Mathematical modeling of CO2 separation from gaseous-mixture using a Hollow-Fiber Membrane Module: Physical mechanism and influence of partial-wetting

Journal of Membrane Science 474 (2015) 64–82

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Mathematical modeling of CO2 separation from gaseous-mixture using a Hollow-Fiber Membrane Module: Physical mechanism and influence of partial-wetting Nikhil Goyal, Shishir Suman, S.K. Gupta n Department of Chemical Engineering, Indian Institute of Technology Delhi, Hauz Khas, 110016 New Delhi, India

art ic l e i nf o

a b s t r a c t

Article history: Received 21 June 2014 Received in revised form 17 September 2014 Accepted 18 September 2014 Available online 28 September 2014

The present study describes a steady-state phenomenological model for CO2 separation via reactive absorption into aqueous Diethanolamine (DEA) solution using a micro-porous Poly-propylene (PP) Hollow-Fiber Membrane Module (HFMM). The developed model is based on the fundamental mechanisms of molecular diffusion, bulk convection and liquid-phase chemical reaction, and simultaneously accounts for the consequences of ‘partial-wetting’ phenomenon. Furthermore, a physically-consistent wetting mechanism has been formulated assuming that the membrane pores may be modeled as a bundle of straight cylindrical capillaries with distinct radii (characterized by the membrane pore-size distribution) and equal lengths, while keeping in mind the various pore-scale micro-physical phenomena. Under the simplifying parameterizations of the Finite-Volume Method (FVM), the source-code for discretized equations was compiled and implemented using C þ þ Language for a co-current module operation with aqueous DEA solution flowing inside the fiber-lumen and CO2–N2 gaseous mixture passing through the shell-side. A Benchmarking Analysis revealed an excellent agreement between the model predictions and the experimental data reported in open-literature, thereby validating the current model formulation, and rendering it fundamentally relevant with respect to the wetting-phenomenon. In addition, the module performance in terms of CO2 flux, Overall Mass-Transfer Coefficient (MTC), and Removal-Efficiency, has been systematically analyzed pertaining to the physical influence of other operating variables such as absorbent concentration, hydrodynamics, pressure, temperature, and membrane characteristics. From a modeling standpoint, it may be concluded that the present model successfully captures various observations vis-à-vis the process of CO2 separation using micro-porous HFMMs, reported previously in the literature. Moreover, for a given gas-phase hydrodynamics, the current set of results suggest the existence of a unique liquid-phase hydrodynamic regime, bounded by a minimum and a maximum permissible pressure, under which the module can be effectively operated without any dispersive losses. Besides, the currently developed model has been demonstrated to explain the reduction in CO2 flux over time by allowing for morphological changes, including an enlargement in the average pore-size and a broadening of the pore-size distribution. & 2014 Published by Elsevier B.V.

Keywords: CO2 separation Mathematical modeling Partial-wetting mechanism Non-dispersive hydrodynamics Morphological changes

1. Introduction The global carbon emissions have increased considerably in the past three decades, mainly due to our dependence on fossil-fuels to meet our energy requirements. Therefore, novel separation techniques have attracted a lot of attention recently to rein in

Abbreviations: HFMM, Hollow-Fiber Membrane Module; FVM, Finite-Volume Method; MTC, Mass Transfer Coefficient; PP, Polypropylene; TMPD, Transmembrane Pressure Difference n Corresponding author. Tel.: þ 91 11 2659 1023; fax: þ91 11 2658 1120. E-mail address: [email protected] (S.K. Gupta). http://dx.doi.org/10.1016/j.memsci.2014.09.036 0376-7388/& 2014 Published by Elsevier B.V.

CO2, the major greenhouse gas responsible for global warming. Amongst various alternatives, the use of Hollow Fiber Membrane Modules (HFMMs) for chemical absorption into aqueous alkanolamines, represents a high-efficiency method to capture CO2 [1]. Micro-porous HFMMs provide a higher surface area per unit volume for continuous gas–liquid contact as compared to packed beds and correspondingly high mass-transfer rates. With this behavior staying unviolated in a wide range of flow rates, the common problems of loading and flooding with packed towers can be avoided by the use of HFMMs for CO2 absorption [2]. However, the phenomenon of gradual membrane wetting by absorbent over longer time-scales increases the overall mass-transfer resistance

N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

and needs to be addressed during process intensification studies [3]. Therefore, the choice of membrane material is critical to the efficiency of this process, and preference must be given to hydrophobic polymers like poly-vinylidene-fluoride (PVDF), polypropylene (PP), poly-tetrafluoro-ethylene (PTFE) and poly-sulfone (PS) in order to avoid membrane wetting. Numerous modeling and experimental investigations [2–26] have been conducted previously to understand CO2 separation using HFMMs. The application of PS and PP HFMMs for CO2 separation was experimentally investigated by Kreulen et al. [4,5]. For the case of physical absorption in water, the authors [4] discussed advantages of HFMMs over traditional bubblecolumn contactors, the effect of mal-distribution of phases and correlated the results using the heat transfer analog of Graetz– Leveque solution. In a complementary study of chemical absorption in aqueous NaOH, the authors discussed the experimental results in parallel with a mathematical model formulated under the assumption of zero-wetting [5]. Likewise, Yan et al. [6] experimentally demonstrated the use of aqueous Potassium Glycinate (PG) as an absorbent for CO2 separation using non-wetted PP HFMMs. The process of CO2 absorption in aqueous Potassium Glycinate (PG) using a PTFE HFMM has been further explored by Eslami et al. [7] through a mathematical model developed using COMSOL software package with non-wetting physics. Similarly, Zhang et al. [8] deployed MATLAB as a mathematical tool to study CO2 absorption in water and aqueous Diethanolamine (DEA) using PP HFMMs under non-wetting conditions. At this point, one should note that although these studies [4–8] ignored the repercussions of wetting phenomenon, ‘partial-wetting’ of hydrophobic HFMMs cannot be avoided when the liquid-phase pressure is higher than its gas-phase counterpart, a primary condition that must be met in order to avoid any hydrodynamic inter-dispersion of the two phases [9,10]. Over the years, several authors [11–20] have reported the occurrence of wetting during experimental studies with HFMMs. For instance: refer to the work of Kreulen et al. [11], who determined the mass-transfer resistance of both wetted and non-wetted flat microporous membranes, and attributed the corresponding inconsistencies to the phenomenon of ‘partialwetting’. Analogously, Rangwala [12] studied CO2 absorption in water, DEA and NaOH using PP HFMMs, and ascribed the discrepancy between the theoretically and experimentally determined MTC values to be an implicit manifestation of ‘partialwetting’. Similar observations of reduction in CO2 flux owing to the ‘partial-wetting’ of HFMMs have been discussed by other authors [13–20] as well. A particularly interesting experimental study of CO2 removal from natural gas at high gas-phase pressures using aqueous Mono-ethanolamine (MEA) was conducted by Faiz and Al-Marzouqi [20] who reported a good agreement with the theoretical predictions corresponding to a 0.5% ‘pseudo-wetting’. Although such a small extent of membrane-wetting makes sense, especially with the gas-phase pressure reaching as high as 50 bars, the authors’ [20] efforts may possibly be clouded by the complexity of hydrodynamics at such high operating pressures i.e. the overlooked dispersive interactions between the two phases. To appreciate this, one may refer to the work of Kreulen et al. [5], who observed the formation of small gas-bubbles in the liquid-phase on account of an inevitable rise in the gas-phase pressure with increasing velocity. In light of the aforementioned arguments, it becomes essential to study the potential implications of wetting phenomenon on the module performance under diverse operating conditions. Till date, only a few theoretical and experimental attempts have been made in this regard. Vis-à-vis the assumptions of ‘no-wetting’ and ‘complete-wetting’, a mathematical model has been discussed and experimentally validated in the work of Karoor and Sirkar [21] for

65

gas-absorption in water using microporous HFMMs. As to the phenomenon of ‘partial-wetting’, practically all prior studies [10,16,20,22,23] available under open-literature have suggested the use of a wetting fraction, representative of the extent to which the membrane may be wetted, as a fitting parameter to match the theoretical predictions to experimental results. Distinct from this seemingly dominant perspective, Malek et al. [24] described a mathematical model to analyze dissolved O2 removal from water using microporous HFMMs under a loose assumption of ‘partialwetting’. Loose in the sense that for points along the fiber length where the transmembrane pressure difference (TMPD) exceeded the critical wetting-pressure (determined by the Laplace Equation), the authors [24] assumed ‘complete-wetting’ for simulation purposes and a ‘no-wetting’ mode otherwise. Furthermore, the authors [24] assumed a constant value for the critical wetting-pressure, based on the average pore-size, similar to the work of Rangwala [12], which may not be valid for membranes with a non-uniform pore-size distribution. While distinctively notable attempts [25,26] have been made previously to relax these assumptions, these are restrictive at the same time with respect to a few irregularities. For instance: Lu et al. [25] made efforts to formulate a wetting mechanism by accounting for the influence of transmembrane pressure difference (TMPD), capillary pressure and a log-normal pore-size distribution. However, the expression for the wetting fraction reported by Lu et al. [25] seems mathematically inconsistent vis-à-vis a normalized poresize distribution, further details of which have been explained later in this study. Their work was succeeded by the work of Boributh et al. [26], who adapted Lu et al.’s mechanism for further mathematical analysis. Therefore, it is quite evident that despite the several prior attempts, large uncertainties remain in the current state of understanding on wetting phenomenon in HFMMs. Additionally, a combination of multifaceted pore-scale micro-physical phenomena may render the gas–liquid interface immobilized inside membrane pores, thereby causing a ‘partial-wetting’ of the HFMM. Inclusive attempts, towards the quantification of mass-transfer resistance, prevailing in the membrane under a given set of operating conditions, have not been reported so far in the openliterature. Likewise, the phenomenon of ‘partial-wetting’ has not been studied in detail, compared to the customary assumptions of ‘no-wetting’ and ‘complete-wetting’ reported in earlier studies. The present work is an effort en-route to the mechanistic modeling of reactive CO2 absorption using aqueous DEA solution in a PP HFMM. A wetting-mechanism, formulated through a physical coupling of the log-normal pore-size distribution along-with the well-established capillary bundle representation of membrane pores, has been described for incorporation into the developed mathematical model. Furthermore, a systematic analysis of the relative importance of different parameters in the determining the module performance: Flux, Overall MTC and Removal Efficiency, has been discussed as a part of a larger effort to quantify the physical effects of hydrodynamics, temperature, pressure, and membrane characteristics.

