Characteristics of Gas Permeation Behaviour in Multilayer Thin Film Composite Membranes for CO2 Separation

Porous, porous/gutter layer and porous/gutter layer/selective layer types of membranes were investigated for their gas transport properties in order to derive an improved description of the transport performance of thin film composite membranes (TFCM). A model describing the individual contributions of the different layers’ mass transfer resistances was developed. The proposed method allows for the prediction of permeation behaviour with standard deviations (SD) up to 10%. The porous support structures were described using the Dusty Gas Model (based on the Maxwell–Stefan multicomponent mass transfer approach) whilst the permeation in the dense gutter and separation layers was described by applicable models such as the Free-Volume model, using parameters derived from single gas time lag measurements. The model also accounts for the thermal expansion of the dense layers at pressure differences below 100 kPa. Using the model, the thickness of a silicone-based gutter layer was calculated from permeation measurements. The resulting value differed by a maximum of 30 nm to the thickness determined by scanning electron microscopy.


Introduction
The development of a membrane production technology is a key element for transferring the potential of novel, high performance membrane materials into technical application. A thin film composite membrane for gas separation can contain several layers with different permeation properties. The layer thicknesses of multilayer membranes differ from tens of micrometres for the support layers, to tens of nanometres for the ultrathin separation layers. During the penetration through the membrane, the gas molecules interact with all layers of the membrane, where the gas transport characteristics can differ by several orders of magnitude. Thereby, for thin film composite membranes (TFCM) not only the thickness of the separation layer, but also the layer's interaction with its adjoining layers affects the total membrane performance. For the design of new high-performance gas separation membranes with transport characteristics dominantly governed by the properties of the selective layer material, the accurate prognosis of the gas transport parameters of the entire multilayer structure of the membrane is necessary. V P,i of a single gas i at transmembrane pressure differences ranging from 1 kPa to 60 kPa were performed in turn for each of the aforementioned feed pressure ranges. The use of pressures instead of fugacities is justified since the absolute pressures involved are sufficiently low.
Gas transport properties of thick isotropic films were characterised by a "time-lag" experiment implementing a constant volume, variable pressure method using an in-house developed measurement instrument as described elsewhere [24]. The single gas permeability of each sample was determined in the temperature range from 20 to 80 • C. Each measurement was repeated at least 3 times at feed pressures of (64, 50, 40,30) kPa. The permeability coefficient P (mol·m·m −2 ·s −1 ·Pa −1 ) of a single gas i was determined as: where V P,i is the constant permeate volume (m 3 ), δ is the film thickness (m), A is the effective area of membrane (m 2 ), R is the gas constant (8.314 Pa·m 3 ·mol −1 ·K −1 ), T is the temperature (K), p F is the feed pressure, p P1 and p P2 are the permeate pressures (Pa) at the start and at the end of the pressure increase, respectively, and ∆t is the time for permeate pressure increase from p P1 to p P2 (s). The gas permeances for the TFCMs were measured using the pressure increase facility, designed and built at HZG [25], where the aforementioned constant volume, variable pressure method is realized [26]. Single gas permeation data were determined at (50, 75, 100) kPa feed pressures and in the temperature range from 20 • C to 80 • C.
The permeance L (mol m −2 ·s −1 ·Pa −1 ) of each layer for the gas i can be calculated employing the following equation for a TFCM consisting of a porous support and one or more dense layers: Membranes 2019, 9, 22 4 of 19 where p F , p 0 , and p P(t) (Pa) are the pressures of the feed, of the permeate side at the beginning, and of the permeate side at the end of measurement, respectively, and ∆t (s) is the time of the measurement between permeate pressures p 0 and p P(t) . The permeance of component i in layer j is calculated by [27]: The thickness of the layer is assumed to be dependent on the penetrating gas i, since different gases will cause different degrees of swelling.
Two sets of circular membrane samples (called Stamps in the following) were prepared. Their designations are given in Table 1.

Model Development
For the mathematical description of mass transport in multilayer composite membranes, various models can be used. Since the mass transport in porous membranes is based on a mechanism different from those in dense membranes, the two membrane types are considered separately.

Gas Transport in Porous Layers
A membrane consisting of a non-woven polyester and a PAN porous layer was used to experimentally study and simulate the transport processes occurring in the TFCM support structure. The DGM developed by Mason and Malinauskas [16] and described in detail by Krishna and Wesslingh [28] was employed in this paper. The DGM describes the pore wall as large motionless "dust" molecules that are uniformly distributed in space ( Figure 1).
The thickness of the layer is assumed to be dependent on the penetrating gas i, since different 138 gases will cause different degrees of swelling. 139 Two sets of circular membrane samples (called Stamps in the following) were prepared. Their 140 designations are given in Table 1. 141

Model Development 143
For the mathematical description of mass transport in multilayer composite membranes, various 144 models can be used. Since the mass transport in porous membranes is based on a mechanism different 145 from those in dense membranes, the two membrane types are considered separately. 146

Gas Transport in Porous Layers 147
A membrane consisting of a non-woven polyester and a PAN porous layer was used to 148 experimentally study and simulate the transport processes occurring in the TFCM support structure. 149 The DGM developed by Mason and Malinauskas [16] and described in detail by Krishna and 150 Wesslingh [28] was employed in this paper. The DGM describes the pore wall as large motionless 151 "dust" molecules that are uniformly distributed in space ( Stefan diffusion equation [29]. The linear form of the DGM for single gas transport through the 158 porous layer is: 159 This model takes Knudsen diffusion, molecular diffusion in case of multicomponent mixtures and convective flow including the porous medium effect into account and is based on the Maxwell-Stefan diffusion equation [29]. The linear form of the DGM for single gas transport through the porous layer is: where: p a,i = 0.5 · p F,i + p P,i , where · n " PS,i is the molar flux through the membrane divided by area (mol·m −2 ·s −1 ), ε is the accessible fractional void volume of the porous medium, τ is the tortuosity factor characterising the porous matrix, ∆p i is the difference of pressure between feed p F,i and permeate side p P,i of the layer (Pa), δ ps is the thickness of the porous support (m), d pore is the pore diameter (m), M i is the molecular weight of gas (kg·mol −1 ), p a,i is the average pressure (Pa) and η i is the dynamic viscosity of the gas (Pa·s).
The non-woven supporting layer is not included into the DGM calculation because the mass transfer resistance of the non-woven can be assumed to be negligible due to extremely open structure formed by the polymer fibres ( Figure 2a). where: 160 p a,i = 0.5 · p F,i + p P,i , where n PS, i ʹʹ is the molar flux through the membrane divided by area (mol·m −2 ·s −1 ), ε is the accessible 161 fractional void volume of the porous medium, τ is the tortuosity factor characterising the porous 162 matrix, Δpi is the difference of pressure between feed pF,i and permeate side pP,i of the layer (Pa), δps is 163 the thickness of the porous support (m), dpore is the pore diameter (m), Mi is the molecular weight of 164 gas (kg·mol −1 ), pa,i is the average pressure (Pa) and ηi is the dynamic viscosity of the gas (Pa·s). 165 The non-woven supporting layer is not included into the DGM calculation because the mass 166 transfer resistance of the non-woven can be assumed to be negligible due to extremely open structure 167 formed by the polymer fibres ( Figure 2a). 168

