According to solution-diffusion approach, permeance, J, including vapours and permanent gases in a mixture are ordinarily expressed as shown by Leemann et al. (1996) as:
Also, temperature-dependent diffusivity D and sorption coefficient G follow the Arrhenius relationship as described below:
Combining Equations (5), (6), and (7), the permeance can be expressed as:
where Ep refers as apparent activation energy including activation energy for a diffusion step Ed and the heat of sorption Δhs.
The selectivity for a pair of gases in a mixture is:
For vapour-selective membrane materials, the permeances for a vapour–permanent gas mixtures strongly depend on the temperature and feed composition (Pinnau and He (2004) and Jiang and Kumar (2005)). Moreover, we noted (Jiang and Kumar, 2006) that the Ep of hydrocarbon is also influenced by its feed concentration in addition to the physical properties of the penetrant and polymer matrix as well as the chemical structure of the polymer. Due to the coupling effect, the Ep of permanent gas in the same mixture is affected by total feed hydrocarbon concentration at varying operating temperatures to a different extent. For hydrocarbon and nitrogen mixtures in the present study, these could be described by first-order polynomial expressions as:
The exception was nitrogen where:
For each component in a mixture, J0 refers to the permeance when temperature T is extremely high and feed concentration x approaches zero. Apparent activation energy E principally indicates that it is an endothermic permeation process (positive) or an exothermic permeation process (negative). Theoretically, permeance increases with decreasing temperature in an exothermic permeation process, conversely permeance decreases with decreasing temperature in an endothermic permeation process. Apparent interaction parameter “a” shows sensitivity to the feed concentration quantitatively due to the fact that the membrane was plasticized by sorbing penetrant(s) to a different degree of swelling. The values of J0, E and “a” can be obtained through the Marquardt-Levenberg algorithm for non-linear regression using experimental data on the transmembrane flux at various feed concentrations, temperatures, and pressures. Calculated Ji and JN2 principally are independent of the operating pressure as will be discussed later in the Section “Error Analysis.”
Table 1 lists all parameters of Equations (14) and (15) for eight different mixtures using same membrane and experimental set-up as described in our previous work (Jiang and Kumar, 2006). Most multiple correlation coefficients (R1) were near 1, indicating that the equations were good descriptions of the relations between the independent and dependent variables. The probabilities of being wrong (PW) in concluding that the fitted parameters are not zero were less than 0.05 in most cases. Therefore the independent variables (T, xi and xt) can be used to predict the dependent variables (Ji and JN2) without significant errors. Coincidently with experimental results reported earlier (Jiang and Kumar, 2006), the E-values of propylene and propane were always negative in the mixtures containing propylene, propane, or both. This implies that these strong sorbing penetrants loosened segmental chains in the polymeric matrix. Moreover, “a” values of both propylene and propane increased considerably with the increase of the component number for a mixture, showing that C Quaternary and C2 Quaternary mixtures plasticized PDMS coating to the highest degree of swelling in the mixtures including C3 ternary, C binary and C3 binary. This can also be confirmed from E and “a” values of ethylene and ethane in these two mixtures compared with other mixtures containing no propylene and/or propane. For example, “a” values of ethylene and ethane in the mixtures of C Quaternary and C2 Quaternary were much higher than those in the mixtures of C Binary, C2 Binary, and C2 Ternary due to strong coupling effects. Simultaneously their E-values became negative, indicating that ethylene and ethane were in exothermic permeation processes in the presence of C3 components. Based on the positive E-values, it can be concluded that there was no plasticization for C Binary and C2 Binary mixtures, and slight plasticization occurred for C2 Ternary mixture since only ethane E was negative and closed to zero. Also, nitrogen E and “a” values in all eight mixtures showed an endothermic permeation process. However, its permeances were significantly impacted by the swelling degrees which were demonstrated by the magnitudes of “a” values, indicating that the nitrogen permeances were elevated adequately in a larger “a” value gas mixtures. So, it appears that nitrogen “a” values in the mixtures of C Binary, C3 Binary, and C3 Ternary were much larger than those in the mixtures of C Binary, C2 Binary, and C2 Ternary. It is interesting to note that nitrogen “a” values in the mixtures of C Quaternary and C2 Quaternary were even smaller than those in the mixture of C Binary, C3 Binary, and C3 Ternary. This is due to the fact that weakly condensable ethylene or ethane in the same mixtures competed for the limited activated sites. Choosing C Binary and C Binary mixtures as examples, Figures 1 and 2 show the distinction between plasticization and no-plasticization permeance and selectivity with calculated and experimental data at varying temperatures and feed concentrations.
