeate stream.

To meet H_{2} purity specification, membrane modules cannot have 100% full recovery. While the latter would be ideal for the process industry to reduce product losses, the advantage of using inorganic membranes is associated with separation of gases at high temperatures. To counter balance the losses of H_{2}, membrane systems attract major gains by dispensing the requirement of conventional energy-intensive cooling down gas stream processes to separate H_{2}. Further comparison of _{2} purity (~0.5%), H_{2} yield (~2%) and H_{2} recovery (~2%). Although these values are modest, in terms of long-term production for large processing plants, this may translate into millions of dollars in savings in production costs.

The space velocity of gases inside a vessel or reactor is an important parameter in process design, particularly related to the sizing of a membrane module. In this case, space velocity correlates with the feed flow rate. This means that the faster the feed flow rate, the higher the space velocity or a lower retention time is, and _{2} purity, as the surface area of the membrane module increased to a point of being over-designed. One strategy to increase H_{2} purity is to increase the feed flow rate or space velocity. Hence, the simulation was carried out for each type of membrane module by increasing the feed flow rate from 1 NL min^{−1} to 5 NL min^{−1}. In _{2} purity increased slightly with an increasing feed flow rate. Notably, the H_{2} yield rose significantly. However, the H_{2} recovery deceased as the retention time in the module was reduced.

H_{2} processing performance as a function of feed flow rate and number of membranes in a module: (_{2} purity; (_{2} yield; (_{2} recovery.

These results clearly show that using one membrane with a 1 NL min^{−1} feed flow rate delivered higher H_{2} purity, ~94%, as compared with the five membrane module, which reached up to ~83%. However, again, H_{2} recovery was compromised, as this high H_{2} purity was associated with low H_{2} recovery of ~50%. When the feed gas fraction is constant, it is not possible to have both high H_{2} purity and H_{2} recovery at the same time. However, it must be said that an H_{2} purity of ~84% with a recovery rate of ~85% is within industrial targets; in particular, this separation process is aimed at being achieved at 500 °C.

The H_{2} feed molar fraction also influences the overall separation performance of membrane modules. The variation of the H_{2} feed fraction provides different amounts of the H_{2} fraction along the length of the membrane module and, in turn, affects the overall driving force accordingly. In _{2} feed fraction is increased gradually from 0.3 to 0.7, whilst other operating conditions are kept constant, namely feed flow rate (1 NL min^{−1}), membrane length (1 m), number of membrane tubes (5) and temperature (500 °C). It is observed that increasing the H_{2} feed fraction resulted in an increase of H_{2} purity, H_{2} yield and H_{2} recovery. H_{2} yield shows a linear correlation to H_{2} feed fraction, due to the fact that H_{2} partial pressure in the feed domain is proportional to the H_{2} feed fraction. However, both H_{2} purity and H_{2} recovery are convex functions to H_{2} feed fraction, and essentially, these process parameters will level off and converge to a single value of one.

H_{2} processing performance as a function of the H_{2} feed molar fraction: (_{2} purity; (_{2} yield; (_{2} recovery.

In terms of process, these results suggest that the H_{2} feed molar fraction is effective in controlling the driving force. If H_{2} purity cannot be achieved in a single pass, as the H_{2} feed fraction is too low, then the permeate stream can be fed in a second pass by another membrane module as a cascade system. For instance, in a first pass, 30% of the H_{2} feed fraction will be processed to 68% H_{2} purity in the permeate stream, which can be used as a feed stream in a second pass delivering an H_{2} purity of 88%. Similarly, H_{2} purities and H_{2} yields will also be affected accordingly. All of these parameters must be traded off to achieve the optimal membrane performance of an H_{2} product specification for industrial process separation.

This study presents a model to simulate the membrane separation performance in a scale-up single and multi-tube membrane module arrangement. Cobalt oxide silica membranes were used to validate the model for H_{2} separation at 500 °C. For constant feed flow conditions, longer membrane tubes increased H_{2} yield and H_{2 }recovery, but did not deliver a higher H_{2} purity. Multiple membranes in parallel enhanced the H_{2} yield and H_{2 }recovery compared to the single membrane, but the H_{2} purity decreased. Increasing the feed flow rate (and space velocity) avoided the reduction of H_{2} purity in the parallel arrangement, but at the expense of H_{2} recovery. Meanwhile, the H_{2} yield and H_{2 }recovery are observed to be higher in a multi-tube configuration. Increasing the H_{2} feed fraction resulted in an increase in all the performance parameters. When the tubes are too long or there is sufficient large surface area to deplete the H_{2} concentration in the feed domain, then the driving force for H_{2} permeation across the membrane is greatly reduced. As a result, the permeation of the other gases increased slightly, resulting in the reduction of H_{2} purity. To improve the H_{2} purity, it is possible to increase the space velocity (feed flow rate), which, in turn, reduces H_{2} recovery. Hence, all these parameters are interlinked, of which the process conditions and product specification will dictate the optimal process conditions to be deployed in the industry.

total molar concentration (mol·m^{−3})

Fick diffusivity in gas phase (m^{2}·s^{−1})

Maxwell-Stefan diffusivity in the membrane (m^{2}·s^{−1})

_{i}

Maxwell-Stefan single gas diffusivity in membrane (m^{2}·s^{−1})

_{ij}

Inter-exchange coefficient between component ^{2}·s^{−1})

_{ii}

self exchange coefficient (m^{2}·s^{−1})

permeable area (m^{2})

molar permeate flow rate across membrane (mol·s^{−1})

_{H2}

molar permeate flow rate across membrane of component H_{2} (mol·s^{−1})

computational volume (m^{3})

flow rate (mol·s^{−1})

flux (mol s^{−1}·m^{−2})

_{H2}

permeate flux across membrane of component H_{2} (mol s^{−1}·m^{−2})

_{Ar}

permeate flux across membrane of component Ar (mol s^{−1}·m^{−2})

Henry’s constant (mol·m^{−3}·Pa^{−1})

axial coordinate (m)

grid size (m)

the number of grid

pressure (Pa)

concentration of adsorbed gas (mol·m^{−3})

gas constant (8.314 J·mol^{−1}·K^{−1})

radial coordinate

source term (mol·s^{−1}·m^{−3})

_{1}

pressure (Pa)

temperature (K)

time

molar fraction

_{1}

H_{2} molar fraction

coefficient matrix in Maxwell-Stefan equation

inversed matrix of [

matrix of flux across membrane

matrix of pressure gradient

viscosity (Pa·s)

chemical potential (J·mol^{−1})

_{0}

chemical potential in the chosen standard state (J·mol^{−}^{1})

fractional occupancy of adsorption

Guozhao Ji gives special thanks for the scholarship provided by the University of Queensland and the China Scholarship Council. The authors acknowledge funding support from the Australian Research Council (DP110101185).

The authors declare no conflict of interest.

_{2}PSA process using layered beds

_{2}separation membranes

_{2}separation

_{2}purification