Modeling strategies of membrane contactors for post-combustion carbon capture: A critical comparative study

Chemical Engineering Science 87 (2013) 393–407

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Modeling strategies of membrane contactors for post-combustion carbon capture: A critical comparative study E. Chabanon, D. Roizard, E. Favre n LRGP-CNRS UPR 3348, Universite´ de Lorraine, 1, rue Grandville, 54001 Nancy, France

H I G H L I G H T S c c c c c

Simple models perform equally well than complex models when MEA is in excess. Significant MEA conversion required for model discrimination. Membrane mass transfer coefficient exerts a high parametric sensitivity. Prediction of membrane mass transfer coefficient is illusory. Effective membrane mass transfer coefficient may depend on operating conditions and axial position.

a r t i c l e i n f o

abstract

Article history: Received 7 July 2012 Received in revised form 7 September 2012 Accepted 17 September 2012 Available online 3 October 2012

Membrane contactors are considered as a promising process for intensification purposes of gas–liquid absorption units. CO2 post-combustion capture is one of the currently intensively investigated applications and experiments are most often performed on lab scale hollow fiber modules with a monoethanolamine (MEA) aqueous solution, usually in a large excess, as chemical solvent. Different mathematical models, showing a broad range of intrinsic complexity, have been already proposed in order to simulate or predict the separation performances of membrane contactors for this application. Unfortunately, no systematic comparison of the different modeling approaches has been performed yet. This study addresses this issue through a series of experiments performed on the two main types of hollow fiber membrane contactors (microporous PTFE and dense PMP skin composite membrane) under different sets of operating conditions. Four different types of models (constant overall mass transfer coefficient, 1D resistance in series, 1D and 2D convection–diffusion models) have been compared with the membrane mass transfer coefficient as the only adjustable parameter. It is shown that the different models lead to comparable predictions of the experimental results, with slightly similar membrane mass transfer coefficient values. This result addresses key questions in terms of strategy for model validation and with regard to the current trend of increasing model complexity. Interestingly, a different situation holds when MEA is significantly converted at the liquid outlet: in that case, a 1D model approach is required (variable mass transfer coefficient), and leads to results which are comparable to 2D models. Guidelines for a relevant model comparison strategy are finally proposed. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Absorption Mass transfer Membrane Contactor Carbon capture Separations

1. Introduction The control of CO2 anthropogenic emissions is currently one of the main challenges for environmental policies to limit global climate change (Davidson and Metz, 2005). Generally speaking, different strategies are potentially available to mitigate CO2 emissions: reduce energy consumption, improve energy efficiency, reduce the use of fossil fuels, promote renewable energies and capture CO2 for storage purposes (CCS for Carbon Capture and Storage)

n

Corresponding author. Tel.: þ33 383 175 390. E-mail address: [email protected] (E. Favre).

0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.09.011

(Bounaceur et al., 2006). For the latter, the development of capture processes which are efficient and not too expensive is obviously a key issue. Among the different capture integration possibilities (such as oxycombustion or precombustion for instance), the direct CO2 capture from flue gases, referred as post-combustion, is an attractive strategy because it applies in principle to any type of emission and allows the retrofit of current power or industrial plants. One difficulty of post-combustion capture is that, given the CO2 volume fraction (typically 4–15%) and total pressure of the flue gases (typically atmospheric pressure), CO2 partial pressure is low. In that case, gas–liquid absorption in alkanolamine solutions, typically a 30 wt% monoethanolamine (MEA) aqueous solution,

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is considered as the reference technology (Abu-Zahra et al., 2007; Mansourizadeh et al., 2010; Wang et al., 2010). Intensive research efforts are performed in order to minimize the capital and operating expenses of the carbon capture chain. For the latter, the specific energy requirement is a key issue; numerous strategies are explored in this direction based on new solvents or novel integrated processes. In terms of capital cost, packed columns are the reference equipment for absorption and stripping; the minimization of the size of the installations (occasionally referred as process intensification) is considered as a major target for research. To that respect, membranes contactors are one of the most promising intensification strategies, due to the increased gas–liquid interfacial area provided by the membranes (usually hollow fibers) (Chabanon et al., 2011; Keshavarz et al., 2008; Kumar et al., 2002). Additionally, the limitations of packed columns such as channelling, foaming and flooding are circumvented because no direct contact takes place between the gas and the liquid phase in a membrane contactor (Abu-Zahra et al., 2007; Mansourizadeh et al., 2010; Wang et al., 2010). Numerous studies are continuously reported on CO2 capture with membrane contactors in order to better evaluate the applicability, the performances and the limitations of this technology. The influence of membrane type (hydrophobic porous polymers such as PP, PVDF, PTFE, or composite membranes), module configuration (flat, spiral or hollow fiber modules) and operating conditions (gas and liquid velocities) is experimentally investigated and reported in a large numbers of publications (Chen et al., 2011; deMontigny et al., 2006; Kim and Yang, 2000; Kumar et al., 2002; Mavroudi et al., 2003; Nguyen et al., 2011; Qi and Cussler, 1985a and 1985b; Takahashi et al., 2011). In most cases, the possibility to fit the experimental data by theoretical models is also tested. The identification of the most efficient model strategy, which would ideally combine a rigorous approach without excessive complexity is obviously a key issue in this area. Similarly to the resolution of any chemical engineering problem, this represents an essential prerequisite for the prediction of the process performances under a given set of operating parameters and for scale-up studies. Unfortunately, despite the increasing numbers of publications dedicated to the modelling of membrane contactors for CO2 capture (Al-Marzouqi et al., 2008a, 2008b; Faiz et al., 2011; Khaisri et al., 2010; Lu et al., 2005; Malek et al., 1997; Qi and Cussler, 1985a, 1985b), the situation is still unclear and can be summarized as follows:  Publications most often report experimental results obtained on one type of membrane contactor, almost systematically at laboratory scale (mini hollow fiber modules).  A dedicated modeling approach is proposed and generally a good agreement with experimental data is shown.

 The previous conclusion is surprisingly obtained in several publications for very different modeling strategies, which range from a simple constant overall mass transfer coefficient K, to sophisticated 2D simulations (including the contribution of convection, diffusion and reaction phenomena). This study intends to develop a critical comparison of the different mathematical approaches which have been already proposed for CO2 absorption with membrane contactors. A series of experimental data covering the two main types of membranes (microporous hydrophobic and dense skin composite hollow fibers), operated under different conditions (solvent flowing in or out the fibers, variable gas and liquid velocities) has been achieved for that purpose. The prediction capacity of the major different modeling approaches, from the simplest (constant K) to the more elaborate ones (2D) is compared. Based on the literature review, the achievement of a perfect prediction (i.e. without any fitting parameter) remains unrealistic, whatever the model type. Consequently, a single adjustable parameter, namely the membrane mass transfer coefficient (km), has been taken for all the models tested. The key questions which will be addressed concern the differences in terms of prediction efficiency for the different types of models and the identification of the most relevant level of complexity, depending on membrane type and/or operating conditions.