2. Mathematical model The schematic of a hollow-fiber with a co-current flow of liquid inside the fiber-lumen and gas through the shell-side is displayed in Fig. 1. The characteristic dimensions of a commercially available HFMM: MiniModules 1  5.5 Liqui-Cel Membrane Contactor, procured from Membrana (Celgard LLC – Charlotte, USA), described in Table 1, have been used in the current study. The independent formulation of transport model on a single fiber basis requires the shell-side dimensions to be explicitly defined. This is accomplished by use of Happel’s Free-Surface Model [27] to

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estimate the outer radius (Re ) of fluid envelope surrounding the fiber using Eq. (1), assuming no interactions amongst different fibers. Ro Rm Re ¼ pffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffi nF 1ϕ

ð1Þ

The shell-side N2 concentration has been assumed constant because of its negligible solubility in aqueous DEA solution. Hence, it has been excluded from the subsequent model development. The mechanism for CO2 separation, via absorption into aqueous DEA solution, involves four consecutive steps: (1) diffusion of CO2 molecules from shell-side to the gas–liquid interface residing in the membrane; (2) dissolution followed by liquid-phase chemical reaction inside membrane pores; (3) diffusion into the liquidphase bulk (fiber-lumen) across the inner surface with simultaneous chemical reaction. Accordingly, the coupled transport through liquid-phase, gas-phase and the microporous membrane has been modeled subsequently.

2.1. Reaction kinetics

Fig. 1. Schematic of a hollow-fiber.

Table 1 Characteristic dimensions and properties of MiniModules 1  5.5 membrane contactor. Parameter

Value

Inner radius of fiber (Ri )

110  10  6 m 6

Outer radius of fiber (Ro )

150  10

Fiber length (LF ) Number of fibers in HFMM (nF ) Module diameter (Rm ) Membrane porosity (ϵm ) Membrane tortuosity (τm )

120  10  3 m 2300

Minimum pore-size (r min p )

1  10  12 m

m

127  10  4 m 0:40 3:5

Average pore-size (r avg p )

3  10  8 m

Maximum pore-size (r max p ) Geometric standard deviation (σ) Packing fraction (ϕ) Active surface area

1  10  5 m 1:2 0:679 18  10  2 m2

The reaction kinetics of CO2 absorption in DEA has been a subject of numerous investigations [28–33] based on different experimental techniques, as summarized briefly in Table 2. Due to wide variations in the findings of these studies, a considerable research activity has been focused on unraveling the reaction mechanism between CO2 and DEA in the last three decades or so. Of particular interest here is the widely cited Zwitterion Mechanism proposed by Danckwerts [34], which has been generally accepted to be the most accurate, and may be described through the following equations: CO2 þ R2 NH-R2 NCOOH

ð2Þ

R2 NCOOH þ R2 NH-R2 NCOO  þ R2 NH2þ

ð3Þ

Some studies have recommended the incorporation of an additional term in the final rate expression, describing the effect of hydroxyl ions on the de-protonation of zwitterion in aqueous phase [31]. However, the contribution of hydroxyl ions to the zwitterion’s de-protonation is extremely difficult to measure due to experimental inaccuracies, and its effect may be neglected when the experimental conversion exceeds 1% [32]. With these considerations, and the principle assumption of a pseudo steady state for the zwitterion formation, the following rate expression may be easily derived:

Table 2 Literature data on CO2–DEA reaction kinetics. Temperature range (1C)

  1 DEA concentration range mol l

5.8–40.3

0.174–0.719

25

0.460–2.880

11

0.500–2.000

25

0.393–2.308

25

0.393–2.308

25 Results from [31] re-evaluated:

considering the effect of OH  ions,

1

sec  1 Þ

  2 exp 12:41  2775 TðKÞ cCO2 cDEA cCO2 cDEA ð1=1410Þ þ ð1=1200cDEA Þ cCO2 cDEA ð1=890Þ þ ð1=560cDEA Þ cCO2 cDEA ð1=5800Þ þ ð1=5:34cH2 O þ 7:05  104 cOH  þ 228cDEA Þ cCO2 cDEA ð1=7300Þ þ ð1=3:7cH2 O þ 8:52  104 cOH  þ 479cDEA Þ cCO2 cDEA ð1=5790Þ þ ð1=3:61cH2 O þ 534cDEA Þ cCO2 cDEA ð1=3240Þ þ ð1=1:71cH2 O þ 707cDEA Þ

0.086–8.5 a

Fitted rate expressionðmol l

b

neglecting the effect of OH  ions

Deployed routine

Source

Rapid mixing method

[28]

Gas absorption

[29]

Stirred cell

[30]

Gas absorption

[31]

Re-evaluationa [32] Re-evaluationb Gas absorption

[33]

N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

boundary conditions is applicable for the solution of aforesaid governing equations:

Table 3 Kinetic parameters (Eq. (4)). Parameter

Value

k1

3:24 m3 mol

k2

1:71  10  6 m6 mol

2

s1

k3

7:07  10  4 m6 mol

2

s1

a) Inlet Boundary Condition: 1

s1

z ¼ 0; 8 r A ½0; Ri  :

r CO2

ð4Þ

In Eq. (4), ξ represents a dummy variable, which can be set to either 0 or 1 to study the case of CO2 absorption with or without chemical reaction respectively. For the current study, the kinetic parameters (k1 , k2 and k3 ) reported in Table 3, have been adapted from the study of Versteeg and Oyevaar [33] as they are applicable in a wide range of DEA concentration. Besides, the dependence of water concentration (cH2 O ) on the liquid-phase DEA concentration at fiber inlet (cin BL ) and the density of aqueous DEA solution (ρL ) may be written as follows: c H2 O ¼

  MM DEA cin ρL BL 1 MM H2 O ρL

cAL ¼ cin AL

and

cBL ¼ cin BL

ð10aÞ

b) Axisymmetric Flow: r ¼ 0; 8 z A ½0; LF  :

 ξc c   CO2 DEA  ¼ r DEA ¼  1=k1 þ 1=k2 cH2 O þ k3 cDEA

67

ð5Þ

∂cAL ¼0 ∂r

and

∂cBL ¼0 ∂r

ð10bÞ

∂cBL ¼0 ∂z

ð10cÞ

c) No Dispersion at Fiber-Outlet: z ¼ LF ; 8 r A ½0; Ri  :

∂cAL ¼0 ∂z

and

d) Continuity of Concentration and Flux at Inner Surface: r ¼ Ri ; 8 z A ½0; LF  :

cAL ¼ cAiLM and cBL ¼ cBiLM     ∂cAL ∂cALM ¼ DALM and r ¼ Ri ; 8 z A ½0; LF  : DAL ∂r ∂r     ∂cBL ∂cBLM ¼ DBLM ð10dÞ DBL ∂r ∂r

2.2. Liquid-phase mass transfer Although a HFMM can handle high flow rates, the actual liquid flow rate for a single hollow-fiber corresponds to a low (  100) Reynolds Number. As a result, the liquid-phase hydrodynamics can be safely assumed to be characterized by a fully-developed Newtonian Laminar-Flow as follows:  2 ! r vL ðr Þ ¼ 2vavg 1  ð6Þ L Ri Since the flow-average liquid-phase velocity at fiber inlet depends on the cumulative pressure drop along the module length, the pressure at a given axial distance may be estimated using the Hagen-Poiseuille Equation as illustrated by Eqs. (7) and (8).   out R2i P in L  PL avg ð7Þ vL ¼ 8μL LF P L ¼ P in L 

8μL vavg L z R2i

¼ P out L þ

8μL vavg L ðLF  zÞ R2i

2.3. Gas-phase mass transfer Considering DEA to be non-volatile, the gas-phase transport of CO2 on the module shell-side is governed by convection (axial) and molecular diffusion (axial þradial). However, the gas-phase diffusivity of CO2 is 3 to 4 orders of magnitude higher than its liquid-phase counterpart and the viscosity of CO2 –N2 mixtures is invariably low ( 10  5 Pa s) [35]. Therefore, the gasphase hydrodynamics may be simply denoted using an assumption of a Plug-Flow with negligible drop in operating pressure along the fiber length. Clearly, this may seem like a gross oversimplification, but a number of previous studies [16,22,23] have utilized and validated the plug-flow assumption for CO2 absorption modeling. Based on the development presented by Zhang et al. [16], the gas-phase hydrodynamics

ð8Þ

The steady-state transport of CO2 and DEA is governed by molecular diffusion in radial and axial directions, convection in axial direction and chemical reaction in the liquid-phase. It should be noted that although convection is the predominant transport mechanism axially, molecular diffusion may also become important at low liquid-phase velocities. With the subscripts A and B denoting CO2 and DEA respectively, the differential form of Continuity Equation (Eqs. (9a) and (9b)) involving these mechanisms is as follows:  2  ∂cAL ∂ cAL ∂2 cAL 1 ∂cAL vL  DAL ¼ r AL þ þ ð9aÞ r ∂r ∂z ∂z2 ∂r 2 vL

 2  ∂cBL ∂ cBL ∂2 cBL 1 ∂cBL DBL ¼ r BL þ þ r ∂r ∂z ∂z2 ∂r 2

ð9bÞ

These equations need to be solved simultaneously because the chemical reaction terms (r AL and r BL ) are functions of both CO2 and DEA concentrations. The following set (Eqs. (10a) – (10d)) of

Fig. 2. Axis-symmetrically discretized hollow-fiber cross-section.

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N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

relevant to the present study is as follows:       DAG dcAG D dc DAG dcAG r^ þ cAG vG  AG AG z^   r^ þ ðcAG vG Þz^ jCO2 ¼  dr dz dr ð11Þ  

dnAG ¼ π R2e  R2o ðcAG vG Þz  ðcAG vG Þz þ dz

ð12Þ

The axial-diffusion component of the flux may be dropped in Eq. (11) as compared to the bulk convection term, and the amount of CO2 transferred from gas-phase (dnAG ) may be estimated using Eq. (12) from the absolute change in gas-phase concentration and velocity across the discrete segments shown in Fig. 2. Furthermore, the Ideal-Gas Law and the assumption of a uniform operatingpressure may be applied to relate the change in velocity to the gasphase CO2 concentration as shown below: dnAG ¼

P G dvG AS RT

out P G ffi P in G ffi PG

ð13Þ ð14Þ

It should be noted that although the module operation at a non-zero gas-phase velocity inherently implies a finite change in the gas-phase pressure across the fiber length, Eq. (14) still holds as the molecular viscosity of gaseous-mixture is low enough to ignore the axial pressure-drop for computational simplicity. 2.4. Mass-transfer across membrane 2.4.1. Wetting mechanism As already mentioned, the liquid-phase pressure has to be kept higher than the gas-phase pressure [10] at all points along the fiberlength to provide effective gas–liquid contact without any hydrodynamic dispersion of the two phases [9]. Hence, the HFMMs deployed for this purpose are generally hydrophobic [4,21] and do not allow the penetration of liquid absorbent until the transmembrane pressure difference exceeds the critical wetting pressure [1], the magnitude of which is determined by the greatest pore size in the membrane [36]. As the transmembrane pressure difference is increased beyond this critical magnitude, either via an increase in liquid-phase pressure or a reduction in gas-phase pressure, the membrane pores begin to wet spontaneously starting from the largest pore-size to the smallest one [25]. Nevertheless, this wetting is rarely complete, and the liquid is able to penetrate only a fraction of the entire pore-length due to viscous dissipation. Hence, the extent of wetting is usually studied [26] in terms of a wetting ratio (x, Fig. 2) defined in the subsequent manner: x¼

Vw Vp

ð15Þ

For the present study, V w and V p are symbolic of the porous-volume occupied by liquid absorbent and the total porous-volume respectively.

