On the use of molecular-based thermodynamic models to assess the performance of solvents for CO2 capture processes: monoethanolamine solutions.

Predictive models play an important role in the design of post-combustion processes for the capture of carbon dioxide (CO2) emitted from power plants. A rate-based absorber model is presented to investigate the reactive capture of CO2 using aqueous monoethanolamine (MEA) as a solvent, integrating a predictive molecular-based equation of state: SAFT-VR SW (Statistical Associating Fluid Theory-Variable Range, Square Well). A distinctive physical approach is adopted to model the chemical equilibria inherent in the process. This eliminates the need to consider reaction products explicitly and greatly reduces the amount of experimental data required to model the absorber compared to the more commonly employed chemical approaches. The predictive capabilities of the absorber model are analyzed for profiles from 10 pilot plant runs by considering two scenarios: (i) no pilot-plant data are used in the model development; (ii) only a limited set of pilot-plant data are used. Within the first scenario, the mass fraction of CO2 in the clean gas is underestimated in all but one of the cases, indicating that a best-case performance of the solvent can be obtained with this predictive approach. Within the second scenario a single parameter is estimated based on data from a single pilot plant run to correct for the dramatic changes in the diffusivity of CO2 in the reactive solvent. This parameter is found to be transferable for a broad range of operating conditions. A sensitivity analysis is then conducted, and the liquid viscosity and diffusivity are found to be key properties for the prediction of the composition profiles. The temperature and composition profiles are sensitive to thermodynamic properties that correspond to major sources of heat generation or dissipation. The proposed modelling framework can be used as an early assessment of solvents to aid in narrowing the search space, and can help in determining target solvents for experiments and more detailed modelling.


Introduction
Carbon dioxide (CO 2 ) emissions are generally considered to play a major role in climate change and particularly in global warming. Fossil fuel power plants are the major fixed point-source emitters of CO 2 . In response to the threat posed by global warming, the Roadmap for 2050 set by the European Commission in 2011 suggested a reduction of greenhouse gas emissions in Europe to 25% by 2020 and to 80% by 2050 1 . In the recent Paris climate conference (COP21), an agreement was made between participating parties to cut greenhouse gas emissions to a level that limits the global average temperature to "well below" 2 • C above pre-industrial levels and to "pursue efforts to limit the temperature increase to 1.5 • C" 2 . In this context, the development of carbon capture systems must be addressed in the short term, and amine-based post-combustion capture processes are seen as the most promising near-term technology in terms of development and applicability, e.g., see Mac Dowell et al. 3 . In this technique, absorption is achieved both physically and chemically, so that significant CO 2 removal can take place even at the challenging low partial pressures of the greenhouse gas. The major advantage of this technology is that it can be retrofitted to existing power plants.
There are however several concerns with this technology, in particular the large energy requirements associated with solvent regeneration, the degradation of the solvent, which is exacerbated by the presence of oxygen in the flue gas, and the environmental and health impact that may result from solvent losses and solvent degradation products. These issues are particularly important because of the scale of deployment required to have a meaningful impact on CO 2 emissions. There are significant experimental programmes to identify new solvents [4][5][6][7][8][9][10][11][12] and several pilot-plant studies are under way [13][14][15][16][17][18][19][20][21] . Modelling studies can play an invaluable and complementary role in addressing some of these issues, including the choice of solvent and operating conditions that yield optimal performance.
A key challenge in realising the benefits of a model-based approach to design is the development of models that can accurately predict the behaviour of the CO 2 capture process under different conditions and for a range of solvents. This is particularly difficult in the case of CO 2 absorption due to the complex reaction chemistry that occurs and the large number of ionic species present in the process. For example, in the case of absorption of CO 2 using the most common solvent, an aqueous solution of monoethanolamine (MEA, HOC 2 H 4 NH 2 ), the key reactions, excluding the speciation of water, are 22 terization of the speciation, reaction mechanism, equilibria and kinetics for mixtures relevant to CO 2 chemisorption are, however, required before detailed models can be developed, and this necessitates extensive experimental investigation. The types of data that are required include data specific to the solvent involved, such as reliable physicochemical (e.g., vapour-liquid equilibrium (VLE) and liquid phase speciation) and kinetic (including reaction rate constants and diffusion coefficients) data, and information regarding the effect of the column specifications, for instance the type of packing material employed. This presents a significant barrier to the rapid development of improved processes for carbon capture.