2. Materials and methods 2.1. Membranes and modules Two different types of hollow fiber membranes, corresponding to the two major types of materials for membrane contactor applications, have been studied: one based on microporous polytetrafluoroethylene (PTFE) fibers and one based on composite fibers with a dense skin of polymethylpentene (PMP) coated on a microporous support in polypropylene (PP). Additionally, the influence of packing fraction has been studied with two different PTFE hollow fiber modules (showing a packing fraction of 0.13 and 0.59, respectively). The characteristics of the fibers and modules are summarized in Table 1. 2.2. Experimental setup Fig. 1 shows the experimental setup used for CO2 absorption experiments. CO2/N2 mixtures (5 or 15 vol% CO2) are prepared thanks to two mass flow meters (Model 5800S, Brooks). For the composite fibers (PMP on PP), the gas phase flows on the lumen side (‘‘liquid out’’ configuration) in order to put the dense skin into contact with the liquid phase. For the microporous PTFE

Table 1 Characteristics of the three different hollow fiber membrane materials and membrane contactors tested in this study. PTFE

PTFE

PMP

Hollow fiber membrane materials and structure Membrane type Microporous Polymer PTFE Fiber supplier Polymem (Toulouse, France)

Microporous PTFE Polymem (Toulouse, France)

Composite PMP (dense skin) coated over porous PP Membrana (OxyplusTM)

Membrane contactor modules Fiber external diameter, de (mm) Fiber internal diameter, di (mm) Fiber length, l (m) Fiber number (dimensionless) Specific interfacial area, a (m2/m3) Packing fraction (dimensionless) Operating conditions

870 430 0.30 119 1331 0.59 Liquid inside the fibers

380 200 0.24 210 1889 0.18 Liquid outside the fibers

870 430 0.30 27 302 0.13 Liquid inside the fibers

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Fig. 2. Evaluation of the measurement reproducibility of PTFE fibers. Comparison of the evolution of the CO2 capture ratio (Z) as a function of the gas velocity (ug). Two sequential measurements on fresh fibers (first and second set) are compared to the results obtained after vacuum drying (third set). wMEA ¼ 30 wt%, ul ¼1.0  10  2 m s  1, yin CO2 ¼ 15% vol. Fig. 1. Experimental setup used for the determination of membrane contactors CO2 absorption performances over time. A CO2/N2 gas mixture prepared by two mass controllers (MFC) flows in the membrane contactor. A MEA solution flows countercurrently. P stands for pressure gauge.

fibers, similarly to a large majority of studies on membrane contactors, the gas flows on the shell side (‘‘liquid in’’ configuration). A solution of high purity MEA (Sigma Aldrich Reagent grade, purity 499%) in distilled water (20 or 30 wt%) is prepared as a chemical solvent. The liquid phase flows in the module countercurrently to the gas phase. Liquid flowrate is controlled by a precision pump (HPLC type 515, Waters; 520S, Watson Marlow). An infra-red analyzer (MGA-3000 Multi-gas analyzer ADC) is used to continuously monitor the CO2 volume fraction in the gas phase at the inlet and outlet of the module thanks to a by-pass system. The CO2 capture ratio of the process is experimentally determined from the inlet and outlet gas flowrates and CO2 concentrations, according to

Z¼

in out out Q in g C CO2 ,g Q g C CO2 ,g in Q in g C CO2 ,g

ð1Þ

out the where Z is the CO2 capture ratio (dimensionless), Qin g and Qg in 3 1 inlet and outlet gas flowrate (m s ), respectively, and C CO2 ,g and C out CO2 ,g the CO2 concentration in the gas phase respectively at the inlet and outlet of the membrane contactor (mol m  3). Similarly to numerous studies on CO2 absorption by membrane contactors, experiments are performed with a large excess of MEA in the liquid phase, under steady state, isothermal conditions (ambient temperature) and atmospheric pressure at gas phase outlet. For each set of operating conditions, steady state hypothesis is experimentally checked by a constant CO2 volume fraction over time in the gas phase outlet of the module.

experimental results obtained just after the second set, when the membrane module is dried under vacuum before use. The comparison between the second and the third set enables the incidence of membrane wetting on data reproducibility to be estimated. More generally, it can be seen that a very good reproducibility is achieved with microporous PTFE fibers. The stability vs. time and the reproducibility of the performances under a given set of operating conditions is an indication that no rapid and massive wetting of the fibers occurred. This is an essential condition to undertake a comparison of the different models. Performance stability over time and reproducible mass transfer performances are indeed key postulates which greatly simplifies modeling studies. Moreover, each set of experiments are performed randomly, the operating conditions for the first trial being repeated at the end. This strategy enables to check that the changes of performances over time or hysteresis effects are negligible. The data reported in Fig. 2 show that no significant evolution of the separation performances is observed over the limited time scale of the experiments. Nevertheless, they cannot be simultaneously used to precisely evaluate the experimental reproducibility, because the capture ratio remains too close to 100%. Consequently, several set of operating conditions have been repeated, for the experiments performed under different operating conditions. Based on these data (shown after in Figs. 8 and 10), a 6% relative error is estimated. Additionally, dense skin composite fibers have been shown to offer an excellent resistance to wetting problems (Chabanon et al., 2011). No wetting phenomena occur on the short time scale used in this study (about 10 h). Consequently, an excellent stability over time is obtained. A similar experimental strategy to the one used for microporous PTFE fibers has been applied (the operating conditions of the first trial is repeated at the end of the series). For composite fibers, a 7% relative error is estimated (Fig. 9), similar to the relative error obtained on PTFE fibers.

2.3. Experimental protocol validation and reproducibility

3. Modeling approaches of membrane contactors

In order to evaluate the protocol and check the reproducibility, a series of CO2 absorption experiments is first repeated several times under the same operating conditions. The reproducibility of the experimental results for PTFE hollow fibers is shown in Fig. 2. Three sets of data are presented: the first one corresponds to the initial results obtained with the fresh membrane module; the second set is obtained under the same operating conditions as the first set but after a daily use under different operating conditions lasting 1 month; the last set presents the

The prediction of the separation performances as a function of operating conditions is obviously the major target of modeling studies for membrane contactor applications. For CO2 capture, this objective requires the development of a mass transfer model for CO2 absorption in a chemical solvent (MEA), based on a series of hypotheses. Depending on the assumptions, different set of equations, covering a broad range in terms of complexity have been already proposed for CO2 absorption by MEA in membrane contactors. Table 2 summarizes the main types of modeling strategies, which will be detailed in the

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Table 2 Overview of the different types of modeling strategies reported in the literature and the associated assumptions. Modeling strategy

Assumptions

References

Constant overall mass transfer coefficient (KOV) model

   

T, P, KOV, Qg, a constant Low C CO2,l Plug flow in the gas phase Adjusted parameter KOV

Bottino et al. (2008) Qi and Cussler (1985a, 1985b) Wickramasinghe et al. (1992)

One-dimensional (1D) model

    

T, kg, kl constants CMEA, DC, P variable Plug flow in the gas phase and the liquid phase Chemical reaction considered Adjusted parameter: km

Rode et al. (2012)

One-dimensional-two-dimensional (1D–2D) model

    

T, P, kg constant Plug flow in the gas phase Axial convection and radial diffusion in the liquid phase Chemical reaction considered Adjusted parameter: km

Dindore et al. (2005) Gong et al. (2006) Lu et al. (2005) Lu et al. (2007)

Two-dimensional (2D) model

 T, P constant  Axial convection and radial diffusion in the liquid phase and the gas phase