Table 1. Parameters in Equations (14) and (15) for eight mixtures.
|Mixture||Component||J0, GPU||E, J/mol||a||R1|
|C2= Binary||N2||24 013||11 863||0.13||0.994|
| ||C2H4||23 186||7385||0.89||0.951|
|C2 Binary||N2||24 786||12 207||0.12*||0.990|
|C2 Ternary||N2||17 278||11 409||0.16||0.995|
Figure 2. Comparison of calculated and experimental data of permeance and selectivity as the function of hydrocarbon concentration in feed for C Binary and C Binary. Lines and symbols represent calculated and experimental data, respectively. Feed pressure: 393 kPa (a). Permeate pressure: 101.0 kPa (a). Temperature: −8°C.
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The spiral wound membrane module consists of many flat membrane envelopes, which are wrapped around a central pipe with fluid-conductive spacers inside and outside envelopes. The feed under high pressure is introduced from one end of the shell-side. As feed gas passes along the length of the membrane module, a portion of feed gas as permeate penetrates into the membrane envelopes with low pressure. It travels perpendicularly to the feed and spirals to a central collecting pipe. The rest of the feed gas leaves at another end of the shell-side as residue. Figure 3 shows the diagram of a flat membrane envelope with the flow configuration and the computational scheme. The numerical integration was adopted for modelling the separation of multi-component mixtures, which permitted the incorporation of pressure, composition, and temperature-dependent permeances as represented by Coker et al. (1998) and Thundyil and Koros (1997). The membrane envelopes were divided into a series of N sections in the axial direction of the central pipe. Boundary conditions included feed composition, feed flow-rate and pressure, permeate pressure, and operating temperature. Following assumptions were made for formulating the numerical models:
Gas flows were at steady state in the membrane at isothermal conditions.
The deformation of membrane under pressure was negligible.
There was no gas mixing of shell and permeate sides in the directions of bulk flows.
The pressure build-up in shell and permeate sides were negligible.
Feed composition change in any selected infinitely small section k was ignored.
In each section, component i transmembrane flux of a mixture from feed to permeate sides across the membrane was described by Ghosal and Freeman (1994) as:
where vk and yk,i are the mass flow-rate and concentration of component i that leaves the membrane skin surface in section k. According to Equation (16) concentration yk,i principally depends on the permeance, feed concentration, feed and permeate pressures. Therefore, the composition of this transmembrane flux does not relate with any other gas flow-rates which are obtained by the permeation in adjacent sections on the permeate side. Equation (16) can be rewritten as:
In this model, the permeances for each component in this section were determined by the operating temperature and feed composition located at section k − 1 using Equations (14) and (15), then composition and temperature-dependent permeances were placed into the model.
To obtain the value of yk,i, initially vk was set to zero and then checked if . This procedure was iterated by an increment of vk until the inequality was satisfied. These vk and yk,i values were the transmembrane flux and their concentrations produced by the permeation in section k.
Applying this straightforward crossflow model in series from 1 to N section with given boundary conditions at feed, the profiles of feed flow-rate and composition along the axial direction were obtained, which naturally showed the permeance variations in each section. At the end, the compositions and flow-rates of permeate and residue, located at section N, were calculated accurately for a large membrane area to be used in a module.
The membrane process in this paper refers to only membrane modules. It can be module-in-series, module-in-parallel, or a combination as described by Baker et al., (1998). This arrangement depends on the optimum design required for a particular application. Finally, the total recovery of hydrocarbon i for whole membrane process was represented as: