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Typical output curve of a time lag measurement performed on a constant volume/pressure increase instrument in a regime where the permeate pressure is negligible compared to the feed pressure.
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The most widely used method to measure the transport properties of dense polymeric membranes is the time lag method in a constant volume/pressure increase instrument. Although simple and quick, this method provides only relatively superficial, averaged data of the permeability, diffusivity, and solubility of gas or vapor species in the membrane. Th...
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Context 1
... complete evacuation of the membrane for a sufficiently long time, it is suddenly exposed to the gas of interest, from the feed side of the membrane cell, after which the pressure is recorded in the permeate side with constant volume. A typical curve can be divided into three distinct regions (Figure 1): in the penetration region, the gas is absorbed at the feed side of the membrane and it starts diffusing across the thickness of the film, but it does not reach the permeate side; in the transient region, the first gas molecules start desorbing from the membrane at the permeate side, and the rate gradually increases until it becomes constant. These two phases are related to the diffusion coefficient, D. Finally, in the stationary state, which is used to determine the permeability coefficient, P, a constant flux across the membrane takes place. ...
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... two phases are related to the diffusion coefficient, D. Finally, in the stationary state, which is used to determine the permeability coefficient, P, a constant flux across the membrane takes place. If Fickian diffusion takes place, and the solubility of the gas and its diffusion coefficient in the polymer are both constant, then the diffusion coefficient can be determined from the membrane thickness, l, and the time lag, Θ, which is the intersection of the tangent to the steady-state permeation curve and the horizontal axis (Figure 1), allowing the determination of the diffusion coefficient as follows: ...
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... type of permeation curve displayed in Figure 1 where pt is the permeate pressure at time t, p0 is the permeate pressure at the starting of the measurement, (dp/dt)0 is the baseline slope, which is related to the eventual presence of micro-defects in the membrane or leaks in the system, R is the universal gas constant, T is the absolute temperature, Vp is the permeate volume, Vm is the molar volume of the penetrant gas in standard conditions, A is the exposed surface area of the membrane, l is the thickness, pf is the feed pressure, S is the solubility coefficient, D is the diffusion coefficient. ...
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Citations
... The single gas permeances of TFCM samples were determined using a "constant pressure/variable volume" (PI) facility developed at Helmholtz-Zentrum Hereon and described elsewhere [24,39,41]. PI measurements conducted at a constant temperature and low pressure were also used for the estimation of Knudsen-type dependence for the porous UF-PAN membranes. ...
The gas transport properties of thin film composite membranes (TFCMs) with selective layers of PolyActive™, polydimethylsiloxane (PDMS), and polyoctylmethylsiloxane (POMS) were investigated over a range of temperatures (10–34 °C; temperature increments of 2 °C) and pressures (1–65 bar abs; 38 pressure increments). The variation in the feed pressure of condensable gases CO2 and C2H6 enabled the observation of peaks of permeance in dependence on the feed pressure and temperature. For PDMS and POMS, the permeance peak was reproduced at the same feed gas activity as when the feed temperature was changed. PolyActive™ TFCM showed a more complex behaviour, most probably due to a higher CO2 affinity towards the poly(ethylene glycol) domains of this block copolymer. A significant decrease in the permeate temperature associated with the Joule–Thomson effect was observed for all TFCMs. The stepwise permeance drop was observed at a feed gas activity of p/po ≥ 1, clearly indicating that a penetrant transfer through the selective layer occurs only according to the conditions on the feed side of the membrane. The permeate side gas temperature has no influence on the state of the selective layer or penetrant diffusing through it. The most likely cause of the observed TFCM behaviour is capillary condensation of the penetrant in the swollen selective layer material, which can be provoked by the clustering of penetrant molecules.
... One of the models that better explain the wood-water interactions during sorption processes is the Sorption Site Occupancy (SSO) model (Willems 2014(Willems , 2015. Finally, lagtime water vapor transmission experiments are an easy way to evaluate wood permeability and diffusivity (Al-Ismaily et al. 2012;Fuoco et al. 2020). ...
... where l and A are the thickness and area of the sample at 0% RH, respectively, and Δp is the difference in water partial pressure. A commonly used approach for the diffusion coefficient D determination in moisture transmission experiments is the one using the following expressions (Al-Ismaily et al. 2012;Fuoco et al. 2020): where θ is the lagtime calculated from the crossing of the extrapolated linear permeability region at m/ m 0 = 1 (linear approach) or θ = M w /ṁ (exponentiallinear approach). ...