 Chemical reaction considered  Adjusted parameter: km

Al-Marzouqi et al. (2008a, 2008b) Khaisri et al. (2010) Kumar et al. (2002) Zhang et al. (2008)

Fig. 3. Schematic diagram of the membrane contactor used in the mathematical models.

following section and critically compared to experimental results afterwards. For the different modeling approaches, a cylindrical representation, shown in Fig. 3 is proposed, in order to reflect hollow fiber geometry. In terms of mass transfer representation, three different domains, shown in Fig. 4, can be considered: CO2 transfer from the bulk of the gas phase to the gas–membrane interface, CO2 transfer in the membrane to the liquid–membrane interface and finally CO2 absorption in the liquid phase, where the chemical reaction with MEA takes place. Each of the previous zones (gas, membrane and liquid) may contribute to a certain extent to mass transfer resistance. The evaluation of the effective gas–liquid mass transfer coefficient requires a set of hypotheses to be defined. This target can be achieved through different strategies. The different types of approaches proposed in previously reported studies are detailed hereafter. 3.1. Constant overall mass transfer coefficient (KOV) model The simplest expression for membrane contactor mass transfer modeling is based on a constant overall mass transfer coefficient KOV. This approach has been historically proposed in the first publications dedicated to membrane contactors by different authors (Bottino et al., 2008; Qi and Cussler, 1985a, 1985b; Wickramasinghe et al., 1992). It shows the advantage to include the different mass

Fig. 4. Schematic representation of mass transfer resistances for membrane contactor modeling.

transfer mechanisms into a single overall parameter (KOV) which enables process performances to be quickly and easily assessed. The approach is based on the following set of assumptions:

 Plug flow for the gas phase in the membrane contactor.  Constant gas velocity (ug).  Constant overall mass transfer coefficient KOV between the inlet and the outlet of the membrane module.

 Negligible CO2 concentration in the liquid phase. This assump-

 

tion can be considered as acceptable for a large excess of MEA in the liquid phase throughout the contactor. It offers the opportunity to simply express the local driving  force for CO2 as the CO2 concentration in the gas phase C CO2 ,g . Isothermal conditions. Total gas pressure is constant (negligible pressure drop).

According to the above set of assumptions, CO2 mass balance on the gas phase can be expressed as ug

dC CO2 ,g ¼ K OV a C CO2 dz

ð2Þ

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where ug is the interstitial gas velocity (m s  1), C CO2 ,g the CO2 concentration in the gas phase (mol m  3), KOV the overall mass transfer coefficient (m s  1) which is assumed to be constant, a the specific interfacial area (m2 m  3) and z the axial contactor coordinate (m). For a contactor of effective length L, the CO2 capture ratio can be easily expressed, after integration, as   K OV aL Z ¼ 1exp ð3Þ ug

397

total gas pressure (pressure drop) can also be taken into account. The set of equations which is solved is detailed hereafter:

 CO2 mass balance in the gas phase:   d Q g C CO2 ,g ¼ KaC CO2 ,g Sshell dz

ð5Þ

 Global mass balance in the gas phase: Q in g

  P in  Pg  g 1yin 1yCO2 CO2 ¼ Q g RT RT

ð6Þ

1

Interestingly, the product KOV a (s ) corresponds to the inverse of the CO2 transfer time in the absorption unit. This is a key process performance parameter, which is useful when it is desired to compare between different gas absorption technologies for instance. The term L/ug (s) is homogeneous to a gas–liquid contact time. The ratio between these two times is dimensionless and controls the process mass transfer performance. The constant overall mass transfer coefficient model (KOV) is simple to use, as soon as the evolution of the capture ratio (Z) with gas velocity (ug) is experimentally determined. In that case, KOV being the only unknown is used to fit the data.

The other modeling strategies systematically separately consider the three different mass transfer domains shown in Fig. 4 in order to determine the effective local mass transfer coefficient K. A resistance in series expression based on film theory is classically used in that case. For a flat geometry, the local overall mass transfer coefficient K can be obtained through ð4Þ

with kg, km and kl are the mass transfer coefficient in the gas phase, in the membrane and in the liquid phase respectively (m s  1), E the enhancement factor (dimensionless), m the CO2 gas–liquid partition coefficient (dimensionless). The calculation of the different mass transfer coefficients is performed based on classical correlations detailed in Appendix A. Each of the three terms on the right hand side of the above equation can be in principle evaluated. For instance, correlations making use of Sherwood number can be proposed for the local gas and liquid mass transfer coefficients kg and kl (Gabelman and Hwang, 1999; Lin et al., 2009). Similarly, the local enhancement factor E can be determined based on theoretical expressions making use of the Hatta number. If radial dispersion effects are neglected in the bulk gas and liquid zones, a one dimensional model can be proposed and has been recently investigated for CO2 absorption through a parametric study (Rode et al., 2012). More specifically, the key assumptions of the 1D model which is proposed hereafter are the following:

 Film theory (diffusion in boundary layer) applies in the liquid    

between CO2 and MEA:     d Q l C MEA,l ¼ 2d Q g C CO2 ,g

ð7Þ

In the lumen side, the pressure drop is calculated by the Hagen–Poiseuille correlation: 

DP dz

¼ r 2e



8m   1 d=r e Þ4 jv

ð8Þ

In the shell side, the pressure drop is expressed according to the Kozeny equation:

3.2. One-dimensional (1D) model

1 1 1 1 ¼ þ þ K kg km mEkl

 Stoichiometric constraint due to the chemical reaction

and gas phase. Membrane mass transfer (km) coefficient is constant. Plug flow of the bulk gas and liquid phase. Thermodynamic equilibrium at the gas–liquid interface. Isothermal conditions.

Thus, compared to the constant KOV approach, the 1D model clearly takes into account the evolution of the local mass transfer coefficients through the axial coordinate, and, from this, of K. Additionally, changes in gas velocity (due to CO2 absorption), and



DP dz

¼

4m j2   v r 2e 1j 2

ð9Þ

k ¼ 150 j4 314:44 j3 þ241:67 j2 83:039 j þ 15:97

ð10Þ

2

where Sshell is the module area without the fibers (m ), Pg the gas phase pressure (Pa), R the ideal gas constant (8.314 J mol  1 K  1), Qg and Ql the flowrate respectively of the gas phase and the liquid phase (m3 s  1),yCO2 the CO2 volume fraction in the gas phase (dimensionless), T the temperature of the process (K), CMEA,l the MEA concentration in the liquid phase (mol m  3), DP the pressure drop (Pa), m the viscosity (Pa s), v the superficial velocity (m s  1), re the external fiber radius (m), k the Kozeny coefficient (dimensionless), j the packing factor (dimensionless) defined by j ¼nfib(re/rshell)2 QUOTE , nfib the fiber number (dimensionless) and rshell the internal module radius (m). It should be noted that the choice of the most appropriate pressure drop equation for shell side hollow fiber packings can be tricky (Wu and Chen, 2000; Lively et al., 2011). Nevertheless, the pressure drop systematically remains below 50 mbar for all the experimental conditions tested. Consequently, the impact of the pressure drop estimation on the computation remains minor. The differential equation system is modified to be dimensionless thanks to four variables: C nCO2 ,g ¼ NUTg ¼

C CO2 ,g C in CO2 ,g

,

L kmax a , u0g

C nMEA,l ¼ zn ¼

C MEA,l C in CO2 ,g

,

P ng ¼

Pg P in g

,

z L

where NTUg is the number of transfer units (dimensionless), kmax the maximal mass transfer coefficient (m s  1) defined by kmax ¼ mEmax kl ¼ m Ha kl ¼ m

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C nMEA,l kr Dl

ð11Þ

where Emax is the maximal enhancement factor reached when E¼Ha, Ha the Hatta number (dimensionless), kl the liquid mass transfer coefficient (m s  1), kr the rate constant of the chemical reaction which occurs between CO2 and MEA (m3 mol  1 s  1), Dl the CO2 diffusion coefficient in the liquid phase (m2 s  1).

