Modeling for design and operation of high-pressure membrane contactors in natural gas sweetening

Over the past decade, membrane contactors (MBC) for CO2 absorption have been widely recognized for their large intensification potential compared to conventional absorption towers. MBC technology uses microporous hollow-fiber membranes to enable effective gas and liquid mass transfer, without the two phases dispersing into each other. The main contribution of this paper is the development and verification of a predictive mathematical model of high-pressure MBC for natural gas sweetening applications, based on which model-based parametric analysis and optimization can be conducted. The model builds upon insight from previous modeling studies by combining 1-d and 2-d mass-balance equations to predict the CO2 absorption flux, whereby the degree of membrane wetting itself is calculated from the knowledge of the membrane pore-size distribution. The predictive capability of the model is tested for both lab-scale and pilot-scale MBC modules, showing a close agreement of the predictions with measured CO2 absorption fluxes at various gas and liquid flowrates, subject to a temperature correction to account for the heat of reaction in the liquid phase. The results of a model-based analysis confirm the advantages of pressurized MBC operation in terms of CO2 removal efficiency. Finally, a comparison between vertical and horizontal modes of operation shows that the CO2 removal efficiency in the latter can be vastly superior as it is not subject to the liquid static head and remediation strategies are discussed. © 2018 The Authors. Published by Elsevier B.V. on behalf of Institution of Chemical Engineers. This is an open access article under the CC BY license (http://creativecommons. org/licenses/by/4.0/).


Introduction
Natural gas (NG) is presently the third most-utilized form of fossil fuel energy and is widely used for both electricity production and transportation. In the reference case of the latest International Energy Outlook [IEA, 2017], the world's NG consumption is expected to increase by 69% between 2012 and 2040, accounting for 29% of the energy consuming market, and surpassing coal as the second most utilized fuel by 2030. NG consists of a mixture of combustible hydrocarbon gases typically from methane (CH 4 ) to pentane (C 5 H 12 ), with impurities such as carbon dioxide (CO 2 ). Removal of CO 2 from separation. Chemical solvent absorption remains the most widely adopted technology to capture CO 2 , using conventional packed, spray or bubble column absorption towers. About 90% of the acid gas treating processes in operation use alkanoamines solvents, such as methylethanolamine (MEA), diethanolamine (DEA), and methyldiethanolamine (MDEA), due to their versatility and ability to remove acid gases to ppm levels [Paul et al., 2007]. Nonetheless, conventional absorption towers have a high capital cost and a large physical footprint, and they are subject to operational problems such as flooding, channeling, foaming and liquid entrainment [Gabelman and Hwang, 1999]. Membrane technology has been applied since the 1980s for CO 2 removal in large-scale applications due to their potential to reduce the footprint and the capital and operating costs [He and Hägg, 2012]. However, the usual gas separation membranes have a relatively low permeability, and their low selectivity can lead to large product loss. By combining membranes and solvents, membrane contactors (MBC) offer a unique way to perform gas-liquid absorption [Gabelman and Hwang, 1999]. The microporous membrane acts as a non-selective phase barrier, allowing the liquid and gas phases to contact with each other, yet without the dispersion of one phase into the other. This barrier prevents flooding or foaming issues from happening, thereby making MBC simple to operate. Their packaging into hollow-fiber membrane (HFM) modules offers a higher mass transfer area compared with conventional packed columns, giving MBC a high intensification potential [Rangwala, 1996;Favre, 2011]. This modularity also empowers a more flexible design and scale-up. The effectiveness of MBC for both preand post-combustion CO 2 capture, as well as NG sweetening and dehydration, has been extensively studied and compared with conventional techniques over the past decades; see, e.g., review papers by [Mansourizadeh and Ismail, 2009;Favre, 2011;He and Hägg, 2012;Hoff and Svendsen, 2014]. In NG sweetening for instance, Hoff and Svendsen (2013) reported a 75% size reduction using MBC compared to conventional packed columns. This technology has been embraced by industry too, with recent patents on acid gas removal from NG granted to Petroliam Nasional Berhad (PETRONAS) [Quek et al., 2015] and the Gas Technology Institute (GTI) [Zhou and Meyer, 2014].
Notwithstanding their high potential, MBC can present rather low mass-transfer coefficients under undisturbed, laminar flow of the liquid phase. Moreover, hydrophobic membranes can be wetted by organic solvents, either partially or fully. The non-wetted mode of MBC operation (Fig. 1a) is preferred since it presents a higher CO 2 absorption rate. Operating with partially-or fully-wetted membranes (Fig. 1b, c) can decrease the mass-transfer flux quite drastically, mainly due to a lower gas diffusivity in the wetted membrane phase. For instance, Wang et al. (2005) reported that a change in the degree of wetting as small as 5% could lead to a 20% reduction in mass transfer rate. In principle, membrane wetting could be prevented by keeping the operating pressure below a critical value, the so-called breakthrough pressure; but partial-wetting may still occur in practice due to nonuniform membrane pore-sizes, with bigger pores being more easily wetted. Many factors are known to affect the degree of wetting, including the membrane properties and various operating parameters, such as the inlet liquid pressure, liquid velocity, liquid temperature, and amine concentrations [Lu et al., 2008;Mosadegh-Sedghi et al., 2012;Rongwong et al., 2015]. Moreover, the variation in liquid pressure along the length of the HFM can affect membrane wetting considerably, e.g. due to pressure drops or the static head in a vertical MBC.
Mathematical models provide an effective tool to help understand the CO 2 removal mechanisms in MBC, and thus enable a better assessment and optimization of their performance. Many modeling and simulation studies in the literature focus on the non-wetted mode of operation, e.g. for MBC operating at or near atmospheric pressure [Hoff et al., 2004;Al-Marzouqi et al., 2008;Rezakazemi et al., 2011;Hoff and Svendsen, 2014]. When partially-or fully-wetted operation is considered, the degree of membrane wetting (or a related parameter) is typically used as a tuning parameter in order for the model predictions to fit given experimental data. For instance, Chabanon et al. (2013) compared several modeling approaches for predicting CO 2 absorption in MBC, ranging from constant mass-transfer coefficient models to 1-d or 2-d convection-diffusion models, and with the membrane masstransfer coefficient used as the single tuning parameter. They found that the use of convection-diffusion models is justified to obtain accurate predictions, in particular when large amine solvent conversions take place. Other studies showing a good agreement between experimental data and the predictions of a 2-d convection-diffusion model by adjusting the degree of membrane wetting can be found in Lu et al. (2008) and Cui et al. (2015). As far as high-pressure MBC operation is concerned, e.g. for NG sweetening applications, limited modeling studies have been published to date. For instance, Faiz and Al-Marzouqi (2010) studied the removal of CO 2 from NG in high-pressure MBC between 10 and 50 bar. They reported a good agreement between a 2-d model and experiments when adjusting the degree of wetting, which also they found to be highly sensitive.
Because of this high sensitivity of the membrane wetting to different operating conditions, analyzing or optimizing the performance of an MBC under the assumption of a constant membrane wetting could lead to erroneous (over-optimistic) results. In response to this, approaches to predicting the degree of membrane wetting in MBC have started to appear in recent years. The studies by Boributh et al. (2011) and Goyal et al. (2015) in particular, exploit knowledge about a membrane's pore-size distribution to predict the degree of wetting in MBC operating at atmospheric pressure.
The main objective of this paper is the development of a predictive mathematical model of high-pressure MBC for NG sweetening applications, on the basis of which modelbased parametric studies and optimization can be conducted. Building upon previous modeling studies, we consider a combination of 1-d and 2-d mass-balance equations to predict the CO 2 absorption flux, whereby the degree of membrane wetting itself is calculated using the Laplace-Young equation based upon knowledge of the membrane pore-size distribution, and real gas behavior too is accounted for at high-pressure operation. The predictive capability of this model is tested against data from two experimental settings: a lab-scale MBC module, where the purification is conducted using binary gas mixtures of CH 4 /CO 2 and N 2 /CO 2 at 11 bar; and a pilot-scale MBC module operated under industrially relevant conditions at a natural gas processing plant in Malaysia. All of the experiments were conducted with aqueous mixtures of MDEA and piperazine (PZ) as the chemical solvent. This ability to predict the variation in membrane wetting along the fiber length makes it possible to analyze the effects of various design and operational decisions on the MBC performance, including the membrane properties, module characteristics, gas and liquid flow rates, and operating pressures and temperatures.
The rest of the paper is organized as follows. The mathematical model of a counter-current hollow-fiber MBC module is presented in Section 2, followed by a description of both experimental set-ups, corresponding model parameters, and computational methods in Section 3. Results of the experimental model verification are presented and discussed in Section 4. Then, a model-based analysis is conducted in Section 5, with a view to quantifying the effects of module orientation and high-pressure operation on a module's CO 2 removal efficiency. Finally, Section 6 concludes the paper and discusses future research directions.