To overcome this difficulty, the task of identifying solvents that lead to improved CO 2 absorption processes can be subdivided into two main steps. The first is the rapid identification of a list of promising solvents by assessing a wide solvent search space as fully as possible and analyzing key performance indicators that relate closely to process performance, energetics, environmental impact, and solvent degradation. To minimize the reliance on experimental data and accelerate the search for new solvents, models that make it possible to predict physical and chemical properties from molecular structure are highly desirable. The motivation for this first step is to reduce the number of solvents to be considered in more detail, and the models used should therefore offer broad predictive capabilities, which may require making simplifying assumptions in model development. In the second step, promising solvents can be further analyzed and some of the assumptions made in the first step can be re-assessed. Once a list of candidate solvents is obtained, experiments can be conducted on a subset of these solvents, with the aim of obtaining the information required for a more detailed evaluation of each solvent and the corresponding carbon dioxide process.
To explore the space of possible solvents, there is a need to develop models that offer adequate predictive capabilities without exhaustive reliance on experimental data, and that can provide a quantitative insight into the behaviour of the process; the use of a thermodynamic model to capture the phase and chemical equilibria of mixtures of carbon dioxide, water, and alkanolamine within a process model is investigated in our current paper, as a means of obtaining an estimate of the performance of the absorption. As an initial assessment of the method, we focus on MEA because as a ubiquitous solvent for CO 2 capture there is extensive experimental data available to test the validity of the predictions. Before presenting the model, we first place the proposed approach in the context of other modelling work in the area.
There is an extensive body of literature concerning the process modelling and simulation of CO 2 absorption in packed columns. The modelling approaches that have been proposed to date differ in the choice of thermodynamic and kinetic models, and, where appropriate, heat-and mass-transfer models. Most of the effort has focused on the performance of aqueous monoethanolamine solutions, due to their widespread industrial use and the availability of pilot-plant data, although there have been some models developed for other solvents, notably aminomethylpropanol (AMP) 16,[32][33][34][35] . In modelling an absorber, the column is usually divided into hypothetical stages, each representing a (sometimes infinitesimal) section of packing in the column 36,37 . Each stage can be modelled using either an equilibrium or a rate-based model. In an equilibrium model, vapour-liquid equilibrium is assumed at each stage, everywhere on the stage. A rate-based model accounts for heat-and mass-transfer limitations. For chemisorption processes with fast reaction kinetics, as is the case for the process of interest 28,38 , a rate-based process is more reliable. Indeed, Lawal et al. 39 have compared the two approaches using the same physical property model and concluded that the rate-based model provide a better description of the pilot-plant 14 temperature profiles where MEA was used as a solvent. A similar comparison was made by Afkhamipour and Mofarahi 35 for CO 2 absorption in AMP solution, and a rate-based model was found to give a better prediction of the temperature and composition profiles of the pilot-plant runs 13 .
When using a rate-based approach, an important aspect in model development is the choice of approach to treat heat-and mass-transfer phenomena. The concentration and temperature profiles across hypothetical films in a two-film model can be imposed, taking into account the effect of chemical reactions on mass transfer with an enhancement factor, defined as the ratio of the amount of gas absorbed in a reacting liquid to the amount which would be absorbed if there were no reaction [40][41][42] . The enhancement factor varies along the length of the column and can often be adjusted to pilotplant data. This is the route followed in most models of CO 2 absorption (e.g., Tontiwachwuthikul et al. 13 , Sivasubramanian et al. 43 , Pintola et al. 44 , Pacheco and Rochelle 45 , Al-Baghli et al. 46 , Tobiesen et al. 47 , Faramarzi et al. 48 , Khan et al. 49 , Neveux et al. 50 , Saimpert et al. 51 , Jayarathna et al. 52 , Afkhamipour and Mofarahi 53 ). A comparative review of the rate-based models that have been used to specifically treat CO 2 absorption in aqueous MEA solutions can be found in reference 54 .