In principle, the quantity V w depends on the membrane’s pore-size distribution, transmembrane pressure difference and the equilibrium configuration of liquid inside the pore. The transmembrane pressure difference along the fiber-length may be computed through Eq. (16), resulting from Eqs. (8) and (14) described earlier. in ΔP TMPD ðzÞ ¼ P L  P G ¼ P in L  PG 

8μL vavg L z

ð16Þ

R2i

To understand the wetting phenomenon on pore-scale, the mechanical equilibrium of forces associated with capillary-pressure and operating-pressure fields at the gas–liquid interface is depicted in Fig. 3. It is apparent that, for θ 4 901, the transmembrane pressure difference (ΔP TMPD or P L  P G ) provides the necessary driving force for the penetration of liquid inside a pore against the capillary pressure (P C ). Moreover, in this case, the gas–liquid interface is concave with respect to the liquid-phase. Conversely, with θ o 901, the force due to capillary pressure is reversed, and wetting would occur spontaneously even with no external driving force (ΔP TMPD ¼ 0). Additionally, for the membrane pores located at a given distance, a useful insight into the wetting behavior is provided by the rearranged form of Laplace–Young Equation: r cp ¼

 2γ L cos θ ΔP TMPD ðzÞ

ð17Þ

which states that for a given transmembrane pressure difference (ΔP TMPD ), only the pores larger than a critical-size (r cp ) can undergo wetting. Unfortunately, Eq. (17) does not provide any insight into the extent of this wetting, i.e. the role of transmembrane pressure difference (ΔP TMPD ) and capillary pressure (P C ) in determining the liquid-penetration length for each ‘wet-able’ (size Z r cp ) pore. The estimation of V w requires further details of the membrane morphology and associated pore sizes. For the present study, the membrane pore-size is assumed to be characterized by a LogNormal Distribution of the form given below 0   pffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 avg  ln ðr =r Þ 1 þ σ2 p p   1 B C   ffi ð18Þ w r p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp @ A   2 ln 1 þ σ 2 r p 2π ln 1 þσ 2 The rationale, that a Log-Normal Distribution (Eq. (18)) may provide a consistent description of pore-size even with the additional positivity constraint, is essentially based on the arguments of Zydney et al. [37,38]. For a cylindrical pore, the liquid-occupied volume can be written as πr 2p lw τm with lw τm embodying the effectively-wetted pore-length. In addition, the number-fraction of pores in the size

  range: r p ; r p þ dr p is given by w r p dr p as per the definition of pore-size distribution. Hence, the liquid-occupied porous volume may be representatively computed through the summation of liquid-occupied volume for all ‘wet-able’ (size Zr cp ) pores as shown in Eq. (19). On similar lines, the total porous-volume, representative of the membrane porosity, may be symbolically calculated using the integral in Eq. (20), with lp τm signifying the effective total pore-length (lw r lp , for all pores). Z Vw ¼

rcp

Z Vp ¼

  ðπr 2p lw τm Þw r p dr p

ð19Þ

  ðπr 2p lp τm Þw r p dr p

ð20Þ

r max p

r max p r min p

It should be noted that the values of r min and r max p p , described in   R rmax p Table 1, have been chosen in such a way that rmin w r p dr p ffi 1 p

Fig. 3. Wetting behavior of membrane pores.

(0.9999, to be precise).

N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

By substituting Eqs. (19) and (20) into Eq. (15), the wetting ratio (x) at any point (z) along the fiber-length may be written as follows:   r 2p lw w r p dr p xðzÞ ¼ R rmax   p r 2 l w r p dr p r min p p R rmax p r cp

ð21Þ

p

The evaluation of xðzÞ using Eq. (21) is subject to knowledge on the constitutive relationship involving r p ; lw and lp . While lp is generally an independent parameter, one may anticipate lw to be functionally dependent on r p , particularly when the well-known necking phenomenon cannot be neglected due to irregularities in pore-shape. Hence, it becomes imperative to consider the physical balance of different forces acting on gas–liquid interface, for the estimation of lw from a porescale perspective. Assuming that membrane pores may be modeled as a bundle of straight cylindrical capillaries with distinctly uniform diameters and equal lengths, the wetting behavior may be described using the Newton-Dynamics Equation as follows:     d dl 8 dl 2γ cos θ l þ 2 μL l þ ρL gl ¼ ΔP TMPD þ L For r p Z r cp : ρL dt dt dt rp rp ð22Þ At t ¼ 0 :

l¼0

and

dl ¼0 dt

ð23Þ

Eq. (22) incorporates the balance of transmembrane pressure difference (P L  P G ), capillary pressure (P C ), viscous dissipation, gravitational force and kinetic effects [39]. For liquid penetration in a capillary of radius r p (Z r cp ), Eq. (22) may be solved subject to the initial conditions described in Eq. (23) to obtain the long-time asymptotic solution:     r cp   1 2γ cos θ ΔP TMPD ðzÞ ΔP TMPD ðzÞ þ L 1 l1 r p ; z ¼ ¼ ρL g rp ρL g rp

ð24Þ

which may then be substituted as lw in the numerator of Eq. (21) to determine the wetting ratio (x) at any point (z) along the fiber-length. Note that in Eq. (22), the contact angle (θ 4 901) has been assumed to be constant, irrespective of the variations that may arise due to the dynamics of wetting behavior. Nonetheless, one may argue that the above analysis would be valid only in presence of the gravitational force, which cannot be established straight away for all pores due to their random orientations, and that the resulting estimate of wetting ratio (x) might not be always correct. At this point, we invoke the assumption of an equal and complete accessibility of cylindrical pores to the liquid-phase, which ensures that lw ffilp for all ‘wetable’ (sizeZ r cp ) pores. This assumption may be justified by taking the following example, where at some given location (z) along the fiber-length, a pore is subjected to a transmembrane pressure difference (ΔP TMPD ) of 2 bars which is just below the critical ΔP TMPD required for the pore to undergo wetting. Or equivalently, consider a pore at this location (z) with a size (r p ) nearly equal to the critical pore-size (r cp ) given by Eq. (17) corresponding to ΔP TMPD ¼ 2 bars. Now, one may easily notice that even for the case where the transmembrane pressure difference is increased only by 0.1% over its value of 2 bars, the resulting estimate of l1 using Eq. (24) would be nearly equal to 2 cm. More broadly, a generic argument may be made with respect to the term ΔP TMPD ðzÞ=ρL g in Eq. (24), that it would be in range of a few centimeters or even a few meters depending on the transmembrane pressure difference. However, at the same time, since the HFMM is microporous, such large pore lengths would be obviously non-existent, and this essentially means the liquid would completely go through a pore (size Zr cp ) in spite of the presence or absence of gravity. Accordingly, it may be safe to assume that lw ffilp for all ‘wet-able’ (size Zr cp ) pores, as a result of which Eq. (21) is reduced to

Eq. (25) as follows:   R rmax p r 2p lp w r p dr p rc xðzÞ ffi R rpmax   p r 2 l w r p dr p r min p p

69

ð25Þ

p

Now the only information precluding the calculation of x is the knowledge of pore-length (lp ), and its plausible variation with pore-size (r p ). Various assumptions for the functional dependence of lp on r p , for instance: lp ¼ constant; lp p r p and lp p r p 1 , have been outlined by Dullien [40], with the conclusion that the wetting ratio (volumetric saturation) predicted through a linear relationship (lp p r p ) seems to be too high, while that estimated via an inverse relationship (lp p r p 1 ) seems to be too low [40]. In light of these observations, a constant value of lp (for all pores) may provide a comparatively better description of the wetting phenomenon. Note that such an argument also finds manifestations during the determination of pore-size distribution through Mercury-Intrusion Porosimetry experiments [41]. In view of that, Eq. (25) further resolves to Eq. (26), which has been used in the current mathematical analysis.   R rmax p r 2p w r p dr p r cp xðzÞ ffi R rmax ð26Þ   p r 2 w r p dr p r min p p

At the moment, it becomes important to highlight the physical and mathematical inconsistencies in the conceptually similar expressions for wetting ratio (x) described in earlier studies. As discussed previously, although Lu et al. [25] formulated an analogous mechanism, but without any pressing arguments so as to why any distinction was not assumed for liquid-intrusion into pores of different lengths. The mathematical expression for wetting ratio, referred to as the degree of pore-wetness by authors [25], is as follows:   R rmax p 2 r c ðπr p τm ÞðRo  Ri Þ r p dr p   xðzÞ ¼ p ð27Þ π R2o  R2i ϵm LF From Eq. (27), one can easily notice that while the authors [25] estimated V w in a fashion similar to this work, they erroneously  took V p to be the actual porous-volume given by π R2o R2i ϵm LF for a single hollow-fiber. This is erroneous in the sense that the integral in Eq. (19) is only symbolic of the porous volume occupied by liquid-phase and for V w to be actually equal to the same, it must be multiplied with the total number of pores which cannot be determined since the pore-size distribution is normalized. Therefore, on parallel lines, V p must be computed through appropriate considerations to the pore-size distribution as outlined in Eq. (20), and cannot be taken as the actual porous-volume. It may be further noticed that the authors [25] went to assume lw (numerator, Eq. (27)) as the membrane thickness (Ro  Ri ) without any reinforcing justifications. Similar loopholes may be found in the work of Boributh et al. [26], who re-worked Lu et al.’s [25] mechanism to study the effects of membrane characteristics on CO2 absorption in water using HFMMs. Along with that, these authors [26] may have used a flawed definition of Log-Normal Distribution to characterize the membrane pore-size. 2.4.2. Mass transfer: wetted-membrane phase The steady-state transport of CO2 and DEA across the liquidfilled membrane pores is governed by molecular diffusion and chemical reaction. Eqs. (28a) and (28b), given below, have been used to capture the mathematical statement of these phenomena.  2  ∂ cALM 1 ∂cALM DALM ¼ r ALM þ ð28aÞ r ∂r ∂r 2

70

DBLM

N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

 2  ∂ cBLM 1 ∂cBLM ¼ r BLM þ 2 r ∂r ∂r

ð28bÞ

Note that the axial molecular diffusion has been neglected in keeping with the observations of Keller and Stein [42], who established that molecular diffusion in microporous membranes with porosity greater than 10% can be assumed to be onedimensional. The effective diffusivity of CO2 and DEA in wetted membrane pores is given as follows: ϵm ð29aÞ DALM ¼ DAL τm DBLM ¼