Modeling of high-pressure hollow-fiber MBC
This section of the paper describes a comprehensive mathematical model for predicting the CO 2 removal efficiency in a high-pressure, hollow-fiber MBC module, as depicted on the left panel in Fig. 2. The NG gas mixture containing CO 2 flows through the membrane fibers, while the amine solvent flows inside the shell, in a counter-current arrangement. The gas mixture diffuses from the tube side through the fiber walls into the shell, where CO 2 dissolves in the solvent before reacting with the solvent in order to enhance the removal rate.
The shell area between the fibers depends on the packing density, ∅ defined as where N is the number of fibers in the MBC module; R m [m] is the inner radius of the module; and r 2 [m] is the outer radius of the fibers. Following Happel (1959), the domain of fluid surrounding each fiber can be conveniently approximated by a cylinder with cross-section radius, r 3 [m] given by r 3 = r 2 1 ⁄∅. (2) It is thus sufficient to consider a piece of hollow fiber in order to model the MBC module, as shown on the center panel in Fig. 2. In order to describe the degree of membrane wetting, we introduce the so-called wetted radius, r w [m], a conceptual variable representing the average fraction of membrane pores filled with liquid; see Section 2.1 below for details. The nonwetted and fully-wetted modes of operation thus correspond to r w = r 1 and r w = r 2 , respectively, with r 1 the inner radius of the fibers. On exploiting symmetry, the spatial domain to model a piece of hollow fiber can be taken as (r, z) ∈ [0, r 3 ] × [0, L], which is further partitioned into four subdomains as shown on the right panel in Fig. 2: (i) tube, 0 ≤ r ≤ r 1 ; (ii) membrane-dry, r 1 ≤ r ≤ r w (z); (iii) membrane-wet, r w (z) ≤ r ≤ r 2 ; and (iv) shell, r 2 ≤ r ≤ r 3 . Notice that the geometry of the wetted and non-wetted membrane subdomains is complicated by the dependence of the wetted radius on the axial position, z. In this counter-current configuration, the solvent and the gas are fed at z = 0 and z = L, respectively. Mass conservation equations and a corresponding set of boundary conditions are detailed in Section 2.2.

Modeling of membrane wetting
In partially-wetted operations, a hydrophobic membrane with a non-uniform pore size distribution exhibits a range of breakthrough pressures, with larger pores filled first, followed by smaller ones. According to the Laplace-Young equation, a pore is wetted when the transmembrane pressure difference P TMPD [Pa], is greater than the breakthrough pressure, P c [Pa] given by where ı [m] denotes the pore radius; [N m −1 ], the surface tension of the liquid phase; and Â [rad], the contact angle of the liquid on the membrane. The transmembrane pressure, namely the difference between the pressures in the liquid and gas phases, P 1 and P g [Pa], is such that where P in g [Pa] is the inlet gas pressure; P out l [Pa], the outlet liquid pressure; P l / z [Pa m −1 ], the pressure gradient along the shell axis; l [kg m −3 ], the liquid density; and g = 9.81 m s −2 is the gravitational acceleration. Notice the additional contribution of the liquid static head in a vertical orientation compared with a horizontal MBC. Moreover, the pressure of the gas phase is considered constant in Eq. (4), i.e. the effects of the pressure drop and static head are neglected. One way of expressing the pressure drop in the shell is by using an analogy with viscous flow through assemblages of cylinders. Under the assumption of a constant pressure drop in the flow direction and a noslip condition on the cylinders, Happel (1959) showed that the pressure drop may be computed as where l [kg m −1 s −1 ] is the dynamic viscosity of the liquid phases; and the average liquid velocity, v l [m s −1 ] is given by with F in l [m 3 s −1 ] the liquid volumetric flowrate at the shell inlet z = 0.
In order to prevent membrane wetting and bubble formation, an MBC should ideally be operated such that for all pore radii and all axis positions. If partial wetting occurs at a given position 0 ≤ z ≤ L, the wet pores are those having a radius larger than ı w [m], given by In practice, the degree of wetting is often described in term so of the wetting ratio, , which represents the ratio between the porous-volume occupied by liquid phase and the total porous-volume. Under the assumption that the wetted pores are completed filled with liquid (see Goyal et al. (2015) for a discussion), the wetting ratio may be computed as for a given pore size distribution (PSD) function f, and a maximal pore radius, ı max [m]. Then, the conceptual wetted radius r w representing the average fraction of a pore filed with liquid at a certain position z (see right panel in Fig. 2) is related to the wetting ratio in the following way

Mass conservation equations
The mathematical model presented hereafter is based on steady-state and isothermal operation in all of the phases-see Section 4 for further discussions about the effect of temperature. The mass conservation equations in the gas, liquid and membrane phases are detailed in the following paragraphs, together with the corresponding boundary conditions and the main modeling assumptions.