Gravimetric vapor sorption experiments were performed on beech wood samples to determine the directional permeability, diffusion and sorption coefficients in the three orthotropic wood directions. Dynamic Vapor Sorption (DVS) experiments allowed for the direct evaluation of the diffusion coefficient from the analysis of the kinetic sorption profile using a double stretched exponential model with values ranging from 0.10 × 10⁻¹⁰ to 1.52 × 10⁻¹⁰ m²/s and depending on the wood direction of the sample and the RH-values. Moisture sorption isotherms (MSIs) were constructed and fitted to a modified Guggenheim-Anderson-de Boer and a Sorption Site Occupancy model, which allowed for the calculation of the sorption coefficient which was found to be between 2.4 and 3.0 mol/(m³ Pa). Dynamic Vapor Transport (DVT) experiments were performed to calculate the permeability coefficient from the vapor flow rate and it ranges between 0.56 × 10⁻¹⁰ and 4.38 × 10⁻¹⁰ mol/(m s Pa) as a function of the flow direction and RH conditions. These results indicate that such an experimental approach is suitable for determining wood–moisture interactions.
... This could be achieved while conducting a set of pervaporation experiments at various concentrations and temperatures of interest without the intermediate drying of the membrane to obtain "wet" diffusion curves, and a similar set of experiments with a thorough intermediate drying of the membrane to obtain "dry" diffusion curves. In this work, the authors will refer to permeance as a function of the feed concentration at a fixed temperature as a diffusion curve (it must be noted that, when applied to gas separation, the term diffusion curve may represent the overall quantity transported over a membrane as a function of the processing time [40] and is used in time-lag methods for the determination of diffusion coefficients in polymeric systems [41]). ...
PyVaporation—a freely available Python package with an open-source code for modelling and studying pervaporation processes—is introduced. The theoretical background of the solution, its applicability and limitations are discussed. The usability of the package is evaluated using various examples of working with and modelling experimental data. A general equation for the representation of a component’s permeance as a function of feed composition, temperature and initial feed composition is proposed and implemented in the developed package. The suggested general permeance equation may be used for the description of an extremal character of permeance as a function of process temperature and feed composition, allowing the description of processes with a high degree of non-ideality. The application of the package allowed modelling experimental points of various sets of hydrophilic pervaporation data and data on membrane performance from independent sources with a relative root mean square deviation of not more than 9% for flux and not more than 5% for a separated mixture concentration. The application of the facilitated parameter approach allowed the prediction of the components’ permeance as a function of feed concentration at various initial feed concentrations with a relative root mean square error of 3–26%. The package was proven useful for modelling isothermal and adiabatic time and length-dependent pervaporation processes. The comparison of the models obtained with PyVaporation with models provided in the literature indicated similar accuracy of the obtained results, thereby proving the applicability of the developed package.
... Recently we have shown that we can also determine the permeation time lag directly from the measured permeate flow rate as a function of time, i.e. from the original signal, which is the mathematical derivative of the time lag curve [69]. The inflection point in this sigmoidal curve corresponds to the peak in the pulse signal, and the inflection point of the signal in a membrane-free test run can be used for the correction of the instrumental lag time. ...
The rapid advancement of membrane gas separation processes has spurred the development of new and more efficient membrane materials, including polymers of intrinsic microporosity. The full exploitation of such materials requires thorough understanding of their transport properties, which in turn necessitates the use of powerful and reliable characterization methods. Most methods focus on the permeability, diffusivity and solubility of single gases or only the permeability of mixed gases, while studies reporting the diffusion and solubility of gas mixtures are extremely rare. In this paper we report the use of a mass-spectrometric residual gas analyser to follow the transient phase of mixed gas transport through a benzotriptycene-based ultrapermeable polymer of intrinsic microporosity (PIM-DTFM-BTrip) and a polydimethylsiloxane (PDMS) membrane for comparison, via the continuous online analysis of the permeate. Computational analysis of the entire permeation curve allows the calculation of the mixed gas diffusion coefficients for all individual gases present in the mixture and the identification of non-Fickian diffusion or other anomalous behaviour. The mixed gas transport parameters were analysed by three different approaches (integral, differential and pulse signal), and compared with the results of the ‘classical’ time lag method for single gases. PDMS shows very similar results in all cases, while the transport in the PIM gives different results depending on the specific method and instrument used. This comparative study provides deep insight into the strengths and limitations of the different instruments and data elaboration methods to characterize the transport in rubbery and high free volume glassy membranes with fundamentally different properties and will be of help in the development of novel membrane materials.
... For a given gas and membrane, D e should be the same, regardless of how it is evaluated. On the other hand, there are examples for which D e determined directly from the time-lag method is different from the value calculated from the ratio of P e and S e [12][13][14][15]. While the differences between the two values can be attributed to inherent experimental errors associated with the direct determination of P e , S e , and in particular D e , the following question arises. ...
... Since all elementary units are identical, symmetry conditions prevail at the four faces of the element parallel to the direction of permeation, as expressed in Eq. (16) for BC [3][4][5][6] . Also, since the nanoparticle is impermeable, the mass flux at each of the six faces of the cuboid nanoparticle is zero (see BC [7][8][9][10][11][12] in Eq. (17)). The latter boundary conditions imply that the concentration of the migrating species inside the particle is zero. ...