Gas Permeation in Dense Layers 173
Generally, the mass transfer of a single component through a dense polymer membrane can be 174 described as a function of temperature and pressure. The following Equation (6) of the Free Volume 175 model (FVM) is applied to dense polymers in which the flux of a penetrant can be described by Fick's 176 first law and its sorption behaviour expressed by Henry's law [30]. 177 where P ∞,i 0 is the permeability for infinite temperature and the pressure approaching zero, Eact,i is the 178 activation energy of permeability (J·mol −1 ), m0,i is the swelling factor at zero temperature (Pa −1 ) and 179 mT,i takes the temperature dependency of swelling into account (K −1 ). The FVM can also be applied to 180 dense layers of a TFCM: 181 where 182

Gas Permeation in Dense Layers
Generally, the mass transfer of a single component through a dense polymer membrane can be described as a function of temperature and pressure. The following Equation (6) of the Free Volume model (FVM) is applied to dense polymers in which the flux of a penetrant can be described by Fick's first law and its sorption behaviour expressed by Henry's law [30].
where P 0 ∞,i is the permeability for infinite temperature and the pressure approaching zero, E act,i is the activation energy of permeability (J·mol −1 ), m 0,i is the swelling factor at zero temperature (Pa −1 ) and m T,i takes the temperature dependency of swelling into account (K −1 ). The FVM can also be applied to dense layers of a TFCM: where is in compliance with Equation (3) where the activation energy and swelling parameters are theoretically identical to Equation (6) for amorphous, rubbery polymers.

Resistance Model for TFCM
The driving forces for the permeation of a gas i through the polymeric membrane can be estimated as the difference of partial pressures between the feed and the permeate sides of the membrane ∆p i at conditions where the ideal gas law can be assumed to be valid. In this case molar flux of gas i through the unit of membrane area can be expressed as: In case a multilayer composite membrane is considered, Equation (9) is valid for each of the membrane layers j: where ∆p j , i is assumed to contain hypothetical partial pressures in-between layers (cf. Figure 2, where the fugacities shown are to be replaced by partial pressure) if neither feed nor permeate partial pressures are involved.
Due to the continuity equation, the overall flow rate · n t,i must be constant throughout all the layers j = 1, 2, . . . , n layers . · n t,i = · n PS,i = · n G,i = · n S,i = . . . = · n n layers,i .
Writing the Equation (11) Thus, the total flux across the porous support is related to the flux · n " pore,i in the pore as: where A pore is the area of the porous region. The flow rate can be expressed as function of a resistance to flow as proposed by Henis and Tripodi in analogy to an electric circuit [1]. The resistance model determines the total partial pressure drop of the gas i across the membrane as the sum of the individual partial pressure drops across the layers j of a TFCM. The resistance to permeate flow R j,i was defined as equivalent to the electrical resistance: For the composite membrane ( Figure 3b) the total resistance R t,i in this work is determined from the resistance-in-series model [5]. We take into account the resistance of the porous support R PS,i , the resistance of the gutter layer R G,i and the resistance of the selective layer R S,i . For the simplicity of the resistance model of the porous medium, we assume that pores are homogeneous through the thickness of the porous support. The porous medium consists of a number of non-interconnected circular capillaries with diameter d pore . The total resistance of a multilayer membrane as shown in Figure 3b can be expressed as: where the resistance R PS,i was determined analogous of parallel electrical circuit: The resistance R W and R pore are resistances of impermeable bulk substrate (pore wall) and permeable pores, respectively. This approach allows for the description of the TFCM by using the permeation characteristics measured separately for the individual building blocks of the membrane.

Resistance Model for TFCM 185
The driving forces for the permeation of a gas i through the polymeric membrane can be 186 estimated as the difference of partial pressures between the feed and the permeate sides of the 187 membrane Δpi at conditions where the ideal gas law can be assumed to be valid. In this case molar 188 flux of gas i through the unit of membrane area can be expressed as: 189 In case a multilayer composite membrane is considered, Equation (9) is valid for each of the 190 membrane layers j: 191 where Δpj,i is assumed to contain hypothetical partial pressures in-between layers (cf. Figure 2, where 192 the fugacities shown are to be replaced by partial pressure) if neither feed nor permeate partial 193 pressures are involved. 194 Due to the continuity equation, the overall flow rate n t,i must be constant throughout all the 195 layers j = 1, 2,…, nlayers. 196 n t,i = n PS,i = n G,i = n S,i = … = n n layers ,i .
Writing the Equation (11) in terms of fluxes for the membrane area A gives [16] (Figure 3a): 197 (a) (b)

200
Thus, the total flux across the porous support is related to the flux n pore,i ʹʹ in the pore as: 201 where A pore is the area of the porous region. 202 The flow rate can be expressed as function of a resistance to flow as proposed by Henis and 203 Tripodi in analogy to an electric circuit [1]. The resistance model determines the total partial pressure 204 drop of the gas i across the membrane as the sum of the individual partial pressure drops across the 205 Figure 3. Schematic diagram of the modeled TFCM: flow rate trough multilayer membrane (a); TFCM as analogy to the electric circuit (b). Membrane area A is divided in A W for pore wall and A pore for pores, with respective resistances R W and R pore .

Results and Discussion
The validity of the model developed above was tested by comparing the layer thickness determined from scanning electron microscope (SEM) micrographs and the thicknesses determined from gas transport parameters of thick films and TFCMs.