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Boundary conditions are expressed as (Fig. 5) n0 n0  At zn ¼ 0 : C CO ,g ¼ 1, P g ¼ 1. nL n  At z ¼ NUTg : C MEA,l ¼ 830ðMEA 30 wt%Þ. 2

The dimensionless differential equation system with the appropriate boundary conditions is solved using the Matlab software, which uses the preprogrammed bvp4c method to assess numerical solutions. The membrane mass transfer coefficient km is used as the only adjustable parameter. 3.3. One-dimensional-two-dimensional (1D–2D) model

resistance; consequently, a more detailed estimation of the CO2 (and MEA) concentration profile in this zone may be of value for a better description of the system. These two observations suggest a 1D–2D approach to be proposed, which has been indeed tested by several authors (Dindore et al., 2005; Gong et al., 2006; Lu et al., 2005, 2007). A typical set of assumptions for the 1D–2D model is the following:

 Axial convection and radial diffusion are considered in the liquid phase.

 A constant velocity and a perfect plug flow is postulated in the gas phase.

 Membrane mass transfer (km) coefficient is constant.  Thermodynamic equilibrium conditions apply at the gas– liquid interface.

An alternative approach to the film model, based on the resolution of the convection–diffusion–reaction equations, can be proposed in order to estimate the gas and liquid mass transfer resistances. In that case, the general form of differential mass balance equations is solved and no correlation is required to evaluate the liquid mass transfer coefficient. More specifically, based on the general statement that mass transfer resistance in the gas phase is almost negligible, a 1D assumption is often proposed for this domain. At the contrary, the liquid phase is often considered to show a significant mass transfer

 Isothermal conditions.  Constant total pressure of the gas phase. The set of differential mass balance equations system for CO2 and MEA in the liquid phase are expressed as  

@C CO2 ,l @C CO2 ,l 1@ r ¼ Dl þRCO2 uz,l ð12Þ r @r @z @r uz,l

 

@C MEA,l @C 1@ r MEA,l þ 2RCO2 ¼ DMEA r @r @z @r

ð13Þ

with the stoichiometric constraint due to the chemical reaction between CO2 and MEA: uz,l

@C CO2 ,g @C MEA,l ¼ 2ug @z @z

ð14Þ

where C CO2 ,l is the CO2 concentration in the liquid phase (mol m  3), RCO2 the reaction rate of CO2 with MEA (mol m  3 s  1) defined by RCO2 ¼ kr C CO2 ,l C MEA,l , DMEA the MEA diffusion coefficient in the liquid phase (m2 s  1), uz,l the interstitial liquid velocity at the axial coordinate z (m s  1) and r the radial coordinate (m). Boundary conditions are detailed in Fig. 6. It can be noticed that the transmembrane flux corresponds to one of the boundary conditions: Dl

Fig. 5. Boundary conditions of the 1D model.

  @C CO2 ,l ¼ kext C CO2 ,g C CO2 ,l,int @r

ð15Þ

where C CO2,l,int is the CO2 concentration at the liquid–membrane interface (mol m  3), kext the external mass transfer coefficient (m s  1)

Fig. 6. Boundary conditions of the 1D–2D model.

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defined by 1/kext ¼1/kg þ1/km QUOTE and kg it the gas mass transfer coefficient (m s  1). The differential equation system is solved using the Comsol software, which uses finite element method to estimate numerical solution. Similarly to the 1D model, membrane mass transfer coefficient km is assumed to be constant and is the only adjustable parameter. 3.4. Two-dimensional (2D) model A more general two-dimensional approach has been also proposed by different authors for membrane contactor modeling purposes (Al-Marzouqi et al., 2008a, 2008b; Khaisri et al., 2010; Kumar et al., 2002; Zhang et al., 2008). This strategy leads to a more complex set of equations than the previous models. Convection and diffusion contributions are taken into account in the gas and liquid phases. The corresponding assumptions are as follows:

Fig. 7. Boundary conditions of the 2D model.

 Membrane mass transfer (km) coefficient is assumed to be constant.

 Thermodynamic equilibrium conditions apply at the gas–  

liquid interface. Isothermal conditions. Constant total pressure of the gas phase.

In this study a 2D model is tested which considers MEA concentration gradient in the liquid phase, CO2 gradient concentration in the two phases, the chemical reaction between CO2 and MEA in the liquid phase, and the decrease of the gas velocity due to CO2 absorption. The corresponding differential equation system is as follows:

 CO2 and MEA mass balances in the liquid phase are similar to the 1D–2D model (Eqs. (12) and (13)).

 Differential CO2 mass balance in the gas phase postulates convection in the axial direction and diffusion in the radial direction:  

@C CO2 ,g @C CO2 ,g 1@ r ¼ Dg ð16Þ uz,g r @r @z @r

 CO2 transfer in the membrane is assumed to result from a strict diffusional contribution. Consequently, mass balance in the membrane can be expressed through Fick’s law:  

@C CO2 ,m 1@ r ¼0 ð17Þ Dmeff ¼ r @r @r with Dmeff ¼(Dme/t)km ¼(Dmeff/d)Dg QUOTE QUOTE the molecular CO2 diffusion coefficient in the gas phase (m2 s  1) and Dmeff the effective CO2 diffusion coefficient in the membrane (m2 s  1), which includes the porosity and tortuosity effects. Based on the Dmeff value (obtained by curve fit), the km value can be simply calculated through km ¼Dmeff/d with Dmeff ¼Dme/t. Fig. 7 details the boundary conditions and the differential equations, which are solved using the Comsol software. Again, the membrane mass transfer coefficient km is taken as the only adjustable parameter the adjust parameter.

section. The objective of the experiments is to investigate the influence of different variables, in order to evaluate the potential difference in terms of description between the four types of models described before. The variables which have been experimentally tested are as follows:  Membrane type (microporous PTFE fiber vs. dense PMP skin composite fiber).  Module packing fraction (comparison between 0.13 and 0.59 fiber fraction j for PTFE modules).  Gas and liquid velocities (ug and ul).  Flow configuration (liquid flows out the fibers for PMP membranes, while it flows inside the fibers for PTFE membranes).  Solvent composition (20 vs. 30 wt% MEA solutions, wMEA). Among these different parameters the gas and liquid interstitial velocities (ug and ul) are clearly most often tested in other studies. MEA mass fraction (wMEA) and module packing factor (j) have been only occasionally investigated. Fig. 8 shows the experimental CO2 capture ratio obtained for PTFE fibers as a function of gas velocity for two different liquid velocities (10  2 and 5  10  2 m s  1). The different curves correspond to the data fit obtained by the 4 different models based on km (or KOV) as the only adjustable parameter. Generally speaking a good agreement between experimental and numerical results is observed and the four models perform almost equally well, whatever their level of intrinsic complexity. The values of the mass transfer coefficients obtained through curve fit for each model are indicated in Table 3. The increase of the liquid velocity by a factor five induces an increase of the overall mass transfer coefficient KOV. This trend has been already reported and is logically interpreted to reflect the increase of the liquid mass transfer coefficient kl consecutive to a larger Reynolds number. A larger kl value increases K (Eq. (4)) to a certain extent, and thus KOV, depending on the respective contributions of the local mass transfer resistances. The different values obtained for km with the 1D, 1D–2D and 2D models are clearly more difficult to explain. According to km classical expression,

