**3. Gas transport mechanism in polymer membranes**

Polymeric membranes are generally non-porous and gas permeation through dense polymer membranes is typically described by the solution-diffusion model. Gas permeation occurs by sorption of gas on the feed side of membranes, then molecular diffusion through the membrane matrix, and evaporation of the gas from permeate side of membrane surface. Based on the solution-diffusion model, the membrane permeability, *Pi* (Barrer), is defined as the product of diffusivity coefficient and solubility coefficient (given by Eq. (1)) at high fugacity differences across the membrane [40]:

*Polymer-Based Membranes for C3+ Hydrocarbon Removal from Natural Gas DOI: http://dx.doi.org/10.5772/intechopen.103903*

$$P\_i = D\_i \times \mathbb{S}\_i \tag{1}$$

where *Pi* is the permeability coefficient measured in Barrer (1 Barrer = <sup>1</sup> � <sup>10</sup>�<sup>10</sup> cm3 (STP) cm/cm2 s cmHg), *Si* is the solubility coefficient (cm3 gas/cm3 polymer cmHg), and *Di* is the diffusion coefficient of the penetrant (cm2 /s). The solubility coefficient is a thermodynamic parameter and is mainly influenced by the condensability of the penetrant gases [41], and it is inversely proportional to gas boiling point or critical temperature [42]. In contrast to gas sorption, the gas diffusion coefficients vary widely depending on the polymer materials, it is a kinetic parameter that measures the overall mobility of the penetrant molecules in the membrane, depending on various factors such as the size and shape of the gas molecules, the cohesive energy density of the polymer, the mobility of the polymer chains and the free volume size and distribution of the polymer [43]. Accordingly, the membrane selectivity α, the best measure of a membrane's ability to separate gases *i* and *j* is defined as the ratio of the permeabilities of penetrant, *Pi* and *Pj* (given by Eq. (2)) [44].

$$a\_{i\bar{j}} = P\_i/P\_j = \left(D\_i/D\_j\right) \times \left(\mathbb{S}\_i/\mathbb{S}\_j\right) \tag{2}$$

As shown by Eq. (2), membrane selectivity (also known as permselectivity) can be separated into diffusivity selectivity, *Di/Dj*, taken as the ratio of diffusivity coefficients of the two gases and solubility selectivity, *Si/Sj*, taken as the ratio of solubility coefficients. Diffusivity selectivity depends primarily on the size-sieving ability of the membrane, the permeation of the smaller molecule is faster as compared to their larger counterparts. While solubility selectivity is largely driven by gas condensability and affinity with the membrane.

In general, glassy polymeric membranes generally tend to permeate smaller molecules, the diffusion coefficients decrease as the molecular size increases; whereas rubbery polymeric membranes permeate more condensable gases, the sorption coefficients generally increase as the condensability increases. Such trends are schematically illustrated in **Figure 7** for glassy and for rubbery polymeric membranes, which shows the gas permeabilities for principal gas components in natural gas with different molecular size (kinetic diameter, *dk*, Å) [13] and condensability (critical temperature,*Tc,* K) [45]. The molecular sizes and relative condensabilities of C3H8 and C4H10, relative to CH4, are highlighted in **Figure 7**. CH4 can be separated from C3+ hydrocarbons by glassy polymers by size-selectivity; however, interaction of heavier hydrocarbons with the polymers chains tends to increase the flux resulting in decrement in C3+/CH4 selectivity in case of rich hydrocarbon natural gas. On the other hand, rubbery polymers (sorption selectivity membranes) are used to separate C3+ hydrocarbons from CH4, because of their condensabilities. For example, the selectivity of hydrocarbon vapors in PDMS membranes is dominated by the sorption component, the C3H8/CH4 and C3H8/N2 sorption selectivities (*Ki*/*K*CH4 and *Ki*/*K*) are 50-fold and 370-fold greater than that of their diffusion selectivities (*Di*/*D*CH4 and *Di*/*D*N2 ) (**Table 6**) [16, 46–48].