Application of DGM to Experimental Data
The permeation of two stamps of a porous PAN membrane was investigated at varying feed and permeate pressure differences and at different temperatures as described above. Stamps (Stamp 1 and Stamp 2) were taken from two different batches of membrane where the non-woven was coated with a layer of porous PAN using the same recipe. The experimental gas transport data was obtained for gases with different molecular weights: H 2 , CH 4 , N 2 , O 2 and CO 2 . The molar flux may be expressed as: where is · V N i the volumetric flow rate of gas i at normal pressure p Ø = 101.3 kPa and temperature T Ø = 0 • C, respectively.
The use of Equation (17) for the molar flux in Equation (4) and expressing the resulting equation in linear form yield: where Calculating the X and Y values from the experimental data of the single gas measurements and plotting them as shown in Figure 4 allowed for determination of of C 0 / δ PS and C 1 / δ 2 PS as linear regression's Y-intercept and the slope, respectively. The thickness δ PS of the porous PAN layer for the composite membrane was obtained from SEM micrographs (Appendix A, Figure A1) and estimated to be 30 µm ± 7 µm. The error of the porous layer thickness determination is likely to originate from the roughness of the polyester non-woven. composite membrane was obtained from SEM micrographs (Appendix A, Figure A1) and estimated 239 to be 30 µm ± 7 µm. The error of the porous layer thickness determination is likely to originate from 240 the roughness of the polyester non-woven. 241

245
The values of the ε/τ and the average pore size dpore for stamp Stamp 1 were calculated from 246 Equations 18 and 19 as 0.056 and 133 nm, respectively. The measurements with the Stamp 2 247 (Appendix A, Figure A2) taken from another PAN membrane batch gave ε/τ and dpore values of 0.099 248 and 119 nm, respectively. The pore diameter visible on the SEM micrograph of the PAN membrane 249 surface ( Figure 5) had a maximum value of 20 nm. This value is six times smaller than the pore size 250 calculated via the DGM. The spongy, asymmetric morphology of the PAN membrane is characterized 251 by high irregularity of the porous structure across the membrane thickness, with pores tapering 252 towards a smaller diameter as they approach the upper, feed side surface of the membrane [20]. 253 Hence, it can be assumed that the pore size values determined using Equation (18)  composite membrane was obtained from SEM micrographs (Appendix A, Figure A1) and estimated 239 to be 30 µm ± 7 µm. The error of the porous layer thickness determination is likely to originate from 240 the roughness of the polyester non-woven. 241

245
The values of the ε/τ and the average pore size dpore for stamp Stamp 1 were calculated from 246 Equations 18 and 19 as 0.056 and 133 nm, respectively. The measurements with the Stamp 2 247 (Appendix A, Figure A2) taken from another PAN membrane batch gave ε/τ and dpore values of 0.099 248 and 119 nm, respectively. The pore diameter visible on the SEM micrograph of the PAN membrane 249 surface ( Figure 5) had a maximum value of 20 nm. This value is six times smaller than the pore size 250 calculated via the DGM. The spongy, asymmetric morphology of the PAN membrane is characterized 251 by high irregularity of the porous structure across the membrane thickness, with pores tapering 252 towards a smaller diameter as they approach the upper, feed side surface of the membrane [20]. 253 Hence, it can be assumed that the pore size values determined using Equation (18)  composite membrane was obtained from SEM micrographs (Appendix A, Figure A1) and estimated 239 to be 30 µm ± 7 µm. The error of the porous layer thickness determination is likely to originate from 240 the roughness of the polyester non-woven. 241

245
The values of the ε/τ and the average pore size dpore for stamp Stamp 1 were calculated from 246 Equations 18 and 19 as 0.056 and 133 nm, respectively. The measurements with the Stamp 2 247 (Appendix A, Figure A2) taken from another PAN membrane batch gave ε/τ and dpore values of 0.099 248 and 119 nm, respectively. The pore diameter visible on the SEM micrograph of the PAN membrane 249 surface ( Figure 5) had a maximum value of 20 nm. This value is six times smaller than the pore size 250 calculated via the DGM. The spongy, asymmetric morphology of the PAN membrane is characterized 251 by high irregularity of the porous structure across the membrane thickness, with pores tapering 252 towards a smaller diameter as they approach the upper, feed side surface of the membrane [20]. 253 Hence, it can be assumed that the pore size values determined using Equation (18)  Based on these results for the PAN porous layer, the first experiment-model comparisons are ven in Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the essure difference across the membrane Δpi. The permeate flux for all gases increases with an crease in pressure difference. The slope of pressure dependence follows the order of molecular eights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular eight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on e molecular weight of the gases because of the diffusive component of mass transfer stemming om molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing e pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent e gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the tire composite membrane, as described in e.g., [32]. For a more accurate prediction of the rmeation behaviour of TFCM, the individual layers made from different polymers with different icknesses have to be considered individually and the FVM parameters have to be determined for ch layer by using Equations (6)- (8).
A thick film sample of PDMS was prepared and gas transport properties were determined using e time-lag method. The temperature dependence of the permeability coefficients for all gases is esented in Figure 7 and served for estimation of the activation energies and the permeabilities at finite temperature for the investigated gases. The values presented in Table 2 are in agreement with e data reported in previous investigations [33][34][35]. At the operation conditions employed in this udy, swelling was not assessed and the respective parameters of the FVM were therefore not termined. The data shown for the CO2 selective block copolymer PolyActive™ also listed in Table were determined as described above for PDMS using a thick film sample. Based on these results for the PAN porous layer, the first experiment-model comparisons are ven in Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the essure difference across the membrane Δpi. The permeate flux for all gases increases with an crease in pressure difference. The slope of pressure dependence follows the order of molecular eights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular eight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on e molecular weight of the gases because of the diffusive component of mass transfer stemming m molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing e pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent e gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the tire composite membrane, as described in e.g., [32]. For a more accurate prediction of the rmeation behaviour of TFCM, the individual layers made from different polymers with different icknesses have to be considered individually and the FVM parameters have to be determined for ch layer by using Equations (6)- (8).
A thick film sample of PDMS was prepared and gas transport properties were determined using e time-lag method. The temperature dependence of the permeability coefficients for all gases is esented in Figure 7 and served for estimation of the activation energies and the permeabilities at finite temperature for the investigated gases. The values presented in Table 2 are in agreement with e data reported in previous investigations [33][34][35]. At the operation conditions employed in this udy, swelling was not assessed and the respective parameters of the FVM were therefore not termined. The data shown for the CO2 selective block copolymer PolyActive™ also listed in Table were determined as described above for PDMS using a thick film sample. Based on these results for the PAN porous layer, the first experiment-model comparisons are ven in Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the essure difference across the membrane Δpi. The permeate flux for all gases increases with an crease in pressure difference. The slope of pressure dependence follows the order of molecular ights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular ight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on e molecular weight of the gases because of the diffusive component of mass transfer stemming m molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing e pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent e gas transport in porous support layers [27].

. Estimation of FVM Parameters for Dense Layers
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the tire composite membrane, as described in e.g., [32]. For a more accurate prediction of the rmeation behaviour of TFCM, the individual layers made from different polymers with different icknesses have to be considered individually and the FVM parameters have to be determined for ch layer by using Equations (6)- (8).
A thick film sample of PDMS was prepared and gas transport properties were determined using e time-lag method. The temperature dependence of the permeability coefficients for all gases is esented in Figure 7 and served for estimation of the activation energies and the permeabilities at finite temperature for the investigated gases. The values presented in Table 2 are in agreement with e data reported in previous investigations [33][34][35]. At the operation conditions employed in this dy, swelling was not assessed and the respective parameters of the FVM were therefore not termined. The data shown for the CO2 selective block copolymer PolyActive™ also listed in Table ere determined as described above for PDMS using a thick film sample. The values of the ε/τ and the average pore size d pore for stamp Stamp 1 were calculated from Equations (18) and (19) as 0.056 and 133 nm, respectively. The measurements with the Stamp 2 (Appendix A, Figure A2) taken from another PAN membrane batch gave ε/τ and d pore values of 0.099 and 119 nm, respectively. The pore diameter visible on the SEM micrograph of the PAN membrane surface ( Figure 5) had a maximum value of 20 nm. This value is six times smaller than the pore size calculated via the DGM. The spongy, asymmetric morphology of the PAN membrane is characterized by high irregularity of the porous structure across the membrane thickness, with pores tapering towards a smaller diameter as they approach the upper, feed side surface of the membrane [20]. Hence, it can be assumed that the pore size values determined using Equation (18) represent an average value for the entire support structure. surface ( Figure 5) had a maximum value of 20 nm. This value is six times smaller than the pore size 250 calculated via the DGM. The spongy, asymmetric morphology of the PAN membrane is characterized 251 by high irregularity of the porous structure across the membrane thickness, with pores tapering 252 towards a smaller diameter as they approach the upper, feed side surface of the membrane [20]. 253 Hence, it can be assumed that the pore size values determined using Equation (18)  Based on these results for the PAN porous layer, the first experiment-model comparisons are given in Figure 6 where the single component permeate flux i is plotted as a function of the pressure difference across the membrane ∆p i . The permeate flux for all gases increases with an increase in pressure difference. The slope of pressure dependence follows the order of molecular weights of gases studied: H 2 , CH 4 , N 2 , O 2 , and CO 2 . Hydrogen, for example has the smallest molecular weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on the molecular weight of the gases because of the diffusive component of mass transfer stemming from molecule-pore wall collisions, i.e., Knudsen diffusion [31]. Thus, the principle of representing the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent the gas transport in porous support layers [27]. Based on these results for the PAN porous layer, the first experiment-model comparisons are 259 given in Figure 6 where the single component permeate flux n i ʹʹ is plotted as a function of the 260 pressure difference across the membrane Δpi. The permeate flux for all gases increases with an 261 increase in pressure difference. The slope of pressure dependence follows the order of molecular 262 weights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular 263 weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on 264 the molecular weight of the gases because of the diffusive component of mass transfer stemming 265 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing 266 the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the 274 entire composite membrane, as described in e.g., [32]. For a more accurate prediction of the 275 permeation behaviour of TFCM, the individual layers made from different polymers with different 276 thicknesses have to be considered individually and the FVM parameters have to be determined for 277  Based on these results for the PAN porous layer, the first experiment-model comparisons are 259 given in Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the 260 pressure difference across the membrane Δpi. The permeate flux for all gases increases with an 261 increase in pressure difference. The slope of pressure dependence follows the order of molecular 262 weights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular 263 weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on 264 the molecular weight of the gases because of the diffusive component of mass transfer stemming 265 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing 266 the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the 274 entire composite membrane, as described in e.g., [32]. Based on these results for the PAN porous layer, the first experiment-model comparisons are 259 given in Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the 260 pressure difference across the membrane Δpi. The permeate flux for all gases increases with an 261 increase in pressure difference. The slope of pressure dependence follows the order of molecular 262 weights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular 263 weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on 264 the molecular weight of the gases because of the diffusive component of mass transfer stemming 265 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing 266 the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the 274 entire composite membrane, as described in e.g., [32]. Based on these results for the PAN porous layer, the first experiment-model comparisons are 259 given in Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the 260 pressure difference across the membrane Δpi. The permeate flux for all gases increases with an 261 increase in pressure difference. The slope of pressure dependence follows the order of molecular 262 weights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular 263 weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on 264 the molecular weight of the gases because of the diffusive component of mass transfer stemming 265 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing 266 the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the 274 entire composite membrane, as described in e.g., [32]. For a more accurate prediction of the 275 permeation behaviour of TFCM, the individual layers made from different polymers with different 276 thicknesses have to be considered individually and the FVM parameters have to be determined for 277

Estimation of FVM Parameters for Dense Layers
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the entire composite membrane, as described in e.g., [32]. For a more accurate prediction of the permeation behaviour of TFCM, the individual layers made from different polymers with different thicknesses have to be considered individually and the FVM parameters have to be determined for each layer by using Equations (6)- (8).
A thick film sample of PDMS was prepared and gas transport properties were determined using the time-lag method. The temperature dependence of the permeability coefficients for all gases is presented in Figure 7 and served for estimation of the activation energies and the permeabilities at infinite temperature for the investigated gases. The values presented in Table 2 are in agreement with the data reported in previous investigations [33][34][35]. At the operation conditions employed in this study, swelling was not assessed and the respective parameters of the FVM were therefore not determined. The data shown for the CO 2 selective block copolymer PolyActive™ also listed in Table 2 were determined as described above for PDMS using a thick film sample.

293
The swelling effect of CO2 on the pure PDMS at pressures up to 1 MPa was reported to have an 294 (Appendix A, Figure A2) taken from another PAN 248 and 119 nm, respectively. The pore diameter visibl 249 surface ( Figure 5)  given in Figure 6 where the single component permeate flux ṅi ' from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus 266 the pore wall as consisting of "giant" molecules in the DGM appears 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by usin 274 entire composite membrane, as described in e.g., [32]. For a mo 275 permeation behaviour of TFCM, the individual layers made from dif 276 thicknesses have to be considered individually and the FVM parame 277 each layer by using Equations (6)- (8).

278
A thick film sample of PDMS was prepared and gas transport pro 279 the time-lag method. The temperature dependence of the permeabil 280 presented in Figure 7 and served for estimation of the activation ene 281 infinite temperature for the investigated gases. The values presented in 282 the data reported in previous investigations [33][34][35]. At the operatio 283 study, swelling was not assessed and the respective parameters of 284 determined. The data shown for the CO2 selective block copolymer Po 285 2 were determined as described above for PDMS using a thick film sa 286 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus 266 the pore wall as consisting of "giant" molecules in the DGM appears 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by usin 274 entire composite membrane, as described in e.g., [32]. For a mo 275 permeation behaviour of TFCM, the individual layers made from dif 276 thicknesses have to be considered individually and the FVM parame 277 each layer by using Equations (6)- (8).

278
A thick film sample of PDMS was prepared and gas transport pro 279 the time-lag method. The temperature dependence of the permeabil 280 presented in Figure 7 and served for estimation of the activation ene 281 infinite temperature for the investigated gases. The values presented in 282 the data reported in previous investigations [33][34][35]. At the operatio 283 study, swelling was not assessed and the respective parameters of 284 determined. The data shown for the CO2 selective block copolymer Po 285 2 were determined as described above for PDMS using a thick film sa 286  [31]. Thus, the principle of representing of "giant" molecules in the DGM appears to be well suited to represent s support layers [27].

meters for Dense Layers
e FVM may be determined directly by using Equations (6) and (7) for the ne, as described in e.g., [32]. For a more accurate prediction of the FCM, the individual layers made from different polymers with different sidered individually and the FVM parameters have to be determined for ons (6)-(8). f PDMS was prepared and gas transport properties were determined using temperature dependence of the permeability coefficients for all gases is served for estimation of the activation energies and the permeabilities at e investigated gases. The values presented in Table 2 are in agreement with ous investigations [33][34][35]. At the operation conditions employed in this assessed and the respective parameters of the FVM were therefore not n for the CO2 selective block copolymer PolyActive™ also listed in Table  ribed above for PDMS using a thick film sample.  The swelling effect of CO 2 on the pure PDMS at pressures up to 1 MPa was reported to have an effect of less than 5% on the thickness [36][37][38]. In this work we assume that the swelling effect of gases on the single layer can be neglected in the investigated pressure range up to 60 kPa [39]. Hence, the FVM (Equations (6) and (7)) simplifies to be an Arrhenius relationship.
From solving Equations (3) to (16) for the porous support and gutter layer, the thickness of the gutter layer δ G can be estimated. If the tortuosity factor is known, the porosity A pore A can be determined. The geometrical definition of tortuosity implies that it is always larger than unity [40]. In our study, we estimated τ to be in the range from 1.6 to 5 in accordance with the guidelines developed for porous media [41].
In the following section, four different scenarios will be defined for calculating the thickness of the gutter layer from the permeation experiments as described in Section 3 by Equations (3) to (16). The scenarios were: 1.

2.
Scenario 2: using gas transport data of Stamp 1-2 with d pore = 133 nm and ε/τ = 0.056. The thickness of the porous support δ PS was decreased from 30 µm to 23 µm in order to examine the influence of the roughness of the porous support.

4.
Scenario 4: using gas transport data of Stamp 2-2 with d pore = 119 nm and ε/τ = 0.099 in order to compare the results of modeling with the results for Stamp 1.
The thickness of the gutter layer estimated from gas permeation measurements according to Equations (3) to (16) for the studied gases corresponds well to the thickness values determined from SEM micrographs [42] as shown in Table 3: 105 nm for Stamp 1-2 and 150 nm for Stamp 2-2 ( Figure 8a,b respectively). If the uncertainty of the porous layer thickness will be taken into account and the thickness δ ps will be reduced by 25% (Scenario 2) the average thickness of the gutter layer δ G will, according to our model, increase from 140 to 150 nm. Since the DGM does not take into account the pore size distribution, the geometric parameter ε/τ of the porous support is causing the variation of the thickness δ ps in Scenario 3. The SD value of the gutter layer thicknesses determined considering transport data of different gases reaches 65 nm for Stamp 1-2. 8a and Figure 8b respectively). If the uncertainty of the porous layer thickness will be taken into 318 account and the thickness δps will be reduced by 25% (Scenario 2) the average thickness of the gutter 319 layer δG will, according to our model, increase from 140 to 150 nm. Since the DGM does not take into 320 account the pore size distribution, the geometric parameter ε/τ of the porous support is causing the 321 variation of the thickness δps in Scenario 3. The SD value of the gutter layer thicknesses determined 322 considering transport data of different gases reaches 65 nm for Stamp 1-2.    Figure 9 shows the temperature dependence of the calculated gutter layer thickness for both Stamps 1-2 and 2-2 using the parameters of Scenario 3 and 4, respectively. The change of thickness with the temperature for both membrane samples correlates with the PDMS thermal expansion coefficient ∆α = 10.9 × 10 −4 K −1 reported in [43]. The thermal expansion coefficients estimated from our experiments as (dV/dT)/V using the assumptions of Scenario 1 and 4 are 15.9 × 10 −4 K −1 and 3.9 × 10 −4 K −1 for Stamp 1-2 and Stamp 2-2, respectively, the values are in the same order of magnitude as the tabulated one. The thermal expansion coefficient was applied to simulate the gas permeances of the TFCM consisting of a porous and a gutter layer ( Figure 10). The permeances modeled with the calculated average thickness δ G = 140 nm at 30 • C and ∆α = 10.9 × 10 −4 K −1 for PDMS gutter layer give less than 10% discrepancy between measured and prognosis values. (Appendix A, Figure A2) taken from another PAN membrane batch gave ε/τ and dpore values of 0.099 248 and 119 nm, respectively. The pore diameter visible on the SEM micrograph of the PAN membrane 249 surface ( Figure 5) had a maximum value of 20 nm. This value is six times smaller than the pore size 250 calculated via the DGM. The spongy, asymmetric morphology of the PAN membrane is characterized 251 by high irregularity of the porous structure across the membrane thickness, with pores tapering 252 towards a smaller diameter as they approach the upper, feed side surface of the membrane [20]. 253 Hence, it can be assumed that the pore size values determined using Equation (18)  sed on these results for the PAN porous layer, the first experiment-model comparisons are n Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the e difference across the membrane Δpi. The permeate flux for all gases increases with an e in pressure difference. The slope of pressure dependence follows the order of molecular s of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on lecular weight of the gases because of the diffusive component of mass transfer stemming olecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing e wall as consisting of "giant" molecules in the DGM appears to be well suited to represent transport in porous support layers [27].

imation of FVM Parameters for Dense Layers
e parameters for the FVM may be determined directly by using Equations (6) and (7) for the composite membrane, as described in e.g., [32]. For a more accurate prediction of the tion behaviour of TFCM, the individual layers made from different polymers with different sses have to be considered individually and the FVM parameters have to be determined for yer by using Equations (6)- (8). thick film sample of PDMS was prepared and gas transport properties were determined using e-lag method. The temperature dependence of the permeability coefficients for all gases is ed in Figure 7 and served for estimation of the activation energies and the permeabilities at sed on these results for the PAN porous layer, the first experiment-model comparisons are n Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the e difference across the membrane Δpi. The permeate flux for all gases increases with an e in pressure difference. The slope of pressure dependence follows the order of molecular s of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on lecular weight of the gases because of the diffusive component of mass transfer stemming olecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing e wall as consisting of "giant" molecules in the DGM appears to be well suited to represent transport in porous support layers [27].

imation of FVM Parameters for Dense Layers
e parameters for the FVM may be determined directly by using Equations (6) and (7) for the composite membrane, as described in e.g., [32]. For a more accurate prediction of the tion behaviour of TFCM, the individual layers made from different polymers with different sses have to be considered individually and the FVM parameters have to be determined for yer by using Equations (6)- (8). thick film sample of PDMS was prepared and gas transport properties were determined using e-lag method. The temperature dependence of the permeability coefficients for all gases is ed in Figure 7 and served for estimation of the activation energies and the permeabilities at sed on these results for the PAN porous layer, the first experiment-model comparisons are n Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the e difference across the membrane Δpi. The permeate flux for all gases increases with an in pressure difference. The slope of pressure dependence follows the order of molecular of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on lecular weight of the gases because of the diffusive component of mass transfer stemming olecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing e wall as consisting of "giant" molecules in the DGM appears to be well suited to represent transport in porous support layers [27].

mation of FVM Parameters for Dense Layers
e parameters for the FVM may be determined directly by using Equations (6) and (7) for the omposite membrane, as described in e.g., [32]. For a more accurate prediction of the tion behaviour of TFCM, the individual layers made from different polymers with different sses have to be considered individually and the FVM parameters have to be determined for er by using Equations (6)- (8). hick film sample of PDMS was prepared and gas transport properties were determined using e-lag method. The temperature dependence of the permeability coefficients for all gases is ed in Figure 7 and served for estimation of the activation energies and the permeabilities at

343
The parameters of the porous layer e.g., ε/τ as well as the interaction of porous and gutter layers 344 should be taken into account to describe the delivery of the gas from the continuous gutter layer into 345 the pores of the porous support, especially in case of highly permeable gases when the permeance of 346 composite membrane was obtained from SEM micrographs (Appendix A, Figure A1) and estimated 239 to be 30 µm ± 7 µm. The error of the porous layer thickness determination is likely to originate from 240 the roughness of the polyester non-woven. 241 242

245
The values of the ε/τ and the average pore size dpore for stamp Stamp 1 were calculated from 246 Equations 18 and 19 as 0.056 and 133 nm, respectively. The measurements with the Stamp 2 247 (Appendix A, Figure A2) taken from another PAN membrane batch gave ε/τ and dpore values of 0.099 248 and 119 nm, respectively. The pore diameter visible on the SEM micrograph of the PAN membrane 249 surface ( Figure 5) had a maximum value of 20 nm. This value is six times smaller than the pore size 250 calculated via the DGM. The spongy, asymmetric morphology of the PAN membrane is characterized 251 by high irregularity of the porous structure across the membrane thickness, with pores tapering 252 towards a smaller diameter as they approach the upper, feed side surface of the membrane [20]. 253 Hence, it can be assumed that the pore size values determined using Equation (18)  composite membrane was obtained from SEM micrographs (Appendix A, Figure A1) and estimated 239 to be 30 µm ± 7 µm. The error of the porous layer thickness determination is likely to originate from 240 the roughness of the polyester non-woven. 241 242

245
The values of the ε/τ and the average pore size dpore for stamp Stamp 1 were calculated from 246 Equations 18 and 19 as 0.056 and 133 nm, respectively. The measurements with the Stamp 2 247 (Appendix A, Figure A2) taken from another PAN membrane batch gave ε/τ and dpore values of 0.099 248 and 119 nm, respectively. The pore diameter visible on the SEM micrograph of the PAN membrane 249 surface ( Figure 5) had a maximum value of 20 nm. This value is six times smaller than the pore size 250 calculated via the DGM. The spongy, asymmetric morphology of the PAN membrane is characterized 251 by high irregularity of the porous structure across the membrane thickness, with pores tapering 252 towards a smaller diameter as they approach the upper, feed side surface of the membrane [20]. 253 Hence, it can be assumed that the pore size values determined using Equation (18)  pressure difference across the membrane Δpi. The permeate flux for all gases increases with an 261 increase in pressure difference. The slope of pressure dependence follows the order of molecular 262 weights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular 263 weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on 264 the molecular weight of the gases because of the diffusive component of mass transfer stemming 265 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing 266 the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the 274 entire composite membrane, as described in e.g., [32]. For a more accurate prediction of the 275 permeation behaviour of TFCM, the individual layers made from different polymers with different 276 thicknesses have to be considered individually and the FVM parameters have to be determined for 277 each layer by using Equations (6)- (8).

278
A thick film sample of PDMS was prepared and gas transport properties were determined using 279 the time-lag method. The temperature dependence of the permeability coefficients for all gases is 280 presented in Figure 7 and served for estimation of the activation energies and the permeabilities at 281 infinite temperature for the investigated gases. The values presented in Table 2 are in agreement with  282 the data reported in previous investigations [33][34][35]. At the operation conditions employed in this 283 study, swelling was not assessed and the respective parameters of the FVM were therefore not 284 determined. The data shown for the CO2 selective block copolymer PolyActive™ also listed in Table  285 2 were determined as described above for PDMS using a thick film sample. 286 pressure difference across the membrane Δpi. The permeate flux for all gases increases with an 261 increase in pressure difference. The slope of pressure dependence follows the order of molecular 262 weights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular 263 weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on 264 the molecular weight of the gases because of the diffusive component of mass transfer stemming 265 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing 266 the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the 274 entire composite membrane, as described in e.g., [32]. For a more accurate prediction of the 275 permeation behaviour of TFCM, the individual layers made from different polymers with different 276 thicknesses have to be considered individually and the FVM parameters have to be determined for 277 each layer by using Equations (6)- (8).