Gas phase in tube, (r, z) ∈ [0, r 1 ] × [0, L]
We describe the gas phase in the lumen using a simple plug flow, assuming homogeneous concentrations in each crosssection and neglecting the gas diffusivity (Péclet > 10 6 ) and pressure drops along the fiber axis. These assumptions have been validated through previous studies in MBC modeling [Rezakazemi et al., 2011;Goyal et al., 2015]. Moreover, due to operation at elevated pressures, we account for real gas behavior using a compressibility factor correction, calculated with the Peng-Robinson (PR) equation of state.
Mass conservation in the tube expresses the fact that the reduction in flux along the fiber axis is equal to the flux passing through the membrane. The differential equations describing the average velocity of the gas phase, v g [m s −1 ] and the average CO 2 concentration, where r + 1 indicates the gas-membrane interface at the pore side; T g [K] is the temperature of the gas; Z [-], the compressibility factor; R = 8.3145 m 3 Pa mol −1 K −1 , the gas constant; and D CO 2 ,md m s −2 denotes the effective CO 2 diffusion coefficient in the dry part of the membrane, such that with D CO 2 ,g [m s −2 ], the diffusion coefficient of CO 2 in the gas phase; and , , the porosity and tortuosity of the membrane, respectively. Initial conditions at the inlet z = L of the fiber are given by where M in g [kg s −1 ] is the inlet mass flowrate of gas; y in CO 2 ,g [−], the inlet molar fraction of CO 2 ; and g [kg m −3 ], the density of the gas phase is the density of gas.

Liquid phase in shell, (r, z) ∈ [r 2 , r 3 ] × [0, L]
The liquid flow in the shell is laminar (Reynolds <10), and we assume here that the velocity profile is fully developed, following Happel's free surface model [Happel, 1959]. Moreover, we neglect axial diffusivity in the liquid phase (Péclet >10 7 ), and we assume that the liquid phase is incompressible and that the reaction between CO 2 and the solvent does not incur any change in volume.
The distribution of the CO 2 and solvent at the shell inlet are supposed to be uniform, with C in sol [mol m −3 ] the solvent concentration in the liquid feed; and f in CO 2 ,l [mol mol −1 ], the CO 2 loading in solvent. At the shell outlet, a no-dispersion condition is defined as Moreover, the condition of axisymmetric flow condition at r = r 3 gives while continuity of the concentrations and fluxes at the membrane surface, r = r 2 impose where the radial positions r + 2 and r − 2 correspond to either sides of the liquid-membrane interface; and D i,mw [m s −2 ] denotes the effective diffusion coefficient of a species i in the wetted part of the membrane, such that The membrane properties are considered to be uniform along the fiber axis, including pore size distribution, tortuosity, porosity, thickness, and hydrophobicity. We assume that mass transfer inside the membrane is driven by radial diffusion only, considering the gas and liquid phases to be stagnant therein, and that Henry's law is applicable at the gas-liquid interface. Furthermore, we neglect the dissolution of gas species other than CO 2 into the liquid phase, and we consider the amine solvent to be non-volatile. The partial differential equations describing the transport of CO 2 and solvent inside the membrane are given by The equilibrium condition at the gas-liquid interface, r = r w (z) inside the membrane is expressed as where r + w and r − w indicate either sides of the gas-liquid interface inside the membrane; H CO 2 [m 3 Pa mol −1 ] stands for Henry's constant for CO 2 in amine solution; and T l [K] is the temperature of the liquid phase. At this interface, the flux of solvent is equal to zero, while the flux of CO 2 is continuous, ∂C sol (r, z) ∂z At the membrane-gas interface, r = r 1 , the CO 2 concentration is continuous, and the CO 2 flux continuity is already expressed in Eq.
(12). The concentration and flux continuity conditions at the membrane-liquid interface, r = r 2 are given in Eqs. (22) and (23) above.