4. Results and discussion km ¼ 4.1. Models comparison A series of CO2 absorption experiments has been performed according to the protocol detailed in the materials and method

Dm e

td

ð18Þ

where Dm is the CO2 diffusion coefficient in the membrane, e the membrane porosity (dimensionless), t membrane tortuosity (dimensionless) and d membrane thickness (m); one would

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ideally expect the same km data to fit the experimental results for a given membrane. km changes slightly from one model to the other and it systematically increases when liquid velocity increases. This behavior is puzzling. It suggests that, similarly to numerous process modeling situations, one adjustable parameter is necessarily required in order to achieve a precise description of the experimental results. At this stage, the variations of km could be interpreted as a result of the uncertainties on the numerous numerical values for the parameters included in the calculations;

Fig. 8. Experimental results and curve fit with the different models for two liquid velocities with PTFE fibers (j ¼ 0.59, wMEA ¼ 30 wt%, yin CO2 ¼ 15 vol%). D experiments at ul ¼10  2 m s  1, D experiments at ul ¼ 5  10  2 m s  1; simulations results with KOV model (full line), with 1D model (large dashed line), with the 1D–2D model (dashed line), with the 2D model (dotted line).

Table 3 Values of the fitted mass transfer coefficients for two different liquid velocities (PTFE hollow fiber module).

KOV 1D model 1D–2D model 2D model

ul ¼ 10  2 m s  1

ul ¼5  10  2 m s  1

KOV ¼ 8.10  10  4m s  1 km ¼ 7.28  10  4m s  1 km ¼ 9.80  10  4m s  1 km ¼ 1.05  10  3m s  1

KOV ¼1.42  10  3m s  1 km ¼2.18  10  3m s  1 km ¼1.88  10  3m s  1 km ¼ 3.00  10  3m s  1

km could play the role of a lumped parameter which smoothes the prediction errors of the all other parameters included in the model. Alternatively, another completely different hypothesis can be proposed: km should not be considered as a constant for a given membrane, but could depend on the operating conditions. These two different interpretations will be discussed further. Fig. 9 shows the comparison between simulated results and experimental results for PMP fibers with two different MEA mass fractions (20 and 30 wt%). A very good agreement is obtained again between fitted and experimental results for the two different MEA mass fractions. Similar to the results shown in Fig. 8 the four different models globally perform equally well. The values of the mass transfer coefficients obtained through curve fit for each model are reported in Table 4. The same general trend as the one observed for Table 3 is obtained: first, KOV increases with an increase of MEA mass fraction. This could reflect an increase of kl through the enhancement factor E (Eq. (11)). For the 1D, 1D–2D and 2D models however, MEA concentration and liquid viscosity are explicitly taken into account in the equations. As a consequence, one would expect the fitted km values to stay unchanged when MEA concentration increases. Surprisingly, the km values for 30% wt. MEA are systematically larger than the one obtained with 20% wt. MEA. The two hypotheses proposed before for Table 3 data apply equally well: km could either be an error compensation parameter or could alternatively be affected by the operating conditions. The experimental results, for two different packing factors (j ¼0.13 and j ¼0.59) and two liquid velocities (ul ¼1.7  10  2 m s  1 and ul ¼1.0  10  2 m s  1 respectively), with PTFE fiber modules are presented in Fig. 10 and the values of the fitted parameters are reported in Table 5. Just as in the last two cases, Table 4 Values of the fitted mass transfer coefficients for two different MEA mass fractions (PMP hollow fiber module).

KOV 1D model 1D–2D model 2D model

wMEA ¼ 20 wt%

wMEA ¼30 wt%

KOV ¼ 5.14  10  4m s  1 km ¼ 7.11  10  4m s  1 km ¼ 6.97  10  4m s  1 km ¼ 7.33  10  4m s  1

KOV ¼ 6.82  10  4m s  1 km ¼1.07  10  3m s  1 km ¼1.01  10  3m s  1 km ¼1.01  10  3m s  1

Fig. 9. Experimental results and curve fit with the different models for two MEA weight fractions with PMP fiber(j ¼ 0.18, ul ¼ 0.5  10  2 m s  1, yin CO2 ¼5 vol%). & experiments at wMEA ¼20 wt%, Kexperiments at wMEA ¼30 wt%; simulations results with the KOV model (full line), with the 1D model (large dashed line), with the 1D–2D model (dashed line), with the 2D model (dotted line).

E. Chabanon et al. / Chemical Engineering Science 87 (2013) 393–407

experimental results and curve fits show a good agreement. No significant difference is observed between the efficiency of description for the different models. The larger packing ratio increases the specific interfacial area (a). Consequently CO2 capture efficiency is greatly improved for a given set of operating conditions. At this point of the study, two first conclusions can be drawn as follows: (1) Whatever the model type and intrinsic complexity, the use of one adjustable parameter, namely the membrane mass transfer coefficient km, enables a globally similar fit of the numerical results for the various process or operating parameters investigated in this study (membrane type, module packing fraction, gas and liquid flowrates, solvent composition). (2) Each model leads to a fitted km value per set of data which is numerically different from the others, but in the same order of magnitude. These two, somehow surprising conclusions are critically discussed in the next section. 4.2. Discussion 4.2.1. What is the relevant complexity level of models? One of the objectives of this study was to evaluate the required level of complexity for modeling CO2 absorption in MEA thanks to membrane contactors. The fact that all four types of models lead to a similar fit efficiency, as soon as one adjustable parameter is allowed, logically suggests to make use of the simplest approach, namely the constant KOV model. This guideline appears to be in strong contrast to the current trend in this field, where models of

401

increasing complexity are continuously proposed. Nevertheless, this peculiar situation can be explained when the experimental CO2 absorption protocol, which is classically used in most studies, is critically reconsidered. The constant KOV model indeed predicts an exponential CO2 concentration profile in the gas phase; along the axial coordinate (Eq. (2)). Fig. 11a shows that this condition holds when fresh MEA is used at the membrane contactor inlet and when a low conversion is obtained at the outlet (i.e. MEA is in large excess). Consequently, the constant KOV model logically offers a good description (Figs. 8–10). These experimental conditions are most commonly chosen in CO2 absorption investigations and in this study for sake of simplicity. In that case, the CO2 concentration in the liquid phase can be considered to be zero. Additionally, the large excess of MEA results in a high and globally constant enhancement factor (E). As a consequence, the local liquid mass transfer coefficient kl can be considered as constant along the axial coordinate. In summary, when the classical experimental protocol is used, all models logically provide a good fit of the results, because the system shows a constant overall mass transfer coefficient. This experimental choice is obviously not adequate when it is aimed to estimate model relevancy and robustness. Moreover, this experimental situation is very different from industrial CO2 capture conditions, which are based on an internal solvent recycling loop. In that case, MEA is no more fresh, but partly converted at the inlet of the absorption unit, and almost completely converted at the outlet. More specifically, the

Fig. 10. Experimental results and curve fit with the different models for two packing factor (0.13 and 0.59) with PTFE fibers. For j ¼ 0.13: ul ¼ 1.7  10  2 m s  1, and for j ¼ 0.59: ul ¼ 1.0  10  2 m s  1, wMEA ¼30 wt%, yin CO2 ¼ 5% vol. ~ experiments for j ¼ 0.59, D experiments for j ¼ 0.13; simulations results with KOV model (full line), with 1D model (large dashed line), with the 1D–2D model (dashed line), with the 2D model (dotted line).