ϵm DBL τm

ð29bÞ

The following boundary conditions, Eqs. (30a) and (30b), are applicable here: a) Continuity of Concentration and Flux at Membrane–Liquid Interface: r ¼ Ri ; 8 z A ½0; LF  : r ¼ Ri ; 8 z A ½0; LF  :

cALM ¼ cAiLM and cBLM ¼ cBiLM     ∂cALM ∂cAL ¼ DAL and DALM ∂r ∂r     ∂cBLM ∂cBL DBLM ¼ DBL ∂r ∂r

cALM ¼ HcAiGL

and

0R

ϵm B DAGM ¼ @ τm

  1 DAG DAK =ðDAG þ DAK Þr 2p w r p dr p C A   R rmax p 2 w r dr r p p r min p

ð34Þ

p

Eq. (31) may be solved subject to the following boundary conditions: a) Equilibrium at Gas–Liquid Interface: r ¼ Ri þ xðRo  Ri Þ; 8 z A ½0; LF  :

cAGM ¼ cAiGL

ð35aÞ

b) Continuity of Concentration at Membrane–Gas Interface: r ¼ Ro ; 8 z A ½0; LF  :

ð30aÞ

cAGM ¼ cAG

ð35bÞ

∂cBLM ¼0 ∂r ð30bÞ

The different physicochemical properties for water, DEA and CO2, used in the current study are described in Table 4. It should be noted that the liquid-phase density (ρL ), which is a function of DEA mass fraction (wBL ), may also be expressed as a function of liquidphase DEA concentration (cBL ) using the following equation: ρL ¼

2.4.3. Mass transfer: non-wetted membrane phase For non-wetted (gas-filled) membrane pores, the CO2 transport is driven solely by molecular diffusion. In Eq. (31), DAGM represents the effective gas-phase diffusivity of CO2 in membrane pores, comprising the effect of both Continuum and Knudsen Diffusion as follows:  2  ∂ cAGM 1 ∂cAGM DAGM ¼0 ð31Þ þ 2 r ∂r ∂r   ϵm DAG DAK DAGM ¼ τm DAG þDAK

rmax p

r min p

3. Physicochemical data

b) Equilibrium at Gas–Liquid Interface: r ¼ Ri þxðRo  Ri Þ; 8 z A ½0; LF  :

distributed log-normally (Eq. (18)), the effective diffusion coefficient (DAGM ) needs to be corrected [43], by means of Eq. (34). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r p 8RT DAK ¼ ð33Þ 3 π MM CO2

ð32Þ

However, the Knudsen diffusion coefficient depends on the membrane pore-size as per Eq. (33), and since the pore-size is

cBL MM DEA wBL

ð36Þ

Hence for a given temperature (T) and DEA concentration (cBL ), the DEA weight fraction (wBL ) in liquid-phase may be calculated through Eq. (37) and utilized for calculation of liquid-phase density and water concentration (cH2 O ) using Eq. (5).     w2BL 0:002T 2  1:412T þ 354:867 þ wBL 878:47 þ 1:201T  0:003T 2  cBL MM DEA ¼ 0

ð37Þ Also, the DEA mole fraction and DEA mass fraction in liquid-phase may be related using Eq. (38), for use in the estimation of surface tension (Table 4). xBL ¼

wBL MM H2 O wBL MM H2 O þ ð1  wBL ÞMM DEA

ð38Þ

Table 4 Physical and chemical properties. Property DAG ρH2 O ðT Þ

Source

Mathematical expression h h

i 1:75 7:774  10  05 TP G m2 s  1

i

Fuller’s equation [35] CO2–N2 mixture Correlation fitted to data [44] (see Appendix A.1)

H L ðT; cBL Þ

866:8 þ 1:202 T  0:003 T kg m i ð0:002wBL  0:003ÞT 2 þ ð1:201  1:412wBL ÞT þ 878:47 þ 354:87wBL kg m  3 h i   2:045  10  06 exp 1816:4 kg m  1 s  1 T h  i  06 0:232cBL þ 1820:1 1:864  10 exp  0:0003cBL kg m  1 s  1 T h  i 2  1 m s 2:35  10  6 exp  2119 T h  i 2:532  10  06 exp  2121:96T 0:186cBL þ 2:4  10  04 cBL m2 s  1 h  i 1:729  10  06 exp  2287:7  19:699  10  05 cBL m2 s  1 T h i     3:54  10  07 RT exp 2044 T  h  i 1:854  7:904  10  05 cBL exp  240 T

γ L ðT; xBL Þ

½21expð  20:15xBL Þþ ð99:234  0:1639T Þexpðð0:0004  0:0003T ÞxBL Þ mN m  1

Correlation fitted to data [48] (see Appendix A.4)

ρL ðT; wBL Þ μH2 O ðTÞ μL ðT; cBL Þ DA  H2 O ðTÞ DAL ðT; cBL Þ DBL ðT; cBL Þ H H2 O ðTÞ

2

3

h

Correlation fitted to data [44] (see Appendix A.1) Correlation fitted to data [45] (see Appendix A.2) Correlation fitted to data [46] (see Appendix A.2) [47] CO2–N2O analogy [47] (see Appendix A.3) [46] [47] CO2–N2O analogy [47]

N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

71

4. Algorithm

5. Results and discussion

The set of governing equations for mass transfer across the four regions: liquid-phase (Eqs. (6)–(10)), wetted membrane phase (Eqs. (15)–(30)), non-wetted membrane phase (Eqs. (31)–(35)) and gasphase (Eqs. (11)–(14)), involves linear and non-linear, algebraic, transcendental, ordinary-differential (OD) and partial-differential (PD) equations. Apart from these, the reaction rates for CO2 and DEA represented by Eq. (4), which feature in the governing equations for mass transfer across the liquid-phase and wetted-membrane phase, are non-linear functions of the DEA concentration. To solve this set of equations, a Finite-Volume Method (FVM) was utilized for the purpose of discretization. While an upwind scheme was applied to resolve the convection terms in Eqs. (9) and (12), the concentration profiles were assumed to be linear between the successive cell-centroids to re-formulate the diffusion terms in Eqs. (9), (28) and (31). The source code for discretized equations, obtained in this manner, was compiled using Cþ þ language on the basis of the methodology presented in Fig. 4. The system of algebraic equations, resulting from the finite-volume discretization of Eqs. (9a) and (9b), was solved iteratively using the Gauss-Seidel Technique. The integrals in Eqs. (26) and (33) were computed using Simpson’s 1/3rd Rule. Besides, the discretized tridiagonal matrix system for Eqs. (28a) and (28b) was solved using the conventional Thomas Algorithm. The computational procedure attains closure using the fact that for each shell-side segment (Fig. 2), the amount of CO2 removed from gas-phase must equal the sum total of CO2 consumed during the liquid-phase chemical reaction in fiber-lumen and wetted membrane pores, and CO2 transported out of the fiber-lumen via convection.

The operating variables for a base case simulation are described in Table 5. As the various conditions were varied in the many simulations whose results have been discussed in the following sub-sections, all other parameter values were kept fixed at the conditions mentioned in this table. Amongst the module properties, only the average pore-size was varied. The value of 8 mol m  3 for gas-phase CO2 concentration at the fiber inlet corresponds to a mole fraction of about 0.2 in the CO2–N2 gas mixture at atmospheric pressure and room temperature. The gasphase pressure (Eq. (14)) was kept and assumed constant at 1 bar. From an industrial standpoint, CO2 flux is of prime importance, because it determines the active surface area required to achieve a

Table 5 Properties of base case simulation. Parameter

Value

Module properties

Table 1a

cin AG

8 mol m  3

cin BL vin G vavg L P in L

1000 mol m  3

a b

0:10 m s  1

b

0:10 m s  1

b

1:2 bar b 1:0 bar 300 K b

PG T

Only average pore-size varied. Unless mentioned otherwise.

Input Module Properties: Table 1 Operating Conditions: , , , ,

,

,

Calculate Physical Properties: Table 4

Sequential Discretization Eqs. (9a)-(9b)

Eqs. (10a)-(10d)

Eqs. (28a)-(28b)

Eqs. (30a)-(30b)

Computation , For each segment: guess & Estimate (Eq. (12)), (Eq. (13)) &

boundedly (Eqs. (31)-(35))

Multi-Grid Iteration Solve Eqs. (28) subject to Eqs. (30a)-(30b) Solve Eqs. (9) subject to Eqs. (10a)-(10d)

Analysis Calculate Rate of CO2 Transport in Liquid Phase: For all segments, check if:

No! Update For all segments, adjust such that ! , , , , and ! Update Estimate (Eq. (12)), (Eq. (13)) & (Eqs. (31)-(35)) Fig. 4. Solution algorithm.

Yes

b

Exit

b

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N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

given separation and the performance of the membrane module as a gas–liquid contactor on the whole. For the present study, CO2 flux (J CO2 ) has been defined as the cumulative change in gas-phase CO2 molar flow rate across the fiber length, per unit area of gas–liquid interface.   in out out π R2e  R2o ðcin AG vG cAG vG Þ J CO2 ¼ R L ð39Þ F 0 2π ½Ri þ ðRo  Ri ÞxðzÞðdzÞ Note that this interfacial area takes the wetting phenomenon into account, and is conceptually different [12] from the active surface area (2πnF LF Ri ), or one should rather say geometric inner-surface area (mentioned in Table 1) utilized by prior studies for reporting the CO2 flux. One may argue that the definition of area used in Eq. (39) is not really useful as it is a tedious task to know, a priori, the true location of gas–liquid interface. Still, such a definition is advantageous vis-à-vis its ability to transform into the inner-surface area for ‘no-wetting’ case and outer-surface area for the case of ‘complete-wetting’. The other important parameter generally used to characterize the module performance is the overall mass transfer coefficient (kO ) which captures the hydrodynamics near the gas– liquid interface. For the present study, the log-mean driving force (ΔcDF ) has been formulated on the basis of mixing-cup liquid-phase CO2 concentration (cAL ) as follows:  out  ðHcin  cAL in Þ  Hcout AG cAL  ΔcDF ¼  AG ð40Þ in out out ln Hcin AG cAL =HcAG  cAL ΔcDF may be further utilized to define the overall MTC (kO ) as follows: kO ¼

J CO2 ΔcDF

ð41Þ

A grid independence study, conducted using three different grid densities, revealed that the converged solution (results not shown) was independent of the cell aspect ratios. Therefore, a 50 : ½0; Ri   25 : ½0; LF   20 : ½0; Ri þ ðRo Ri ÞxðzÞ grid was used for the rest of this paper. In the discussion that follows, model validation against the experimental results reported in open literature is presented, and the influence of various parameters on the module performance has been described in detail. 5.1. Model validation Experimental data for the validation of proposed model has been taken from the work of Zhang et al. [16], where a conceptually similar although restricted (with respect to the wetting mechanism) mathematical model was formulated and experimentally validated, under the intuitive conjecture of an axiallyuniform wetting pattern, for CO2 absorption in water and 2 M aqueous DEA solution. The simulated flux profiles corresponding to identical parameters (taken from [16]) have been depicted in Fig. 5(a) for two distinct cases: ‘no-wetting’ and ‘partial-wetting’, and benchmarked against the experimental data of Zhang et al. [16], for the case of physical absorption. While the ‘no-wetting’ case represents the case where membrane pores are gas-filled, on the other hand, the ‘partial-wetting’ case uses the minimum allowable liquid-phase pressure (P min L ), defined via Eq. (42), to account for membrane wetting through the mechanism described by Eqs. (15)–(26). The reader should be able to derive P min easily from the consideration that P L must be greater than P G L at the fiber-outlet, in order to prevent any gas–liquid dispersive interactions. P min ¼ PG þ L

8μL vavg L LF R2i

ð42Þ

The predicted results for the ‘no-wetting’ case are in excellent agreement with the experimental data, thus confirming the absence of wetting using water as an absorbent, in keeping with the conclusions of Zhang et al. [16]. However, for the ‘partial-wetting’ case, the CO2 flux predicted by the model is significantly lower than the corresponding value for ‘no-wetting’ case. A key observation in this regard is that the flux profile for the ‘partial-wetting’ case deviates from the experimental and ‘no-wetting’ case data largely at increasing liquid-phase velocities, and through a simply examination of data at the highest liquid-phase velocity (Fig. 5(a)), this deviation can be quantified at 20%. Furthermore, for the ‘partial-wetting’ case, the flux is stabilized at a velocity of around 0.3 m/s and exhibits an insensitivity to any further increase in liquid-phase velocity. Such a behavior makes intuitive sense vis-à-vis the diminishing liquid-phase mass-transfer resistance at higher velocities [12,21], which leads to a scenario where the mass-transfer resistance constituted by the liquid-filled membrane pores dominates and as a result, flux stabilization is achieved. Fig. 5 (b) illustrates the simulated flux profiles as a function of gas-phase velocity for CO2 absorption in 2 M aqueous DEA solution. From this plot, one can clearly observe a broad agreement between the predictions of this study for the ‘partial-wetting’ case and experimental data of Zhang et al. [16], which suggests a greater relevance of the developed model under ‘partial-wetting’ conditions. The strict overprediction of CO2 flux for the ‘no-wetting’ case can be noticed to an extent as high as 22% at the highest gas-phase velocity. At this point, one should note that for the ‘no-wetting’ case i.e. xðzÞ ¼ 0; 8 z A ½0; LF  (current notation) or equivalently δ=ðRo  Ri Þ ¼ 0 (Zhang et al.’s [16] notation), the currently predicted CO2 flux is consistent with the authors’ [16] results as depicted in Fig. 6(a). However, the same cannot be said for the reported wetting-profiles. To further emphasize this point, as illustrated in Fig. 6(b), the ‘partial-wetting’ pattern ensuing from the use of minimum allowable liquid-phase pressure (P min L , Eq. (42)) has been benchmarked against the ‘partial-wetting’ profile assumed by Zhang et al. [16] to match their theoretical predictions with the experimental results. With respect to the variation along fiberlength, clear inconsistencies are observed between the currently predicted wetting-profile and that reported by authors [16], which cannot be disregarded even when a volume-averaged wettingfraction (x, Eq. (43)) is used for the purpose of comparison under the excluded effects of axial non-uniformity. This volumeaveraged wetting-fraction (x), a value of 0.096, representative of the simulated wetting-pattern, is observed to be significantly higher than the value of 0.065, reported by Zhang et al. [16] by means of intuitive-fitting. It may seem speculative to the reader, but such a discrepancy can be attributed to the erroneous assumption of a uniform liquid-phase operating pressure by authors in [16], which directly transforms into an axiallyuniform wetting pattern. Above all, the notion of volumeaveraged wetting-fraction is important since it encompasses the usual definition of wetting fraction adopted by prior studies [10,16,20–24]. Z 1 z ¼ LF x¼ xðzÞdz ð43Þ LF z ¼ 0 Two facts are notable at this point. Firstly, the occurrence of socalled axially-uniform wetting, as presumed in numerous prior works [10,16,20–24], cannot be readily concluded with the circumscribed nature of current understanding on this phenomenon. Hence, it is only fair to assume otherwise for the purpose of theoretical analysis, as outlined in the present study. Secondly, the authors [16] argued that liquid-phase velocities were low enough to ignore the axial pressure-drop, and to discount its effect on the module performance, operating pressures for both phases were maintained at atmospheric pressure. But with these effects excluded, no other conceptual basis for the wetting of HFMM

N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

73

3 1 in Fig. 5. Model validation {cin , P G ¼ 1 atm, P L ¼ P min ); (b) CO2 absorption in aqueous DEA solution L , T ¼298 K}: (a) CO2 absorption in water (vG ¼ 0.0544 m s AG ¼8.42 mol m 3 –1 (cin , vin BL ¼2000 mol m L ¼0.15 m s ).

1 Fig. 6. A comparison of current study versus Zhang et al. [16]: (a) theoretical flux in ‘no-wetting’ case for CO2 absorption in water (vin ); (b) partial-wetting G ¼ 0.0544 m s 3 1 pattern for CO2 absorption in aqueous DEA solution (cin , vin ). BL ¼ 2000 mol m L ¼ 0.15 m s

Table 6 Liquid-phase pressure-drop across the module (inlet value, P in L ¼ 1 atm). 1 vin Þ L ðm s

% Reduction in P L (Eq. (8))

0.1 0.2 0.3 0.4 0.5

16.7 32.13 48.20 64.27 80.33

was described, when the membrane material was by all accounts hydrophobic (PP). Table 6 shows the % reduction in liquid-phase pressure, calculated using Hagen-Poiseuille equation (Eq. (8)) in

the range of velocities studied by Zhang et al. [16]. It can be clearly seen that pressure-drop across the module can be substantial even at a low liquid-phase velocity, an observation which contradicts the authors’ [16] assumptions. For the rest of this work, we describe our current findings, with reference to the numerous observations concerning the effect of different parameters on module performance, reported in literature. 5.2. Effect of hydrodynamics In accordance with the observation that avoiding ‘partialwetting’ during the course of experimental studies seems to be impossible [49], a high CO2 flux would always be on top of the wish-list of desired attributes. Consequently, the objective of

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N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

studying the independent effect of hydrodynamics on the process would be best served under conditions which lead to wetting and yet keep it minimal. The use of minimum-allowable liquid phase pressure (P min L , Eq. (42)) permits to achieve this objective effectively, since it gives the lower bound of P L for a fixed P G and vin L , and therefore corresponds to the minimalistic wetting-pattern resultant from such a trans-membrane pressure field. In the open literature, several prior modeling studies [10,16,20–23] may be found where the volume-averaged wetting-fraction (Eq. (43)), commonly referred to as wetting-fraction, has been varied independent of hydrodynamics, to investigate the module performance. Note that one may be mistaken to assume that it should be possible to quantify the effect of wetting on the flux separately, since wetting and hydrodynamics are inherently, though implicitly, related to P L Z P min L . Fig. 7 shows the influence of liquid and gas velocities on the CO2 flux with P L ¼ P min and other parameters kept at their baseL case value (Table 5). One clearly observes an optimum, i.e., a set of liquid-phase velocities (vin L ) corresponding to a set of gas-phase velocities (vin G ), where the flux is the highest. The explanation behind this observation is not straight forward. At lower liquidphase velocities, the depletion of lumen-side DEA concentration is quite significant [13,50] as shown in Fig. 8(a). This, coupled with the conventionally low liquid-phase diffusivity of CO2 [35], leads

to a scenario where the rate of absorption is determined largely by convective transport. Hence, it may be argued that the preliminary outcome of any increase in liquid-phase velocity would be to ease out this depletion of DEA, as evident from a comparative analysis of Fig. 8(a) and (b), thereby leading to a reduction in liquid-phase mass transfer resistance and a simultaneous enhancement of CO2 flux. This behavior widely agrees with the observations reported in literature [13,15,16,18,50,51], but only up to a certain liquid-phase velocity. Any further surge in vin L triggers a regime shift [23] due to a corresponding intensification of the wetting-pattern, demonstrated by Fig. 9, since P min increases with the liquid-phase velocity L (Eq. (42)), which conforms to the experimental findings of Boributh et al. [22]. This shift may be seen (Fig. 7) from vL ¼ 0:4 m s  1 onwards, exceeding which the flux decreases in close correlation to the overall MTC (kO ), depicted in Fig. 10. Additionally, from Fig. 10, one may observe a similar trend at higher liquid-phase velocities due to an increase in the relative contribution of masstransfer resistance [13,16] constituted by liquid-filled membrane pores. An analogous observation, that a maximum overall MTC could be attained with respect to the liquid-phase velocity for the case of ‘partial-wetting’, was reported by Malek et al. [24]. Additionally, the CO2 flux is observed to be a non-linear increasing function of the gas-phase velocity (Fig. 7). Such a dependence can be explained on the basis of an enhanced rate of CO2 replenishment on the module shell-side, as suggested by

in Fig. 7. CO2 flux (J CO2 ) as a function of liquid-phase (vin L ) and gas-phase (vG ) min 3 3 in velocities: {cin ¼8 mol m , c ¼1000 mol m , P ¼ 1 bar, P ¼ P , T ¼300 K}. L G L BL AG

Fig. 9. Influence of minimum-permissible liquid-phase pressure (P min L ) on ‘partialwetting’ pattern: {P G ¼ 1 bar, T ¼ 300 K}.

-3 in Fig. 8. Liquid-phase DEA concentration (cBL ) Profile {cin AG ¼8 mol m , cBL ¼ 1000 mol 1 1 in m  3, vin , P G ¼1 bar, P L ¼ P min ; (b) vin L , T ¼ 300 K}: (a) vL ¼0.05 m s G ¼ 0.1 m s L ¼ 0.2 m s  1.

in Fig. 10. Overall MTC (kO ) as a function of liquid-phase (vin L ) and gas-phase (vG ) 3 3 velocities: {cin , cin , P G ¼ 1 bar, P L ¼ P min L , T ¼ 300 K}. BL ¼ 1000 mol m AG ¼ 8 mol m

N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

the gas-phase CO2 concentration profile displayed in Fig. 11, together with the fact that a higher value of vin G directly implies a higher J CO2 in accordance with Eq. (39). On the contrary, the overall MTC (kO ) is found to exhibit a decreasing trend (Fig. 10) with respect to the gas-phase velocity. Although this may seem counter-intuitive at the first instance, but the key to this observation lies in the definition of kO (Eq. (41)) involving the log-mean driving force (ΔcDF , Eq. (40)), which increases in a one to one fashion with the gas-phase velocity (vin G ), as an indirect consequence of the reduced CO2 depletion rate. 5.3. Effect of operating conditions 5.3.1. DEA concentration The influence of liquid-phase DEA concentration (cin BL ) on CO2 flux has been demonstrated in Fig. 12, for three distinct gas-phase CO2 concentrations (cin AG ), using base-case simulation parameters (Table 5) and P L ¼ P min L . From Fig. 12, one can clearly notice that any increase in DEA concentration marks an analogous enhancement in CO2 flux, with this effect being more pronounced at higher gasphase CO2 concentrations. Equivalent trends have also been

Gas-Phase CO2 Concentration: cAG (mol m-3)

8 7 6 5 4 3 vin = 0.06 m s -1 G 2

vin = 0.12 m s -1 G vin = 0.18 m s -1 G

1 0

0

0.02

0.04

0.06

0.08

0.1

0.12

Axial Distance: z (m)  3 in Fig. 11. Gas-phase CO2 concentration (cAG ) Profile: {cin , cBL ¼ 1000 mol AG ¼ 8 mol m 1 m  3, vin , P G ¼ 1 bar, P L ¼ P min L , T ¼300 K}. L ¼ 0.1 m s

in Fig. 12. CO2 flux (J CO2 ) as a function of CO2 (cin AG ) and DEA (cBL ) inlet concentrations: 1 1 {vin , vin , P G ¼ 1 bar, P L ¼ P min L , T ¼300 K}. L ¼ 0.1 m s G ¼ 0.1 m s

75

acknowledged in the work of Rongwong et al. [18,23] for two distinctive definitions of CO2 flux: one based on the notion of overall driving force, and the other conceptually similar to the one used in the present study (Eq. (39)). It may be noted that a similar trend for CO2 flux vis-à-vis the influence of cin AG has been mistakenly ascribed to an enhanced driving force by Atchariyawut et al. [15], although the CO2 flux was defined in a fashion, similar to the present study. Parallel observations vis-à-vis the influence of liquid-phase DEA concentration, stated in previous works [2,12,18,26,50], have been attributed to a net higher rate of reaction in the liquid-phase and wetted membrane pores at greater values of cin BL . One must note that a high DEA concentration is also poised to have a significant impact on the different physicochemical properties (Table 4). More specifically, the liquid-phase surface tension (γ L ) decreases, which directly implies a lower break-through pressure (Eq. (17)) for a given pore-size [50], and a higher degree of membrane wetting. Besides, the solution viscosity (μL ) increases, that contributes to an increased axial pressure-drop along the module length (Eq. (8)), thereby entailing a correspondingly lower degree of membrane wetting. Be that as it may, the current results (Fig. 12) suggest the sole-dominance of a net higher reaction-rate in determining the CO2 flux, with no such incidental effect of wetting as manifested in the altered physicochemical properties, which may be accredited to an offsetting interplay of (decreasing) surface tension and (increasing) viscosity. To strengthen the argument further, results for the variation of overall MTC (kO ) and removal efficiency (ηR ), defined in the following way: ! out cout AG vG ηR ¼ 1  ð44Þ in cin AG vG are shown in Table 7. While kO and ηR can be expected to be strongly dependent on cin BL due to a net higher reaction-rate which enhances the CO2 removal as explained above, they are observed to be nearly independent of cin AG . With increasing gas-phase CO2 concentration, any intensification in the CO2 flux seems to be counterbalanced by an equivalent rise in the log-mean driving force (ΔcDF ), leading to a weak dependence of overall MTC (kO ) on cin AG . For the case of removal efficiency (ηR ), such a behavior may be explained by appreciating the fact that the amount of CO2 separable from the gaseous (CO2–N2) mixture, is essentially governed by the given set of operating conditions regardless of the quantity available for separation. Hence, with constant operating conditions of pressure, temperature and hydrodynamics, the currently witnessed correlation between ηR and cin AG makes sense. 5.3.2. Temperature The operating temperature profoundly influences the absorbent viscosity (μL ) as shown in Table 8, which, in turn, has great implications on module performance by altering the driving force for wetting phenomenon (P min  P G ), and the so-called resistance L to wetting phenomenon (P in L  P L ) along the fiber-length. Evidently, any increase in temperature leads to an inevitable reduction in the minimum-allowable liquid-phase pressure (P min L ) as well as the corresponding volume-averaged wetting-fraction (x). An elevated temperature also points to higher CO2 diffusivities (DAGM and DAL ) and a greater equilibrium-solubility of CO2 in liquid-phase. Accordingly, one may expect an enhanced module performance at higher temperatures, which is indeed the outcome of respective simulations, shown graphically in Fig. 13(a). The % intensification in flux, overall MTC and removal efficiency were observed to be 21%, 38% and 20% respectively for an increase in temperature from 300 K to 350 K. Matching trends for the CO2 flux at high operating temperatures have been experimentally reported by Atchariyawut

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Table 7 1 1 Overall MTC (kO ) and removal efficiency (ηR ): (vin , vin , P L ¼ P min L , P G ¼1 atm, T ¼ 300 K) L ¼ 0.1 m s G ¼ 0.1 m s

3 Þ cin BL ðmol m

3 cin AG ¼ 4 mol m

3 cin AG ¼ 8 mol m

kO ðm s  1 Þ  05

4.518  10 4.889  10  05 5.263  10  05 6.315  10  05 7.774  10  05 1.000  10  04 1.454  10  04 1.806  10  04 2.290  10  04

0 10 20 50 100 200 500 1000 2000

kO ðm s  1 Þ

ηR

 05

4.525  10 4.875  10  05 5.230  10  05 6.240  10  05 7.663  10  05 9.865  10  05 1.440  10  04 1.799  10  04 2.280  10  04

0.119 0.142 0.163 0.212 0.271 0.348 0.473 0.549 0.623

Table 8 Surface

tension,

viscosity,

P min L

and

x

as

a

function

of

temperature:

–3 –1 –1 in in (cin BL ¼1000 mol m , vL ¼ 0.1 m s , vG ¼ 0.1 m s , P G ¼1 atm)

TðKÞ

γ L ðN m  1 Þ

μL ðkg m  1 s  1 Þ

  P min N m2 L

x (Eq. (43))

300 310 320 330 340 350

6.397  10  02 6.229  10  02 6.060  10  02 5.891  10  02 5.722  10  02 5.552  10  02

1.290  10  03 1.035  10  03 8.414  10  04 6.928  10  04 5.770  10  04 4.856  10  04

110,235.86 108,209.17 106,675.19 105,496.32 104,577.68 103,852.62

6.377  10  03 3.676  10  03 2.139  10  03 1.260  10  03 7.534  10  04 4.584  10  04

et al. [15] and Mansourizadeh et al. [19] for CO2 absorption in 1M NaOH, and theoretically discussed by Lu et al. [25] and Khaisri et al. [52] for CO2 absorption in aqueous methyl-di-ethanolamine (MDEA) and mono-ethanolamine (MEA) solutions respectively. The profiles depicted in Fig. 13(a) may show substantial variations at higher liquid-phase pressures, where the influence of membrane wetting due to reduction of surface tension would be reasonably more significant as compared to the influence of decreasing viscosity. Moreover, for CO2 absorption in water, various prior studies [15,19, 26] have reported a decline in CO2 flux with temperature. The results for a corresponding validation of this observation using the current mathematical model have been compiled in Fig. 13(b). From a comparison of Fig. 13(a) and (b), one can easily recognize the dissimilarities in flux profiles for CO2 absorption, in aqueous DEA solution and water, to be an indirect consequence of the decreasing CO2 solubility in water at high temperatures [47], in agreement with the experimental findings of Atchariyawut et al. [15]. For example, (CO2–H2O) Henry’s Constant decreased from a value of 0.80 (at T¼ 300 K) to 0.41 (at T¼340 K). The reader may further want to draw parallels from the experimental work of Mansourizadeh et al. [19], where a similar trend for the decline of CO2 flux with temperature was reported for absorption in distilled water. At higher temperatures, it may be therefore argued that for P L ¼ P min L , a higher CO2 flux directly follows from a reduction in the driving force (P min  P G ) for wetting L phenomenon, while for a given P L ð 4 P min L Þ, a lower CO2 flux is the manifestation of diminishing resistance (P in L  P L ) to the wetting phenomenon.

3 cin AG ¼ 16 mol m

ηR 0.120 0.142 0.162 0.211 0.270 0.349 0.478 0.558 0.633

kO ðm s  1 Þ  05

4.539  10 4.852  10  05 5.172  10  05 6.100  10  05 7.446  10  05 9.582  10  05 1.409  10  04 1.783  10  04 2.246  10  04

ηR 0.122 0.143 0.162 0.209 0.269 0.351 0.490 0.579 0.656

maximum flux observed at a liquid-phase velocity of 0.4 m s  1. A higher velocity (0.7 m s  1) entails a greater minimum-allowable liquid-phase pressure (P min L , Eq. (42)), and points to a consistently higher degree of membrane-wetting as established by the trend for volume-averaged wetting-fraction (Eq. (43)) depicted in Fig. 14(b). Also, for a constant liquid-phase velocity, the decline in CO2 flux at higher operating pressures (Fig. 14(a)) can be explained on the basis of an enhanced degree of ‘partial-wetting’ (Fig. 14(b)) leading to a higher mass-transfer resistance, in keeping with the experimental observations of Mavroudi et al. [14], who reasoned analogous trends through the argument of an inevitable rise in liquid-phase pressure at higher flow-rates. In addition, one observes a characteristic convergence of the three flux profiles (Fig. 14(a)) at higher liquid-phase operating pressures. Such a behavior fundamentally indicates the existence of a hydrodynamic regime, where the transformation of a ‘partialwetting’ pattern into a ‘complete-wetting’ profile is imminent, as suggested by the trends for volume-averaged wetting-fraction (Fig. 14(b)), and the process is rendered oblivious to any further hydrodynamic changes. The value of liquid-phase pressure (P L ), at which this convergence is first perceived, can be characterized as the maximum-allowable liquid-phase pressure (P max ), for a safe L experimental operation. In other words, P max denotes that critical L value of the liquid-phase operating pressure, exceeding which the liquid absorbent would start to exit the membrane pores on the module shell-side. Hence, the term ‘safe’ is symbolic of the situation where this loss of absorbent, in the aforementioned manner, can be effectively prevented by operating at P L r P max . L In the previous studies [14,19,22,25,26,53], where the effect of operating pressure on module performance has been outlined, the relevance of P min and P max has been overlooked. Note that one may L L be erroneously lead to believe that it should be possible to realize the true effects of a very high liquid-phase pressure on CO2 flux and overall MTC at a fixed gas-phase pressure. In principle, it is quite obvious that such a study, if at all possible, would only provide clouded results by disregarding the physical-effects of dispersive interactions and an altered hydrodynamics in both phases, owing to a loss of absorbent solution on the module shell-side, governed by the mechanisms [54] of droplet-formation and subsequent surface-spreading. 5.4. Effect of membrane characteristics

5.3.3. Liquid-phase pressure With an aim to discern the operational influence of liquidphase pressure on module performance, simulation results for the variation of CO2 flux, at three different liquid-phase velocities are displayed in Fig. 14(a) with other parameters kept at their basecase values (Table 5). One may notice that the opening-values (corresponding to P L ¼ P min L ) exhibited by these flux profiles are in accordance with the behavior described in Section 5.2, with the

The membrane pore-size features in the governing equation for physical effects of various micro-physical phenomena occurring at pore-scale. More fundamentally, a higher average pore-size implies a lower break-through pressure for a given hydrodynamics, and a consistently greater extent of membrane-wetting. This is clearly evident in the observed trends for volume-averaged wetting-fraction (x), as a function of the liquid-phase pressure,

N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

77

3 1 1 Fig. 13. Effect of Temperature on CO2 flux (J CO2 ), Overall MTC (kO ) and Removal Efficiency (ηR ) {cin , vin , vin , P G ¼ 1 bar, P L ¼ P min L }: L ¼ 0.1 m s AG ¼8 mol m G ¼0.1 m s -3 (a) CO2 absorption in aqueous DEA solution (cin ¼1000 mol m ); (b) CO absorption in water. 2 BL

3 3 1 Fig. 14. Influence of liquid-phase pressure {cin , cin , vin , P G ¼1 bar, T ¼300 K}: (a) CO2 flux (J CO2 ); (b) Volume-averaged wetting BL ¼ 1000 mol m G ¼0.1 m s AG ¼ 8 mol m fraction (x).

shown in Fig. 15(a) for three distinct average pore-sizes. One can clearly discern from Fig. 15(a) that, any increase in either the liquid-phase pressure or the average pore-size is bound to result in an enhanced degree of ‘partial-wetting’ as argued above using physical constraints. This, in turn, indicates a module-underperformance, as evidenced by Fig. 15(b)–(d). The % reduction in the CO2 flux, overall MTC and removal efficiency is observed to be 6.5%, 9.5% and 4% for P L ¼ P min L , following a 3-fold increase in average pore-size from 0:03 μm to 0:09 μm. Besides, these values

intensify (Fig. 15) to 16.8%, 20% and 7.5%, at higher liquid-phase pressures (P L =P min ¼ 1:6), in that order. L Such a functional dependence of CO2 flux on average pore-size, suggested by the current mathematical model, seems to have greater manifestations vis-à-vis the widespread observation of the decline in module performance with operation time, reported independently across numerous experimental gas-absorption studies [13,14,16–19,55]. The reduction in CO2 flux over time can be ascribed to an enhanced degree of membrane-wetting as a

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Fig. 15. (a) Volume-averaged wetting fraction (x); (b) CO2 flux (J CO2 ); (c) Overall MTC (kO ); (d) removal efficiency (ηR ), as functions of average pore-size (r avg p ) and liquid-phase 3 3 1 1 in pressure-ratio (P L =P min , cin , vin , vin , P G ¼ 1 bar, T ¼ 300 K}. L ): {cAG ¼8 mol m BL ¼1000 mol m L ¼ 0.1 m s G ¼ 0.1 m s

consequence of the morphological changes, characterized by an enlarged average pore-size [13,19,53,56–61], an increased surfaceroughness [53,58,60,61] and a reduced surface-hydrophobicity (contact angle) [12,44,58–60], that may ensue from the use of chemical absorbents. However, at this point, such an argument is merely speculative and in order to validate this hypothesis, the day-wise experimental flux-data for CO2 absorption in 2 M DEA solution, reported by Zhang et al. [16], have been benchmarked against the predictions of current mathematical model in Fig. 16. This has been achieved by finding a consistent pore-size distribution, one each for Day-2 and Day-4, corresponding to a match between the model predictions and the experimental results using a trial and error technique. From Fig. 16(a), one can notice the simulated profiles to be accurate estimates of the corresponding experimental data, with an average error of 2.37 %, 1.91 % and 1.88 %, for ‘Day-0’, ‘Day-2’ and ‘Day-4’ respectively. Moreover, the decline in CO2 flux may be

explained reasonably well through a 2-fold enlargement in the average pore-size, and a consistent broadening of the pore-size distribution over 5 days. Nonetheless, one can argue that multiple-realizations of the same flux could be theoretically plausible using different combinations of average pore-size and geometric standard deviation. Therefore, in light of this argument, two pertinent questions arise: first, whether this 2-fold enlargement is realistic or not? Second, if it is, on what conceptual basis, and if not, then what other phenomena could perhaps be accountable? Although the first question cannot be answered with absolute certainty unless relevant experimental studies are conducted, generally speaking, this 2-fold increase in the average pore-size seems reasonable. To appreciate this argument, one is referred to the work of Wang et al. [13], where a significant pore-size enlargement was observed during CO2 absorption in 2 M aqueous DEA solution using PP HFMMs over a period of 4 days.

N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

79

-3 -3 1 in in Fig. 16. CO2 absorption in aqueous DEA solution {cin , P G ¼ 1 bar, T ¼ 298 K}: (a) current predictions of day-wise CO2 flux AG ¼8.42 mol m , cBL ¼ 2000 mol m , vL ¼ 0.15 m s (J CO2 ) versus experimental data of Zhang et al. [16]; (b) fitted temporal-evolution of pore-size distribution.

However, at the same time, one cannot disregard the observations of a prior study by Wang et al. [58], where a substantial (precisely unquantified) expansion in the pore-size was reported for PP HFMMs, this time after 10 days, through membrane immersion experiments in aqueous DEA solution. A parallel study by Wang et al. [61] revealed a 1.5 fold and a 2 fold dramatic increase in the average and the largest pore-size respectively, for PP HFMMs over an immersion period of 40 days in aqueous MEA solution, which might suggest that our current estimate of a 2-fold increase over a period of 5 days is somewhat less realistic. Although noteworthy reductions in module-performance over a period of 3–4 days have been analogously reported in other prior experimental studies [19,55], we leave it open for the reader to interpret the present conclusion of a 2-fold increase in any way deemed correct. If one chooses to believe that such a change is unlikely over a 5-day period, one must also appreciate the reduction in surfacehydrophobicity (contact angle) of the membrane over time which can result in a greater degree of membrane wetting. Such an observation has been commonly ascribed to membrane-absorbent interactions in earlier studies by Rangwala [12], Porcheron et al. [59] and Lv et al. [60]. Accordingly, the prediction of a perceivably unrealistic 2-fold increase in the average pore-size, by the current mathematical model, may be argued through the overlooked physical effects of a reduced surface-hydrophobicity in the present analysis. On the contrary, it becomes important to understand the conceptual basis behind the enlargement in average pore-size ensuing from the exposure to organic solvents vis-à-vis the ‘day-wise’ variation of CO2 flux. One of the earliest attempts in this regard was made by Kamo et al. [56], who argued the formation of liquid-bridges between the membrane micro-fibrils followed by the deformation and adhesion of the adjacent micro-fibrils. Here, the extent of this enlargement was determined by the solvent’s surface tension. On similar lines, for PP membranes with a less-ordered pore-structure, Barbe et al. [57] argued the dominance of lateral forces in displacing the pore-walls due to solvent intrusion. As a result, pores with a larger size were further enlarged, while those with a smaller size, were either completely blocked or partially abridged. The reader might want to relate this argument to the broadening of pore-size distribution with time, as hinted by the current set of results depicted in Fig. 16. Another physically plausible mechanism for the pore-size enlargement could be the solvent-induced swelling of the PP membranes, due to the diffusion of absorbent molecules into the crystal lattice, as outlined by Lv et al. [60].

Finally, we close this discussion by re-iterating the fact that the morphological changes incontrovertibly affect the module performance to a substantial extent, and therefore, must be accounted during the theoretical analysis of mass-transfer in hydrophobic HFMMs.

6. Summary and conclusions A high-rate of mass transfer (vis-à-vis CO2 flux) features on top of the wish-list of attributes desired of HFMMs for their effective deployment in the process of CO2 separation using chemical absorbents. However, the module performance declines over time due to an enhancement in the degree of ‘partial-wetting’, triggered by a complex-cocktail of variations in physicochemical properties and morphological changes, such as an enlarged average pore-size, reduced surface-hydrophobicity and an elevated surfaceroughness. The process exhibits a large sensitivity to the hydrodynamics, absorbent concentration and operating conditions of temperature and pressure. The mathematical model proposed in this work, virtually captures every other observation concerning the gas-absorption process using micro-porous HFMMs, reported independently in various prior modeling and experimental investigations [1–26,50–61]. Furthermore, for a given hydrodynamics and a fixed gas-phase pressure, the authors demonstrate the existence of a hydrodynamic regime, limited by a minimum max (P min ) permissible liquid-phase pressure, L ) and a maximum (P L in which the HFMM can be safely operated, by circumventing any dispersive interactions between the two phases, either in the fiberlumen (gas-bubble formation) or the module shell-side (liquiddroplet formation), and consequently eliminating the situation where any plausible loss of the gas-mixture and/or liquidabsorbent might be imminent. While the notion of P min is based L on ‘minimal-wetting’ conditions, P max is determined essentially L from ‘complete-wetting’ considerations. This entire exercise of operating at P L ¼ P min and P L ¼ P max can therefore be thought of as L L respectively quantifying the lower and higher bounds of the attainable CO2 flux under a given set of other operating conditions. Furthermore, the authors took efforts to integrate a partial-wetting mechanism along-with the currently developed mathematical model via consideration to the physical balance of forces acting on the gas– liquid interface inside a pore, on account of the complex microphysical phenomena. Although a benchmarking study revealed the

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N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

model predictions to be in an excellent agreement with the experimental data from literature, the authors still do not claim a comprehensive validation of the current mathematical model, since it does not account for the occurrence of membrane-wetting at a net-zero transmembrane pressure difference. The authors accredit this to an ‘unaccounted for in the developed model’ reduction in the surfacehydrophobicity (contact angle), either with time due to morphological changes or at higher absorbent concentrations, which may transform the capillary pressure into a net-positive driving force for the penetration of liquid inside membrane pores, even in the absence of any external driving force (ΔP TMPD ¼ 0). In an attempt to explain the observation of a declining module performance with time, the relevant morphological changes of pore-size enlargement and distribution broadening, that the membrane may undergo as a result of exposure to aqueous DEA solution (as reported in the literature), were taken into account to mimic the experimental flux-data (taken from literature) using the present mathematical model. The results for the average pore-size and the geometric standard deviation, obtained in this manner, have been analyzed with respect to the reasonability of enlargement and broadening factors. Attempts at a broader validation of the developed model through experimental investigations and appropriate characterization techniques are currently on the move. Further work from this study should try to elucidate the effect of change in membrane hydrophobicity during the course of experimental operation and the relative importance of various parameters on this change.

Table A.2 Characteristic coefficients (Eq. A.10) Temperature (K)

A

B

C

D

298.15 303.15 308.15 313.15 318.15 323.15

21.00 21.00 21.00 21.01 21.02 21.02

 20.14  20.14  20.15  20.14  20.17  20.16

50.3650:36 49.56 48.76 47.85 47.17 46.24

 0.102  0.103  0.105  0.107  0.108  0.110

squared correlation coefficient of 0.996.   1816:4 μH2 O ðT Þ ¼ exp  13:1 kg m  1 s  1 T

Similarly, for aqueous DEA solution, the viscosity data  measurements (μL ) of Snijder et al. [46] were utilized to plot ln μL versus ð1=TÞ at different DEA concentrations, and the resulting slopes (S) and intercepts (I), reported in Table A.1, were then correlated (Eqs. (A.4) and (A.5)) as linear functions of the DEA concentration (cBL ). The ultimate correlation for the liquid-phase viscosity, as a function of temperature and DEA concentration, can be represented by Eq. (A.6). S ¼ 0:232cBL þ 1820:1

ðA:4Þ

I ¼  0:0003cBL  13:193

ðA:5Þ

 μL ðT; cBL Þ ¼ exp

Acknowledgments The authors would like to thank Dr. Paresh Chokshi (Department of Chemical Engineering, IIT Delhi) for helpful discussions and various clarifications.

S þI T



kg m  1 s  1

ðA:6Þ

3. Diffusivity correlations Using CO2–N2O analogy [47], the diffusivity of CO2 in aqueous DEA solution (DAL ) can be predicted with good accuracy (Eq. (A.7)). 0:8 DAL μ0:8 L ¼ DA  H2 O μH2 O

Appendix A. Physical properties 1. Density correlations The density data for aqueous DEA solutions reported by Han et al. [44], at different temperatures (T) and weight-fractions of DEA (wBL ) have been correlated as a second order polynomial in temperature (Eqs. (A.1) and (A.2)) by means of regression. ρH2 O ðT Þ ¼ 866:8 þ 1:202 T 0:003 T 2 kg m  3

ðA:1Þ

ρL ðT; wBL Þ ¼ ð0:002wBL  0:003ÞT 2 þ ð1:201  1:412wBL ÞT þ 878:47 þ 354:867wBL kg m

ðA:3Þ

3

ðA:2Þ

ðA:7Þ

The diffusivity of CO2 in water (DA  H2 O ) has been reported as Eq. (A.8) by Versteeg et al. [47].   2119 m2 s  1 ðA:8Þ DA  H2 O ¼ 2:35  10  6 exp  T The viscosities of water (Eq. (A.3)) and aqueous DEA solution (Eq. (A.6)) have been correlated to the temperature, previously in Appendix A.2. Hence, the diffusivity of CO2 in aqueous DEA solution (DAL ) can be easily calculated as follows. DAL ðT; cBL Þ ¼ 2:532   2121:96  0:186cBL þ 2:4  10  4 cBL m2 s  1  10  6 exp T ðA:9Þ

2. Viscosity correlations The viscosity data (μH2 O ) measurements for water at different temperatures (T) reported by Korson et al. [45], have been correlated with  an expression of the form given by Eq. (A.3). A plot of ln μH2 O versus ð1=TÞ was found to be a straight line with a Table A.1   Slopes and intercepts for ln μL versus ð1=TÞ Plots

4. Surface-tension correlations The surface-tension data (γ L ) measurements for aqueous DEA solutions reported by Vazquez et al. [48] have been correlated with DEA mole fraction (xBL ) and temperature (T) using a relationship of the form given by Eq. (A.10), and the coefficients shown in Table A.2. γ L ðT; xBL Þ ¼ A expðBxBL Þ þ C expðDxBL Þ mN m  1

DEA concentration (mol m  3 )

Slope (S)

Intercept (I)

10.5 1009.2 1991.8 4011.4

1809.8 2068.5 2285.7 2744.8

 13.101  13.609  13.940  14.488

ðA:10Þ

It is observed that the coefficients A and B are nearly constant in the temperature range of 25–50 1C, whereas the coefficients C and D vary linearly with temperature according to Eqs. (A.11) and (A.12). C ¼ 99:234  0:164T

ðA:11Þ

D ¼ 0:0004  0:0003T

ðA:12Þ

N. Goyal et al. / Journal of Membrane Science 474 (2015) 64–82

Nomenclature Variables   Shell-Side Cross-Sectional Area ðm2 Þ: π R2e  R2o Gas-Phase CO2 concentration at Fiber-Inlet ðmol m  3 Þ cAG Gas-Phase CO2 concentration on Shell-Side ðmol m  3 Þ cAGM CO2 Concentration in Gas-Filled Membrane Region ðmol m  3 Þ cAiGL CO2 Concentration at Gas–Membrane Interface ðmol m  3 Þ cAiLM CO2 Concentration at Membrane–Liquid Interface ðmol m  3 Þ cin Liquid-Phase CO2 Concentration at Fiber-Inlet AL ðmol m  3 Þ cAL Liquid-Phase CO2 Concentration in Fiber-Lumen ðmol m  3 Þ cAL Flow-Average Liquid-Phase CO2 Concentration in Fiber-Lumen ðmol m  3 Þ cALM CO2 Concentration in Liquid-Filled Membrane Region ðmol m  3 Þ cBiLM DEA Concentration at Membrane–Liquid Interface ðmol m  3 Þ cin Liquid-Phase DEA Concentration at Fiber-Inlet BL ðmol m  3 Þ cBL Liquid-Phase DEA Concentration in Fiber-Lumen ðmol m  3 Þ cBLM DEA Concentration in Liquid-Filled Membrane Region ðmol m  3 Þ ΔcDF Log-Mean Driving Force ðmol m  3 Þ cH2 O H2O Concentration in Liquid-Phase ðmol m  3 Þ dnAG Amount of CO2 transferred from Gas-Phase ðmol s  1 Þ DAG Molecular Diffusivity of CO2 in Gas-Phase ðm2 s  1 Þ DAGM Effective Diffusivity of CO2 in Gas-Filled Membrane Region ðm2 s  1 Þ DAK Knudsen Diffusivity of CO2 in Gas-Filled Membrane Region ðm2 s  1 Þ DAL Molecular Diffusivity of CO2 in Liquid-Phase ðm2 s  1 Þ DALM Effective Diffusivity of CO2 in Liquid-Filled Membrane Region ðm2 s  1 Þ DBL Molecular Diffusivity of DEA in Liquid-Phase ðm2 s  1 Þ DBLM Effective Diffusivity of DEA in Liquid-Filled Membrane Region ðm2 s  1 Þ HL Henry’s Constant for CO2-Aqueous DEA Solution System jCO2 Gas-Phase CO2 flux, Eq. (11) J CO2 Average CO2 flux ðmol m  2 s  1 Þ kO Overall Mass-Transfer Coefficient ðm s  1 Þ k1 , k2 , k3 Kinetic Parameters in Rate Expression, Eq. (4) lp Pore Length ðmÞ lw Wetted Pore Length (m) LF Fiber Length ðmÞ 1 MM CO2 Molecular Mass of CO2 ð0:044 kg mol Þ 1 MM H2 O Molecular Mass of H2O ð0:018 kg mol Þ 1 MM DEA Molecular Mass of DEA ð0:105 kg mol Þ nF Number of Fibers packed inside HFMM PC Capillary Pressure ðN m  2 Þ in PL Liquid-Phase Pressure at Fiber-Inlet ðN m  2 Þ PL Liquid-Phase Pressure in Fiber-Lumen ðN m  2 Þ P out Liquid-Phase Pressure at Fiber-Outlet ðN m  2 Þ L As cin AG

P min L P max L P in G PG P out G ΔP TMPD r r avg p r cp rp r max p r min p r AL r ALM r BL r BLM r CO2 , r DEA Ri Ro Re Rm R T vin G vG vavg L vL Vp Vw xBL x x z wBL wðr p Þ

81

Minimum-Allowable Liquid-Phase Pressure ðN m  2 Þ Maximum-Allowable Liquid-Phase Pressure ðN m  2 Þ Gas-Phase Pressure at Fiber-Inlet ðN m  2 Þ Gas-Phase Pressure on Shell-Side ðN m  2 Þ Gas-Phase Pressure at Fiber-Outlet ðN m  2 Þ Transmembrane Pressure Difference ðN m  2 Þ : P L  P G Radial Coordinate ðmÞ Average Pore-Size ðmÞ Critical Pore-Size ðmÞ Membrane Pore-Size ðmÞ Maximum Pore-Size ðmÞ Minimum Pore-Size ðmÞ Rate of Reaction for CO2 in Liquid-Phase ðmol m  3 s  1 Þ Rate of Reaction for CO2 in Liquid-Filled Membrane Region mol m  3 s  1 Rate of Reaction for DEA in liquid-Phase ðmol m  3 s  1 Þ Rate of Reaction for DEA in Liquid-Filled Membrane Region ðmol m  3 s  1 Þ Rate Expressions for CO2 and DEA, Eq. (4) Fiber Inner-Radius ðmÞ Fiber Outer-Radius ðmÞ Happel’s Free-Surface Radius ðmÞ Module Radius ðmÞ 1 Universal Gas-Constant 8:314 J mol K  1 Operating Temperature ðKÞ Gas-Phase Velocity at Fiber-Inlet ðm s  1 Þ Gas-Phase Velocity on Shell-Side ðm s  1 Þ Average Liquid-Phase Velocity at Fiber-Inlet ðm s  1 Þ Liquid-Phase Velocity in Fiber-Lumen ðm s  1 Þ Total Porous-Volume per unit total number of pores Liquid-Filled Porous-Volume per unit total number of pores Liquid-Phase DEA Mole Fraction Wetting-Ratio Volume-Averaged Wetting-Fraction Axial Coordinate ðmÞ Liquid-Phase DEA Mass Fraction Membrane Pore-Size Distribution ðm  1 Þ

Greek Letters γL ϵm ηR θ μH2 O μL ξ ρH2 O ρL σ τm ϕ

Surface Tension of Aqueous DEA Solution ðN m  1 Þ Membrane Porosity CO2 Removal Efficiency Contact Angle Molecular Viscosity of Water ðkg m  1 s  1 Þ Molecular Viscosity of Liquid Absorbent ðkg m  1 s  1 Þ Dummy Variable, Eq. (4) Density of Water ðkg m  3 Þ Density of Aqueous DEA Solution ðkg m  3 Þ Geometric Standard Deviation Membrane Tortuosity   n R2 Module Packing Fraction: 1  RF 2 o m

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