278
A thick film sample of PDMS was prepared and gas transport properties were determined using 279 the time-lag method. The temperature dependence of the permeability coefficients for all gases is 280 presented in Figure 7 and served for estimation of the activation energies and the permeabilities at 281 infinite temperature for the investigated gases. The values presented in Table 2 are in agreement with  282 the data reported in previous investigations [33][34][35]. At the operation conditions employed in this 283 study, swelling was not assessed and the respective parameters of the FVM were therefore not 284 determined. The data shown for the CO2 selective block copolymer PolyActive™ also listed in Table  285 2 were determined as described above for PDMS using a thick film sample. 286 pressure difference across the membrane Δpi. The permeate flux for all gases increases with an 261 increase in pressure difference. The slope of pressure dependence follows the order of molecular 262 weights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular 263 weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on 264 the molecular weight of the gases because of the diffusive component of mass transfer stemming 265 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing 266 the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the 274 entire composite membrane, as described in e.g., [32]. For a more accurate prediction of the 275 permeation behaviour of TFCM, the individual layers made from different polymers with different 276 thicknesses have to be considered individually and the FVM parameters have to be determined for 277 each layer by using Equations (6)- (8).

278
A thick film sample of PDMS was prepared and gas transport properties were determined using 279 the time-lag method. The temperature dependence of the permeability coefficients for all gases is 280 presented in Figure 7 and served for estimation of the activation energies and the permeabilities at 281 infinite temperature for the investigated gases. The values presented in Table 2 are in agreement with  282 the data reported in previous investigations [33][34][35]. At the operation conditions employed in this 283 study, swelling was not assessed and the respective parameters of the FVM were therefore not 284 determined. The data shown for the CO2 selective block copolymer PolyActive™ also listed in Table  285 2 were determined as described above for PDMS using a thick film sample. 286 CO 2 ] and modeled (dashed lines) using Equations (4) to (14) and 17 permeances for Stamp 1-2. Data of TFCM with two layers: porous support and gutter layer. Scenario 1.
The parameters of the porous layer e.g., ε/τ as well as the interaction of porous and gutter layers should be taken into account to describe the delivery of the gas from the continuous gutter layer into the pores of the porous support, especially in case of highly permeable gases when the permeance of the selective layer is close to the one of the porous layers [44]. The DGM takes into account the molecular weight of the fluid only, but not its molecular volume or shape. The significant difference in the calculated PDMS layer thickness for different gases can reflect the effect of penetrant parameters other than molecular weight on penetration through the composite membrane. The paper [45] shows the correlation of the gas kinetic diameter with the diffusion coefficient.
Technically, if the thicknesses of the gutter and support layers are known from SEM investigations, the prognosis of total permeance for the TFCM depending on temperature and at pressures less than 100 kPa can be carried out ( Figure 11). If experimental data of permeances are available, the reciprocal estimation of the thicknesses δ G and δ S is possible. the selective layer is close to the one of the porous layers [44]. The DGM takes into account the 347 molecular weight of the fluid only, but not its molecular volume or shape. The significant difference 348 in the calculated PDMS layer thickness for different gases can reflect the effect of penetrant 349 parameters other than molecular weight on penetration through the composite membrane. The paper 350 [45] shows the correlation of the gas kinetic diameter with the diffusion coefficient. 351 Technically, if the thicknesses of the gutter and support layers are known from SEM 352 investigations, the prognosis of total permeance for the TFCM depending on temperature and at 353 pressures less than 100 kPa can be carried out ( Figure 11). If experimental data of permeances are 354 available, the reciprocal estimation of the thicknesses δG and δS is possible. 355 Based on the successful modeling of a bilayer (PDMS gutter layer on PAN porous support) membrane it is possible to make a new step in direction of modeling of a more complex membrane consisting of aforementioned layers and one additional layer of PolyActive™. This membrane is widely studied for separation of CO 2 from e.g., flue gases [10,32] and it is of tremendous importance to develop a model adequately describing the behaviour of the membrane in various environments and various conditions.
The permeances of PolyActive™ TFCM modeled with average thicknesses δ G = 130 nm and δ S = 90 nm determined by SEM micrographs for the Stamp 1-3 give less than 10% discrepancy between experimental and prognosis values for fast permeating gases and about 3% for slow permeating gases. Similar behaviour was observed for the Stamp 2-3, with no sudden changes. The thermal expansion coefficients ∆α (PDMS) = 10.9 × 10 −4 K −1 , ∆α (PolyActive TM ) = 1.2 × 10 −4 K −1 [46] were applied to simulate the gas permeances of the TFCM consisting of porous, gutter layer and the selective layer ( Figure 12). Equations (3) to (16) were used in combination with values tabulated in Table 2 for PolyActive TM .

245
The values of the ε/τ and the average pore size dpore for stamp Stamp 1 were calculated from 246 Equations 18 and 19 as 0.056 and 133 nm, respectively. The measurements with the Stamp 247 (Appendix A, Figure A2) taken from another PAN membrane batch gave ε/τ and dpore values of 0.09 248 and 119 nm, respectively. The pore diameter visible on the SEM micrograph of the PAN membran 249 surface ( Figure 5) had a maximum value of 20 nm. This value is six times smaller than the pore siz 250 calculated via the DGM. The spongy, asymmetric morphology of the PAN membrane is characterized 251 by high irregularity of the porous structure across the membrane thickness, with pores tapering 252 towards a smaller diameter as they approach the upper, feed side surface of the membrane [20] 253 Hence, it can be assumed that the pore size values determined using Equation (18) Figure A1) and estimate 239 to be 30 µm ± 7 µm. The error of the porous layer thickness determination is likely to originate from 240 the roughness of the polyester non-woven.

245
The values of the ε/τ and the average pore size dpore for stamp Stamp 1 were calculated from 246 Equations 18 and 19 as 0.056 and 133 nm, respectively. The measurements with the Stamp 247 (Appendix A, Figure A2) taken from another PAN membrane batch gave ε/τ and dpore values of 0.09 248 and 119 nm, respectively. The pore diameter visible on the SEM micrograph of the PAN membran 249 surface ( Figure 5) had a maximum value of 20 nm. This value is six times smaller than the pore siz 250 calculated via the DGM. The spongy, asymmetric morphology of the PAN membrane is characterize 251 by high irregularity of the porous structure across the membrane thickness, with pores taperin 252 towards a smaller diameter as they approach the upper, feed side surface of the membrane [20 253 Hence, it can be assumed that the pore size values determined using Equation (18)  Based on these results for the PAN porous layer, the first experiment-model comparisons are 259 given in Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the 260 pressure difference across the membrane Δpi. The permeate flux for all gases increases with an 261 increase in pressure difference. The slope of pressure dependence follows the order of molecular 262 weights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular 263 weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on 264 the molecular weight of the gases because of the diffusive component of mass transfer stemming 265 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing 266 the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the 274 entire composite membrane, as described in e.g., [32]. For a more accurate prediction of the 275 permeation behaviour of TFCM, the individual layers made from different polymers with different 276 thicknesses have to be considered individually and the FVM parameters have to be determined for 277 each layer by using Equations (6)- (8).

278
A thick film sample of PDMS was prepared and gas transport properties were determined using 279 the time-lag method. The temperature dependence of the permeability coefficients for all gases is 280 presented in Figure 7 and served for estimation of the activation energies and the permeabilities at 281 infinite temperature for the investigated gases. The values presented in Table 2 are in agreement with  282 the data reported in previous investigations [33][34][35]. At the operation conditions employed in this 283 study, swelling was not assessed and the respective parameters of the FVM were therefore not 284 determined. The data shown for the CO2 selective block copolymer PolyActive™ also listed in Table  285 2 were determined as described above for PDMS using a thick film sample. 286 Based on these results for the PAN porous layer, the first experiment-model comparisons are 259 given in Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the 260 pressure difference across the membrane Δpi. The permeate flux for all gases increases with an 261 increase in pressure difference. The slope of pressure dependence follows the order of molecular 262 weights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular 263 weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on 264 the molecular weight of the gases because of the diffusive component of mass transfer stemming 265 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing 266 the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the 274 entire composite membrane, as described in e.g., [32]. For a more accurate prediction of the 275 permeation behaviour of TFCM, the individual layers made from different polymers with different 276 thicknesses have to be considered individually and the FVM parameters have to be determined for 277 each layer by using Equations (6)- (8).

278
A thick film sample of PDMS was prepared and gas transport properties were determined using 279 the time-lag method. The temperature dependence of the permeability coefficients for all gases is 280 presented in Figure 7 and served for estimation of the activation energies and the permeabilities at 281 infinite temperature for the investigated gases. The values presented in Table 2 are in agreement with  282 the data reported in previous investigations [33][34][35]. At the operation conditions employed in this 283 study, swelling was not assessed and the respective parameters of the FVM were therefore not 284 determined. The data shown for the CO2 selective block copolymer PolyActive™ also listed in Table  285 2 were determined as described above for PDMS using a thick film sample. 286 Based on these results for the PAN porous layer, the first experiment-model comparisons are 259 given in Figure 6 where the single component permeate flux ṅi '' is plotted as a function of the 260 pressure difference across the membrane Δpi. The permeate flux for all gases increases with an 261 increase in pressure difference. The slope of pressure dependence follows the order of molecular 262 weights of gases studied: H2, CH4, N2, O2, and CO2. Hydrogen, for example has the smallest molecular 263 weight and shows the largest permeate flux. The DGM predicts that the molar flux ratio depends on 264 the molecular weight of the gases because of the diffusive component of mass transfer stemming 265 from molecule-pore wall collisions, i.e. Knudsen diffusion [31]. Thus, the principle of representing 266 the pore wall as consisting of "giant" molecules in the DGM appears to be well suited to represent 267 the gas transport in porous support layers [27].

Estimation of FVM Parameters for Dense Layers 273
The parameters for the FVM may be determined directly by using Equations (6) and (7) for the 274 entire composite membrane, as described in e.g., [32]. For a more accurate prediction of the 275 permeation behaviour of TFCM, the individual layers made from different polymers with different 276 thicknesses have to be considered individually and the FVM parameters have to be determined for 277 each layer by using Equations (6)- (8).

278
A thick film sample of PDMS was prepared and gas transport properties were determined using 279 the time-lag method. The temperature dependence of the permeability coefficients for all gases is 280 presented in Figure 7 and served for estimation of the activation energies and the permeabilities at 281 infinite temperature for the investigated gases. The values presented in Table 2 are in agreement with  282 the data reported in previous investigations [33][34][35]. At the operation conditions employed in this 283 study, swelling was not assessed and the respective parameters of the FVM were therefore not 284 determined. The data shown for the CO2 selective block copolymer PolyActive™ also listed in Table  285 2 were determined as described above for PDMS using a thick film sample. 286  Figure A3).
The separation layer affects the results as the main contributor in the prognosis of permeance values. For the bilayer or trilayer samples, the coupling between sublayer and toplayer can influence their physical properties [47]. However, taking into account the thermal expansion coefficient prevents the eventual increase in discrepancy between experimental and prognosis permeance values with temperature grow. As this study shows, the estimation of geometrical parameters for the subordinated layers and taking into account their gas transport properties can significantly increase the accuracy of the prognosis for TFCM under changing working conditions.

Conclusions
Porous, porous/gutter layer and porous/gutter layer/selective layer types of TFCM were investigated for their gas transport properties. A model describing the individual contributions of the different layers' mass transfer resistances was successfully employed. The porous support structures were described using the Dusty Gas Model whilst the permeation in the dense gutter and separation layers was described by applicable models such as the Free-Volume model, using parameters derived from single gas time lag measurements. The model was employed to calculate the thickness of a silicone-based gutter layer from permeation measurements and compared to the thickness determined by SEM.
The model takes into account the dependence of the total permeance on the properties of the porous layer, as well as the thermal expansion of dense layers at pressures below 100 kPa.
The developed approach allows for the description of gas transport through the multilayer TFCM for variety of gases using performance data of the material that make up the individual layers. For example, the use of the developed model will allow one to obtain comparable selective layer thicknesses assessed from SEM investigation of membrane morphology and from gas transport experiments.

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