Table 5 Values of the fitted mass transfer coefficients for two different module packing fractions (PTFE hollow fiber modules).

KOV 1D model 1D–2D model 2D model

u ¼ 0.13

u ¼ 0.59

KOV ¼ 4.05  10  4m s  1 km ¼2.55  10  4m s  1 km ¼3.23  10  5m s  1 km ¼ 9.00  10  5m s  1

KOV ¼8.10  10  4m s  1 km ¼7.28  10  4m s  1 km ¼9.80  10  4m s  1 km ¼1.05  10  3m s  1

Fig. 11. Evolution of the CO2 concentration profile in the gas phase as a function of axial coordinate. 1D model (full line), 2D model (dashed line): (a) Fresh MEA, ain ¼ 0.000, aout o0.200 and (b) loaded MEA, ain ¼ 0.242, aout ¼0.485.

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recommended MEA loadings are 0.242 at the inlet and 0.485 at the outlet (Abu-Zahra et al., 2007). When these conditions are used, the local mass transfer coefficient in the liquid phase kl significantly changes along the axial direction. CO2 concentration profile in the gas phase is no longer of exponential type (Fig. 11b). Consequently, the constant KOV model is no more adequate and more sophisticated models are required. Thus, one major conclusion of this study is to recommend experimental conditions to be strongly reconsidered when a dedicated model is proposed. An internal MEA recycling loop should be systematically applied, in order to better reproduce the industrial context but also to offer model discrimination possibilities. Unfortunately, we notice that the number of studies in this field which correspond to this framework is scarce.

4.2.2. Toward completely predictive models? One of the major conclusion of this study, the globally similar performances of models of increasing complexity has been tentatively explained in the previous section. The second important outcome of the model comparison concerns the good prediction efficiency of the process performances which can be obtained providing that one adjustable parameter is used. A similar conclusion has been reported by numerous authors (Boributh et al., 2011; Bottino et al., 2008; Lu et al., 2007; Reza-Sohrabi et al., 2011), based on a large variety of models. The membrane mass transfer coefficient, km, has been taken as the adjustable parameter, similarly to the majority of studies in this field. This situation logically addresses the question of the possibilities of prediction of km, in order to build up a completely predictive approach. It is first interesting to notice that the effective values of the membrane mass transfer coefficient km reported in previous studies cover almost four orders of magnitude. A series of data are shown in Fig. 12, obtained by different authors on different membrane materials. The km data which have been obtained by curve fit in this study have been added for comparison purposes. The prediction of km addresses the challenge of prediction of the permeability of a porous or composite medium. This topic has been abundantly discussed in different fields, including catalysts, filtration, reactors and natural environments such as soils for instance. For

membrane application, the evaluation of km is almost systematically performed based on Eq. (18) (Gabelman and Hwang, 1999). We want to stress the general difficulty to precisely predict km through this framework. Three major limitations have to be noticed in terms of prediction accuracy: (i) First, Dm, the gas diffusion coefficient in the porous membrane material has to be precisely estimated. Most authors perform this based on the average pore diameter which is supposed to govern the type of gas diffusion regime. However, the pore size distribution should imperatively be taken into account in order to obtain reasonable estimate of Dm. This, usually unknown function, radically changes the prediction of the membrane mass transfer coefficient km (Lin et al., 2009). Three different pore size distributions are shown in Fig. 13, for illustrative purposes. The following expression has been used (Phattarawik et al., 2003): " f i ¼ exp 

1 SD2log

ln



dpore dpore,av

2 !# ð19Þ

where SDlog is the standard deviation (mm), dpore the pore diameter (mm) and dpore,av the average pore diameter (mm). This function has been reported to give an acceptable representation the pore diameter distribution of microporous membranes. Three different average pore diameters (0.1, 0.2 and 0.5 mm) are shown, with a SDlog value of 1.0 mm. If the average diameter of 0.5 mm is considered, the curve shows a significant quantity of pore diameter between two and 10 times higher than 0.5 mm. Thus, the gas flow in the pore will be a Knusdsen flow or a molecular flow or a combination of the both flow which is really difficult to describe. Considering a mean free path of CO2, l ¼0.55 mm, in a gas phase at 295.15 K, the gas flow in the membrane is as follows: (1) Knudsen type when dpore is lower than the mean free path (0.55 mm), (2) molecular type when dpore is 10 times higher than the mean free path (5.5 mm), and (3) intermediate when dpore is between one and 10 times the mean free path. Hence, in case of PTFE or PP hollow fibers for which the average pore diameter is generally 1.0 mm, the gas flow in

Fig. 12. Order of magnitude of membrane mass transfer coefficient reported in the literature. Shaded area shows the range of km data obtained in this study by curve fit. Atchariyawut et al. (2006), Khaisri et al. (2009) and Matsumoto et al. (1995)

E. Chabanon et al. / Chemical Engineering Science 87 (2013) 393–407

403

Fig. 13. Examples of pore size distribution obtained by Eq. (19) for three different average pore diameters with SDlog ¼ 1 mm.

the membrane pore is a complex combination of Knudsen and molecular flow. (ii) Second, the tortuosity factor has also to be precisely known. t Is however essentially an empirical parameter. A very detailed knowledge of the geometrical characteristics of the membrane would be needed in order to correctly predict this parameter. Depending on the membrane structure, tortuosity factors ranging between 1 and 10 can be obtained (Cussler, 1997). For straight uniform pores aligned perpendicular to the membrane cross section area, a value of 1 can be assumed. Microscopic observations of microporous materials used for membrane contactors applications show a wide diversity of topological structures. It appears to be extremely difficult to correctly predict tortuosity. As a consequence, a one order of magnitude error on km prediction can result from this sole parameter. In some cases, the tortuosity can be estimated through the relation t ¼1/e. Nevertheless, detailed studies on microporous membranes show that more complex expressions are often required (Iversen et al., 1997). (iii) Last but not least, Eq. (18) assumes the pores to be filled by the gas phase only. This non-wetted pore hypothesis has been recently reconsidered in several publications (Chabanon et al., 2011; Keshavarz et al., 2008; Ozturk and Hughes, 2012; Rangwala, 1996; Wang et al., 2005). The partial wetting of the membrane is expressed by a wetting fraction b. If this parameter is known, the effective membrane mass transfer coefficient can be in principle estimated. Similarly to mass transfer problems in heterogeneous porous media, two boundary situations can be proposed for that purpose: a parallel structure (only completely wetted or non wetted pores), or a series structure (uniformly partially wetted pores). The second type of structure is most often proposed in membrane contactor publications based on the following expression: 1 ð1bÞtd btd ¼ þ km eDg emEDl

ð20Þ

where b is the wetting fraction (dimensionless), t the membrane tortuosity (dimensionless)), d the membrane thickness (m), e the

membrane porosity (dimensionless)), and Dg and Dl the gas and the liquid diffusion coefficient (m2 s  1), respectively. The very high sensitivity of the membrane mass transfer coefficient to a small change of the wetting fraction has to be noticed. An example is shown in Fig. 14, where the prediction of the series and parallel wetting situation are compared. CO2 gas diffusion coefficient is supposed to be Dg ¼10  5 m2 s  1, the CO2 liquid diffusion coefficient Dl ¼10  9 m2 s  1, partition coefficient is m¼0.792 and the enhancement factor is E¼1. The fibers geometrical parameter is e/dt ¼10  4 m  1. However, it should be pointed out that probably neither of these approaches is correct. The porous membranes used for membrane contactors applications usually have a wide range of pore sizes. It is reasonable to conclude that capillary action will act on the smallest pores first, thereby causing these to wet out. Thus the completely wet or non-wetted pores assumption is incorrect and the ‘‘uniformly’’ partially wetted pores assumption, proposed by several authors, is also incorrect. It is likely that ‘‘some’’ of the pores will be completely wet (the smallest ones in the distribution) while others are still not wetted (the largest ones in the distribution). In summary, based on the series of previous key parameters which can hardly be precisely estimated, the prediction of the effective km value for a given membrane material appears illusory. This matter of fact suggests to maintain one adjustable parameter for membrane contactors modeling purposes. Given the relatively strong scientific feedback that has been accumulated for gas and liquid mass transfer coefficients, it seems judicious to keep membrane mass transfer coefficient as the fitting variable. This situation should not be considered as too dramatic. It is indeed quite comparable to the prediction possibilities offered by models in several chemical engineering operations such as distillation, gas–liquid absorption in packed columns, heat exchangers or chemical reactors. For these different processes, one (and only one) adjustable parameter is usually left in the model equations set: effective interfacial area in packed columns, surface roughness or fouling factor for heat exchangers, order of the chemical reaction in catalytic systems. However, this (over)simplified analysis of the current situation should not dissuade efforts to improve the prediction efficiency of membrane mass transfer coefficient. Nevertheless, we estimate that, given

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Fig. 14. Influence of wetting fraction on membrane mass transfer coefficient for a series (continuous line) and parallel (dotted line) pore wetting situation. Data used for the calculations: Dg ¼10  5 m2 s  1, Dl ¼ 10  9 m2 s  1, m¼ 0.792, E¼ 1, e/dt ¼ 10  4 m  1 QUOTE .

the actual state of knowledge on most membrane materials used for CO2 absorption, km probably represents the major bottleneck for a completely prediction of membrane contactor performances to be achievable. Besides the difficulty to predict km, we discuss hereafter an implicit hypothesis of all the publications reported in the field of CO2 absorption by membrane contactors.

4.2.3. The constant km hypothesis reconsidered The model comparison study reported in Figs. 8–10 and Tables 3–5 has shown that, while a very good description of the experimental results is obtained by all four equations set; different fitted km values are required for each model (line to line changes in Tables 3–5). This has been interpreted as the result of differences between the simplifying hypotheses required for each model, km possibly playing an error compensation role. Nevertheless, it has also to be noticed that, for a given model, the fitted km value changes when process and/or operating parameters are changed (i.e. column to column changes for the same line in Tables 3–5). This observation is puzzling. According to Eq. (18), one would logically expect km value to depend only on the membrane type, possibly on the model, but systematically to be independent on operating conditions. Moreover, the experimental reproducibility of the experiments for a given set of operating parameters has been carefully checked, and no evolution with time (e.g. a gradual and irreversible wetting of the membrane) has been observed. The changes observed for km for the same membrane and same model could be attributed to some statistical scatter of the data. Nevertheless, the amplitude of the changes as a function of the operating conditions (a factor 3–10 in some cases) seems to significantly overwhelm the experimental error. An alternative interpretation analysis of this observation can be attempted; given the extreme sensitivity of km to wetting (Fig. 14), we address here the hypothesis of a non constancy of km, even when no irreversible changes of the membrane properties take place. The basic picture which would support this hypothesis is sketched in Fig. 15. As soon as one of the operating conditions is modified (e.g. gas or liquid flowrate), the axial pressure drop and transmembrane pressure will change. It might be, in that case, that a minor and local partial wetting phenomenon takes place, depending on the local transmembrane pressure and pore size

Fig. 15. Schematic representation of the gas and liquid axial pressure drop and the corresponding transmembrane pressure for a membrane contactor under steady state absorption conditions. Hypothetical local wetting conditions of the porous membrane have been added for illustrative purposes. Any change in operating conditions (gas and/or liquid velocity) will modify the transmembrane pressure and possibly the wetting distribution.

distribution. Such a moderate, localized and reversible liquid intrusion would possibly quantitatively affect the effective membrane mass transfer coefficient km. Should this hypothetical mechanism be confirmed (for instance by in situ membrane investigation techniques such as NMR or tomography applied on membrane contactors under operation), the effective km value will depend not only on the membrane material, but also on the set of operating conditions, with changes on local km value on the axial direction. Thus, for microporous fibers, a small variation of the MEA mass fraction in the liquid phase can induce some variation of the local km value which is due to a variation, among others, of the contact angle and the surface tension. The confirmation or invalidation of this hypothesis is of major importance for membrane contactors modeling strategies. To our knowledge however, it has never been proposed neither investigated. A systematic analysis of the quantitative changes of km as a function of the operating parameters should be performed for that purpose. In place of the quest of a constant representative km value,

E. Chabanon et al. / Chemical Engineering Science 87 (2013) 393–407

the previous vision would suggest correlations for km prediction to be developed. Coming back to the gas–liquid absorption process, this modeling strategy would correspond to the estimation of the effective gas–liquid interfacial area through correlations, such as the ones used for packed columns.

5. Conclusion This study intended to achieve a critical comparative analysis of modeling strategies for CO2 capture by absorption in MEA thanks to membrane contactors. The conclusions can be summarized as follows: (i) When a classical CO2 absorption experimental protocol is used, the different models which have been compared perform equally well in terms of fit efficiency, whatever their intrinsic level of complexity. In all cases, the membrane mass transfer coefficient (km) is the only parameter used to fit experimental results. This result suggests to strongly modify the classical practice in this field when it is aimed to evaluate the robustness of a given model or to achieve discrimination among different models. The classical operating conditions are quite far from those applied at industrial scale. A significant MEA conversion at the inlet and an almost complete conversion at the module outlet (in order to cope solvent recycling) are strongly recommended for a relevant evaluation of model efficiency. More specifically, the classical constant KOV model does not apply anymore in that case. (ii) Globally, the fitted km values obtained by the different models, although not perfectly equal, are in the same order of magnitude and in the range of effective km data previously reported. (iii) The predictions of a given model strongly depend on km value; a strong parametric sensitivity on km is likely. A precise prediction of km would thus be of major interest in order to achieve a completely predictive approach. (iv) Given the large uncertainties on pore size distribution, tortuosity and membrane partial wetting, the precise prediction of km seems to be illusory, especially for microporous membranes. It is recommended to maintain this key variable as an adjustable parameter in modeling strategies. (v) The question of a km value depending both on operating conditions and more specifically on axial position, even for stable membrane performances over time (i.e. no irreversible wetting phenomenon) is addressed. This hypothesis is of importance for a relevant and representative modeling approach to be built. Experimental efforts should be provided in this direction in order to validate or invalidate the hypothesis of a constant km value for a given membrane material, which, to our knowledge, has never rigorously addressed.