Experimental setups
The lab-scale module experiments were conducted with the binary feed gas mixtures of CH 4 /CO 2 and N 2 /CO 2 at 11 bar, while the pilot module was up-scaled by a factor of about 800 (i.e. membrane area) to be operated under industrially relevant operating conditions at 54 bar in a natural gas processing plant in Malaysia. In both cases, aqueous mixtures of methyldiethanolamine (MDEA) and piperazine (PZ) were used as the chemical solvent. The corresponding operating conditions are summarized in Table 1. The experimental set-ups for lab-and pilot-scale testing are depicted in Figs. 3 and 4, with further details given thereafter.

Lab-scale MBC module
Pressurized gas cylinders were used to supply mixtures of CH 4 /CO 2 or N 2 /CO 2 to the tube side of the lab-scale module, while the lean amine solvent was pumped through the shell side, in a counter-current and vertical configuration. The feed gas and solvent flow rates were controlled using a mass flow controller (MFC) and pump stroke, respectively. The compositions of the feed gas and treated gas were analyzed and recorded using gas chromatography (GC7900, Shanghai Techcomp Instruments Co., Ltd), after the system had reached steady state as indicated by a constant CO 2 composition in the outlet gas stream of the module. The treated gas from the MBC module was then depressurized and vented to a safe location. A transmembrane pressure of P TMPD = 30 kPa was maintained at any point along the fibers in order to prevent gas bubbling [Kang et al., 2017].

Pilot-scale MBC module
The CO 2 -rich natural gas was fed to the tube side of the MBC, while the lean amine was pressurized to ca. 54 bar and fed to the shell side, in a counter-current and vertical configuration. The feed gas and liquid flow rates were controlled using a mass flow controller and a pump stroke, respectively. The flowrate of the treated gas at the MBC outlet was measured with a mass flowmeter before sending it to the flare header. The enriched amine solvent collected from the MBC shell outlet was directed to a flash drum for degassing of any volatile and dissolved light hydrocarbons. The gas compositions in the MBC feed and outlet as well as the flash outlet were analyzed and recorded using gas chromatography (PGC1000 Gas Chromatograph, ABB) after the system had reached steady state. The flash liquid outlet stream was heated by cross exchange with hot lean solvent, and then fed to the solvent regenerator where it was stripped of acid gas by rising stream vapors generated by the reboiler. The acid gas leaving the top of the regenerator was cooled and sent to a reflux drum to separate the condensed water from the acid gas. The condensed water was then returned to the top of the regenerator, while the acid gas was sent to the flare header.

Membrane and module characteristics
The high pressure MBC modules were packed with hydrophobic, polytetrafluoroethylene (PTFE) hollow fiber. Table 2 reports the main characteristics of these membranes, alongside geometrical properties of the MBC modules used for lab-and pilot-scale testing. Following Iversen et al. (1997), a first approximation of the membrane tortuosity in Table 2 was obtained as In order to determine the wetting ratio in Eq. (9) needed for simulating the MBC model, pore-size distribution (PSD) data from the manufacturer were used to fit the following log-normal distribution [Zydney et al., 1994;Lu et al., 2008] f (ı) = 1 2 ln(1 + 2 )ı where ı and stand for the mean pore radius and standard deviation, respectively. The resulting least-squares fit on the left plot in Fig. 5 shows an excellent agreement with the data, and the estimated values for ı and can be found in Table 2.

Thermo-physical, transport, and reaction kinetic data
Temperature-dependent expressions for (i) the macroscopic reaction rates of CO 2 with MDEA and PZ, (ii) Henry's constants for CO 2 in the amine solution, and (iii) the diffusivity coefficients of the various species in gas or liquid mixtures are reported in Appendix A for completeness. Values for the other thermo-physical and transport parameters were obtained by interfacing gPROMS with the property packages 'Advanced Peng Robinson' and 'UNIQUAC-RK'.

Numerical simulation
The mixed set of algebraic, ordinary differential and partial differential equations was implemented in the gPROMS modeling language [Oh and Pantelides, 1996], using ModelBuilder v5.0. A second-order centered finite difference scheme was used to discretize the differential equations, after a rescaling  of the radial dimension in the partial differential equations in order for the membrane dry and wet spatial subdomains to be rectangular. A uniform mesh grid consisting of 70 elements was chosen to perform the simulations herein, which provides solutions within <1% of finer discretizations, while retaining computational tractability. With this discretization, a steady-state simulation takes a few minutes to complete on a desktop computer running Windows 7 with Intel ® CoreTM i7-4790 CPU at 3.60 GHz and 32GB of RAM.

Lab-scale MBC module
A comparison between measured and predicted CO 2 removals in the lab-scale MBC is presented in Fig. 6, in terms of the CO 2 absorption flux, ˚ [mol m −2 s −1 ] computed as The experimental data are for several flow rates of the two gas mixtures N 2 /CO 2 and CH 4 /CO 2 (see Table 1), and also correspond to different flow rates of the amine solvent. Overall, the predictions are found to be in excellent agreement with the measurements, showing errors lower than 5%. Notice that such an agreement is quite remarkable given that none of the model parameters are adjusted here, thereby providing a first validation of the main modeling assumptions.
The MBC model correctly predicts the increase in CO 2 absorption flux with a larger inlet gas flowrate, driven by a larger amount of CO 2 in the gas feed. As far as membrane wetting is concerned, non-wetted operation is predicted for all three N 2 /CO 2 experiments in Fig. 6A-C, whereas a wetting ratio in the range 5-11% is predicted for the CH 4 /CO 2 experiment. This difference between both gas mixtures can be attributed to a higher contact angle in the experiment with N 2 /CO 2 ( = 95.3 • ) compared with CH 4 /CO 2 ( = 92.5 • ).

Pilot-scale MBC module
A comparison between measured and predicted CO 2 removals in the pilot-scale MBC module is presented in Fig. 7, also in terms of the CO 2 absorption flux ˚ defined in Eq. (34). The experimental data correspond to different flow rates of the amine solvent and of the NG mixture (see Table 1). The MBC model predictions are in good agreement with the measurements, albeit showing larger and more systematic errors, up to 10% overestimation, as compared with the lab-scale results in Fig. 6. A possible explanation for such systematic offsets between the predictions and measurements, could be due to the model currently neglecting the heat generated from the reaction between CO 2 and the amines, which causes the liquid temperature to rise. This scenario is analyzed in more details further on. Another contribution to this offset could be underestimating the mass-transfer resistance near the membrane-liquid interface, especially at lower liquid flow rates, e.g. due to the formation of a boundary film. The model correctly captures the improvement in CO 2 absorption flux on increasing the solvent flowrate in Fig. 7A, and it also predicts a corresponding small increase in the membrane wetting. This extra wetting is due to a higher pressure drop in the shell, and therefore a higher transmembrane pressure, as described in Eqs. (4) and (5). On balance however, the effect of a larger wetting remain small in comparison with the effect of a leaner amine (and hence a larger concentration gradient) in terms of the overall CO 2 mass transfer. These results are also a confirmation that the MBC perfor-mance is, to a large extent, dominated by the physicochemical processes taking place in the liquid phase, a behavior that has been reported in the literature previously [Boributh et al., 2011;Goyal et al., 2015].
The model also correctly captures the increase in CO 2 absorption flux for increasing inlet gas flowrates in Fig. 7B, as driven by a larger amount of CO 2 in the gas feed. Also note that, since the pressure drops in the tubes is negligible, see Eq.
(4), increasing the inlet gas flowrate has essentially no effect on the wetting ratio.

Temperature correction
This subsection quantifies the effect of a rise in the solvent temperature due to the exothermic reaction between CO 2 and the amines. An increase in the solvent temperature leads to a reduction in surface tension, therefore causing extra wetting of the membrane according to Eq. (3) and a reduction in the CO 2 flux through the membrane. The temperature difference, T l [K] between the lean amine fed to the MBC module and the enriched amine outlet can be estimated by means of a lumped energy balance under adiabatic conditions [Li et al., 2017], where the term A m [mol s −1 ] describes the overall rate of CO 2 removal; and the specific heat capacity of the amine solvent and the enthalpy of reaction are set to C p = 3600 J kg −1 K −1 [Weiland et al., 1997] and H r = 60 000 J mol −1 [Kabadi, 2007], respectively, in a first approximation. Based on Eq. (35) and the data in Table 1, the solvent temperature in the pilot-scale MBC module is predicted to increase by 10-14 K. This rise, in turn, corresponds to a reduction in the solvent surface tension from 0.046 to about 0.044 N m −1 . The results of the MBC model simulations with the corrected surface tension values are presented in Fig. 8. The predicted CO 2 absorption fluxes are now in close agreement with the measured efficiencies, after an increase in the predicted membrane wetting. By contrast, the solvent temperature in the lab-scale MBC module is predicted to increase by a few degrees only, which does not have as large an impact on the com-puted fluxes in Fig. 6. The larger temperature rise in the pilot-scale module is due to the liquid-to-CO 2 -gas-absorbed (L/G) ratio therein being only one-third of the L/G ratio in the lab-scale module. A low L/G ratio as in the pilot-scale module is comparable to conventional packed columns, and is typically preferred in practice since it reduces the energy needed for solvent regeneration. In sum, this cursory analysis confirms that one should not neglect the temperature rise in the solvent to empower more accurate performance predictions in pilot-scale (or larger-scale) MBC modules. Clearly, this calls for the development of a detailed energy balance alongside the mass balance equations in the MBC model, as part of future work.