The time-lag method is commonly used for the characterization of gas separation membranes. It allows the determination of the diffusion, permeability, and solubility coefficients from a single dynamic gas permeation experiment. This method is widely used for any membrane material, including glassy polymers, polymers with intrinsic microporosity, and mixed-matrix membranes. However, it is understood that for such determined transport coefficients, it is effective rather than intrinsic parameters that are estimated. In this paper, we focus on the applicability of the time-lag method for the characterization of theoretical mixed-matrix membranes (MMMs), in which impermeable particles of different shapes are ideally dispersed in a continuous polymer phase. Dynamic gas permeation experiments were simulated by solving the three-dimensional Fick's second law of diffusion numerically. The generated data allowed the estimation of the effective diffusion coefficient from the time lag and the ratio of the effective permeability and solubility coefficients.
... This allows us to avoid polarization phenomena by having a low stage cut and gives negligible partial pressure in the permeate. Details on the instrument, measurement, and calibration procedures were described in earlier works [47,48]. ...
Poly(ionic liquid)s are an innovative class of materials with promising properties in gas separation processes that can be used to boost the neat polymer performances. Nevertheless, some of their properties such as stability and mechanical strength have to be improved to render them suitable as materials for industrial applications. This work explored, on the one hand, the possibility to improve gas transport and separation properties of the block copolymer Pebax® 1657 by blending it with poly[3-ethyl-1-vinyl-imidazolium] diethyl phosphate (PEVI-DEP). On the other hand, Pebax® 1657 served as a support for the PIL and provided mechanical resistance to the samples. Pebax® 1657/PEVI-DEP composite membranes containing 20, 40, and 60 wt.% of PEVI-DEP were cast from solutions of the right proportion of the two polymers in a water/ethanol mixture. The PEVI-DEP content affected both the morphology of the dense membranes and gas transport through the membranes. These changes were revealed by scanning electron microscopy (SEM), time-lag, and gravimetric sorption measurements. Pebax® 1657 and PEVI-DEP showed similar affinity towards CO2, and its uptake or solubility was not influenced by the amount of PIL in the membrane. Therefore, the addition of the PIL did not lead to improvements in the separation of CO2 from other gases. Importantly, PEVI-DEP (40 wt.%) incorporation affected and improved permeability and selectivity by more than 50% especially for the separation of light gases, e.g., H2/CH4 and H2/CO2, but higher PEVI-DEP concentrations lead to a decline in the transport properties.
... At a different level, computational tools are also used by Fuoco et al. [15] for a better comparison of the experimental data with theoretical transport models. For instance, they discuss two different methods for the analysis of the transient and pseudo-steady state gas transport in dense polymeric or mixed matrix membranes. ...
Computational modelling and simulation form a consolidated branch in the multidisciplinary field of membrane science and technology [...]
The effective diffusivity, permeability, and solubility coefficients of the materials for gas separation membranes are usually determined by the time-lag method in a single dynamic gas permeation experiment. The term “effective properties” implies that they are independent of the thickness of the film used to determine these properties. However, this may not always be the case. In this paper, we modelled time-lag experiments by numerically solving the governing partial differential equation for ideal mixed matrix membranes (MMMs) consisting of uniformly dispersed cuboid particles of the same shape, orientation, and size in a continuous polymer phase. An ideal MMM is comprised of a large number of identical, repeatable elementary units where each elementary unit consists of a centrally-located filler particle embedded in a polymer matrix. Contrary to a neat polymer membrane, the simulated time lag of an ideal MMM is not directly proportional to the square of the membrane thickness, expressed by the number of repeatable layers in the main diffusion direction. The effective diffusivity of MMM from the simulated time lag can be greater or smaller than the constant ratio of MMM’s effective permeability and solubility. The difference depends on the relative transport coefficients of the dispersed particles compared to those of the continuous polymer phase. It decreases with the number of repeatable layers and becomes negligible for five or more layers.
Gravimetric vapor sorption experiments were performed on beech wood samples to determine the directional permeability, diffusion and sorption coefficients in the three orthotropic wood directions. Dynamic Vapor Sorption (DVS) experiments allowed for the direct evaluation of the diffusion coefficient from the analysis of the kinetic sorption profile using a double stretched exponential model with values ranging from 0.10ꞏ10 − 10 to 1.52ꞏ10 − 10 m ² /s and depending on the wood direction of the sample and the RH-values. Moisture sorption isotherms (MSIs) were constructed and fitted to a modified Guggenheim-Anderson-de Boer model, which allowed for the calculation of the sorption coefficient which was found to be between 2.4 and 3.0 mol/(m ³ ꞏPa). Dynamic Vapor Transport (DVT) experiments were performed to calculate the permeability coefficient from the vapor flow rate and it ranges between 0.56ꞏ10 − 10 and 4.38ꞏ10 − 10 mol/(mꞏsꞏPa) as a function of the flow direction and RH conditions. These results indicate that such an experimental approach is suitable for determining wood-moisture interactions.