Nomenclature a C CO2 ,g C CO2 ,l C CO2 ,m CMEA,l dh Dg Dl Dm Dmeff DMEA E

gas–liquid interfacial area (m2 m  3) CO2 concentration in the gas phase (mol m  3) CO2 concentration in the liquid phase (mol m  3) CO2 concentration in the membrane phase (mol m  3) MEA concentration in the liquid phase (mol m  3) hydraulic diameter (m) CO2 gas diffusion coefficient (m2 s  1) CO2 liquid diffusion coefficient (m2 s  1) CO2 membrane diffusion coefficient (m2 s  1) effective CO2 membrane diffusion coefficient (m2 s  1) MEA liquid diffusion coefficient (m2 s  1) enhancement factor (dimensionless)

Emax EN Gz Ha K KOV kext kg kl km kmax kr L m nfib NTUg P DP Qg Ql r re ri rshell R RCO2 Sshell Sh Shlim T u v wMEA yCO2 z Z

405

maximal enhancement factor (dimensionless) limit enhancement factor (dimensionless) Graetz number (dimensionless) Hatta number (dimensionless) local overall mass transfer coefficient (m s  1) constant overall mass transfer coefficient (m s  1) external mass transfer coefficient (m s  1) gas mass transfer coefficient (m s  1) liquid mass transfer coefficient (m s  1) membrane mass transfer coefficient (m s  1) maximal mass transfer coefficient (m s  1) rate constant (m3 mol  1 s  1) fiber length (m) partition coefficient (dimensionless) fiber number (dimensionless) theoretical unit number (dimensionless) pressure (Pa) pressure drop (Pa) gas flowrate (m3 s  1) liquid flowrate (m3 s  1) radial contactor coordinate (m) external fiber radius (m) internal fiber radius (m) internal module radius (m) perfect gas constant (8.314 J mol  1 K  1) reaction rate (mol m  3 s  1) module area without fiber (m2) Sherwood number (dimensionless) limit Sherwood number (dimensionless) temperature (K) interstitial velocity (m s  1) superficial velocity (m s  1) MEA mass fraction (dimensionless) CO2 volume fraction (dimensionless) axial contactor coordinate (m) active fiber length (i.e. module length) (m)

Greek letters

a b d

e Z k j m t

wetting pore fraction (dimensionless) membrane porosity (dimensionless) membrane thickness (m) membrane porosity (dimensionless) CO2 capture efficiency (dimensionless) Kozeny coefficient (dimensionless) module packing factor (volume fraction occupied by the fiber bundle) (dimensionless) viscosity (Pa s) membrane tortuosity (dimensionless)

Acknowledgments This study has been partly funded thanks to the European Commission Seventh Framework Program Integrated Project CESAR (FP7/2007-2011, Grant ]213569) and the ANR (Agence Nationale de la Recherche) Grant CICADI. The useful comments to this manuscript proposed by the reviewers are gratefully acknowledged.

Appendix A. Calculation of the overall mass transfer coefficient (K) The overall mass transfer coefficient is calculated according to a series of equations already reported in the literature (Gabelman

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and Hwang, 1999; Rode et al., 2012; Versteeg and van Swaaij, 1988). Given the laminar condition which prevails in the both side of the membrane contactor, the classical correlations for circular tubes based on the Graetz number and the Sherwood number can be used. The influence of the chemical reaction on the mass transfer coefficient is taken into account thanks to the enhancement factor (E), which depends, among others, on the Hatta number (Ha) (Rode et al., 2012; Tobiesen et al., 2007).

interface): E1 450 Ha

then E ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Ha2  Ha

 The diffusion of the MEA is limiting (a sharp reaction front is formed in the vicinity of the gas–liquid interface): E1 o0:02 Ha

then E ¼ E1

A.1. Calculation of the local mass transfer coefficient (kg, kl)

 Partial diffusional limitation of the MEA (intermediate situaGas phase mass transfer coefficient (kg): Calculation of the Graetz number of the gas phase: Gzg ¼

0:02 o

Dg

then E ¼

2

Boundary condition:  Uniform mass flux at the wall: A¼1.30; Shlim ¼4.36.  Uniform wall concentration: A ¼1.08; Shlim ¼3.66: kg ¼ Shg

Dg d

Liquid phase mass transfer coefficient (kl): Calculation of the Graetz number of the liquid phase (Gzg): Dl ul d

2

For the gas phase flowing in the lumen side, then d ¼2(re  d). For the gas phase flowing in the shell side, then d ¼dh, the hydraulic diameter. Integration of the Graetz equation: If Gzg o0.03 then Shg ¼A Gzg 1/3; if Gzg 40.03, then Shg ¼Shlim, where Shg and Shlim are the Sherwood number and the limit Sherwood number of the gas phase (dimensionless), respectively. Boundary condition:  Uniform mass flux at the wall: A¼1.30; Shlim ¼4.36.  Uniform wall concentration: A ¼1.08; Shlim ¼3.66. kl ¼ Shl

Dl d

A.2. Calculation of Hatta number (Ha) Ha ¼

E1 o 50 Ha

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ha ðE1 EÞ=ðE1 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  tanh Ha ðE1 EÞ=ðE1 1Þ

ug d

For the gas phase flowing in the lumen side, then d ¼2(re  d). For the gas phase flowing in the shell side, then d ¼dh, the hydraulic diameter (m). Integration of the Graetz equation: If Gzg o0.03 then Shg ¼ A Gzg 1/3; if Gzg 40.03, then Shg ¼Shlim, where Shg and Shlim are the Sherwood number and the limit Sherwood number of the gas phase (dimensionless), respectively.

Gzl ¼

tion):

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dl kr C MEA,l kl

A.3. Calculation of the enhancement factor (E) Calculation of the infinite enhancement factor (EN):     C MEA,l Dl 1=3 Dl 2=3 þ E1 ¼ DMEA 2C CO2 ,l-int DMEA Determination of the reaction regime (Rode et al., 2012):

 The diffusion of the MEA is not limiting (the reaction is distributed but located in the vicinity of the gas–liquid

A.4. Calculation of the kinetic rate constant (kr)

 According to Versteeg and van Swaaij (1996), kr ¼ 9:77  107 exp



 4955:16 thus, T

at 295 K, kr ¼ 4:94:

 According to Hikita et al. (1979), kr ¼ 4:4  108 exp

  5400 thus, T

at 295 K, kr ¼ 4:94:

 According to Liao and Li (2002): kr ¼ 7:973  109 exp



 6243 thus, T

at 295 K, kr ¼ 5:14:

A.5. Calculation of the CO2 gas–liquid partition coefficient (m)

 According to Reid et al. (1986) cited in Amann (2007) m ¼0.656 for wMEA ¼30 wt% and m¼0.685 for wMEA ¼20 wt%.

 According to Danckwerts (1970), m¼0.514 for wMEA ¼30 wt% and m¼0.613 for wMEA ¼20 wt%.

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