Model-based analysis
This section of the paper presents a model-based analysis of MBC for natural gas sweetening, with a focus on high-pressure operation and membrane wetting. The analysis is conducted for the pilot-scale MBC module, and applies the temperature correction discussed earlier.

Is high-pressure MBC operation advantageous for CO 2 removal?
The benefits of high-pressure MBC operation in natural sweeting applications, where it can lead to significant savings with regards to compression costs, are well established [Mansourizadeh and Ismail, 2009;Zhang and Wang, 2013]. This subsection investigates whether or not high-pressure operation also presents advantages in terms of the MBC performance for CO 2 removal. Physically, the diffusivity of CO 2 in NG is known to decrease at higher pressure, thus increasing the overall mass transfer resistance; whereas an increase in the CO 2 partial pressure will enhance the mass transfer, and so will a lower gas velocity by increasing the residence time of the gas. Predicting the net effect of raising the operating pressure on the CO 2 absorption flux and the CO 2 removal efficiency is therefore challenging.
The graph in Fig. 9A shows the effect of varying the operating pressure in the (vertical) pilot-scale MBC on the CO 2 removal efficiency, Á [-] given by The MBC model predicts an improvement in CO 2 removal efficiency from about 60% to 80% in increasing both the gas and liquid pressures from 1400 kPa to 5400 kPa (maintaining a transmembrane pressure of P TMPD = 30 kPa along the fibers in order to prevent gas bubbling). This corresponds to an improvement in CO 2 absorption flux ˚ by nearly 40%. On balance, the increase in mass transfer driving force and the longer residence time thus clearly dominate over the lower diffusivity in terms of CO 2 absorption and removal efficiency. These results are consistent with those previously reported by Faiz and Al-Marzouqi (2010), although their model could not predict the effect of varying operating conditions on membrane wetting or the evolution of membrane wetting along a hollow fiber. In particular, the model only predicts small variations in the wetting ratio in operating the MBC at different pressures, and therefore the performance improvement at high pressure is not associated with a reduction in membrane wetting. From a process engineering standpoint, higher efficiency in terms of CO 2 removal in high-pressure MBC compared with atmospheric operation could enable a reduction of the absorber volume (i.e. lower membrane area and module size) and/or a lower solvent flowrate (i.e. lower solvent regeneration and pumping energy) in order to meet a certain CO 2 purity specification.

How large is the effect of module orientation in high-pressure MBC?
The concentration profiles of CO 2 and amines around a single fiber are shown in Fig. 10A, for the pilot-scale module in vertical orientation, operated under 5400 kPa and with other conditions as given in Table 1. The three graphs show that the wetting ratio (or, equivalently, the wetting radius r w ) varies significantly along the fiber axis, from about 6% at the gas inlet to over 64% at the gas outlet. This variation is mainly due to the static head in the liquid phase, while the pressure drops remain small in comparison.
On the left graph in Fig. 10A, the CO 2 concentration in the tube is the highest at the gas inlet, z = 2 m, and decreases along the fiber. In this vertical configuration, the CO 2 removal efficiency is rather low, around Á ≈ 79%, due to significant wetting of the membrane. Recall also that the gas phase is uniform in each tube cross-section per the modeling assumptions in Section 2.2. In the dry part of the membrane, the CO 2 concentration is decreasing slightly in the radial direction due to diffusive transfer limitation, and then shows a discontinu-ity at gas-liquid interface, with a near-zero CO 2 concentration across the liquid phase.
The center and right graphs in Fig. 10A show that the MDEA and PZ amine concentrations in the shell are the highest at the liquid inlet, z = 0 m, and decrease along the fiber axis due to the reactions with CO 2 . It is noticeable that the PZ concentration decreases faster than the MDEA concentration, due to PZ having a higher reactivity with CO 2 than MDEA. These plots also depict steep gradients in MDEA and PZ concentrations across the wetted-part of the membrane, as well as some variations across the liquid phase, which justify the use of 2-d modeling to describe radial diffusive mass transfer in the membrane and shell sections.
For comparison, the concentration profiles shown in Fig. 10B are for the same pilot-scale module, now in horizontal orientation. The wetting ratio is predicted to be much smaller and about constant along the fiber length, around 6%, due to the transmembrane pressure difference no longer being subject to the liquid static head; see Eq. (4). Consequently, the CO 2 removal efficiency in horizontal orientation improves enormously, reaching 99.99% (residual concentration of C CO 2 (0) ≈ 1.5 × 10 −2 mol m −3 ), as compared with a mere 79% removal in vertical orientation. Also notice that a majority of the PZ amine is depleted at the horizontal module outlet. By contrast, the performance of the lab-scale module, whose length is much shorter than the pilot-scale one (see Table 2), does not improve significantly in changing the orientation to horizontal (results not shown).
Overall, this analysis suggests that, for the PTFE hollowfiber membrane at hand, the vertical mode of operation might not be viable in pilot-scale and larger-scale applications where the fibers are several meters long. For such tall modules, the effect of liquid static head could result in the bottom section of the membrane being fully wetted, thereby reducing the CO 2 absorption flux dramatically, or even making it infeasible to reach CO 2 purities down to the ppm level. The vertical mode of operation is nonetheless advantageous in practice since it has a much lower physical footprint than horizontal operation, especially on an offshore platform [Quek et al., 2015]. As part of future work, it will be interesting, to use the MBC model for targeting improvements in the membrane material, amine solvent and operating conditions all together, in order for the vertical mode of operation to become viable in fullscale industrial applications. For instance, the graph in Fig. 9B reports predictions of the CO 2 removal efficiency for (hypothetical) membranes having contact angles with the amine solvent in the range between 91.5 • and 94.0 • . The efficiency increases steeply over this range, as a driven by a large reduction in membrane wetting. Moreover, the MBC model predicts that CO 2 removal efficiencies as large as those obtained in horizontal orientation could be obtained in the vertical pilot-scale module by using a PTFE membranes with higher hydrophobicity/contact angle.

Conclusions and future research directions
This paper has developed a mathematical model of highpressure MBC using chemical solvents, which describes the effect of membrane pore-size distribution and operating conditions on membrane wetting and CO 2 absorption. A verification of the model has been conducted for both lab-scale and pilot-scale MBC modules, showing a close agreement of the predictions with measured CO 2 absorption flux at various gas and liquid flowrate, subject to a temperature correction to account for the heat of reaction in the liquid phase. Next, a model-based analysis of MBC for natural gas sweetening has been presented, with a focus on high-pressure operation and the effect of membrane wetting. The results confirm the advantages of a pressurized operation in terms of CO 2 removal efficiency. Moreover, a comparison between vertical and horizontal modes of operation has shown that the CO 2 removal efficiency in the latter can be vastly superior as it is not subject to the liquid static head. A vertical mode of operation could nonetheless become competitive if membrane materials with improved hydrophobicity were used.
A natural continuation of this work entails the application of systematic, model-based optimization methods to improve the design and operation of full-scale MBC modules. In partic-ular, a comprehensive techno-economic analysis should also include the solvent regeneration unit in the assessment. In addition to optimizing such decisions as the gas and liquid flowrates, operating pressure and temperature, module length and specific area, one could consider optimal solvent selection as well as the optimal arrangement of MBC modules into multistage cascades. Finally, one could also envisage using the MBC model to guide the development of improved membrane materials. The diffusion coefficient of N 2 O in liquid amine solution is estimated using the modified Stokes-Einstein relation [Mandal et al., 2003;Wang et al., 2013]  where the diffusivities of MDEA and PZ in water, and the liquid and water dynamic viscosities are obtained from the package 'UNIQUAC-RK'. The diffusivity of CO 2 in NG was estimated from empirical correlations based on kinetic gas theory [Cussler, 2009;Bird et al., 1960] D CO 2 ,NG = T 1.75 (1/M CO 2 + 1/M NG ) 1/2 10 5 P V CO 2 1/3 + V NG 1/3 2 (A.17) where M CO 2 and M NG stand for the molecular weights of CO 2 and NG; and the summations of atomic diffusion volumes for the species of the CO 2 -NG gas mixture are taken as V CO 2 = 26.9 and V NG = 24.42. Lastly, the diffusivity of CO 2 in N 2 gas was obtained from the package 'Advanced Peng-Robinson'.
The numerical values are reported in Table A1 correspond to the temperature and pressure conditions in the lab-and pilot-scale MBC modules (see Table 1).