As an improvement on film theory, Tobiesen et al. 47 developed a penetration model, where the two films at the interface are described continuously, which was found to describe their own pilot-plant data well. A more rigorous approach is that followed by Kucka et al. 29 , in which the Maxwell-Stefan formalism is used together with film discretization. This more detailed model leads to better predictions of concentration and temperature profiles at the pilot-plant scale 13 than other models, without the need to adjust any parameters to pilot-plant data 29 . The Maxwell-Stefan formalism has also been applied by Lawal et al. 55 and Biliyok et al. 56 . Kale et al. 31 have recently investigated a rate-based model with film discretization, and studied the sensitivity of the calculations to several key parameters. They found that good predictions of column profiles can be obtained with a sufficiently fine discretization.
In all rate-based models, empirical mass-transfer correlations are required to take into account the type of packing used and the operating conditions. The correlations of Onda et al. 57 , 58 and Bravo and Fair 59 were developed specifically for random packing, whereas the correlations of Rocha et al. 60 were developed for structured packing, but can be applied to random packing by using an equivalence relation linking the random packing characteristics to the structured packing. Correlations developed by Billet and Schultes 61 are also available as they apply to both structured and random packing. In a detailed comparison applied to a model of a CO 2 capture pilot plant presented by Faramarzi et al. 48 , it appears that the main operating conditions to consider when choosing a mass-transfer correlation are the flowrates of the flue gas and the lean solvent.
Another important consideration in modelling CO 2 absorption is the representation of the chemical reactions and fluid phase equilibria of the mixture of MEA, CO 2 and H 2 O. In the earliest absorber models, the thermodynamics of the gas and liquid phases were described with the assumption of ideal gas and ideal solution behaviour. This is the case for example of the model developed by Pandya 62 and later used by Tontiwachwuthikul et al. 13 . However, this model was too simplified to describe the complex interactions between the CO 2 and the solvent, and may not be suitable when transferred to other solvents. Two different approaches are typically followed when developing more accurate models of mixtures exhibiting reaction equilibria: those based on physical theories and those based on chemical theories. Most commonly, a chemical approach (e.g., see references 30,[63][64][65][66][67] ) is adopted. In such approaches, all of the reaction species in solution are modelled explicitly, requiring the a priori specification of the relevant reaction schemes and their corresponding temperature-dependent equilibrium constants. Hence, the use of a chemical approach requires experimental data on the concentration of species in solution at various temperatures.
Most models proposed to date that specifically treat CO 2 -amine-H 2 O systems are based on this class of chemical approach; an explicit treatment of the major ionic and non-ionic species formed due to reaction is adopted. Such methods rely on the use of reaction kinetics and equilibrium constants derived from experimental data specific to each reaction 29,30,[68][69][70][71][72][73] . The methodology of combining the electrolyte-NRTL approach 68,74 with a Henry's constant to describe CO 2 solubility has attracted much interest and has been used in recent studies 55 . The eNRTL approach has also been corrected for inconsistencies by Bollas et al. 75 . More recently, Zhang et al. 30 developed a detailed model of the CO 2 -MEA-H 2 O amine system using a chemical approach, for which an eNRTL 66 model was developed for the treatment of the liquid phase, the perturbed chain statistical associating fluid theory (PC-SAFT) 76 was used to obtain the vapour phase fugacity coefficients, and Henry's law constants were obtained to provide a full description of the phase equilibria. While this approach arguably provides the most accurate representation of the behaviour of these mixtures to date, a major drawback of these models is that they contain a large number of parameters, requiring ample experimental data for vapour-liquid equilibrium (VLE) and reaction kinetics, thus making it difficult to use for different conditions and solvents. A similar approach 29 is to combine the Soave-Rechlich-Kwong (SRK) equation of state (EoS) 77 with the eNRTL model. Tobiesen et al. 47 have adapted an activity coefficient model from Hoff 78 to correlate VLE experimental data, using an experimentally-derived equilibrium constant 79 . As an alternative to reduce the number of parameters to be determined and the computational effort needed to solve the vapour-liquid equilibrium, Gabrielsen et al. 80 developed their own thermodynamic model to determine the VLE of MEA, CO 2 and H 2 O. They proposed a simple correlation derived from experimental data of CO 2 solubility in aqueous MEA, which is valid for MEA over the conditions considered: in this case CO 2 loading (defined as the number of moles of CO 2 per mole of MEA in the liquid phase) varied from 0 to 0.5 at 313 K and 393 K. As a simplification, a single chemical equilibrium reaction (carbamate formation) was considered, reducing the number of adjustable parameters required.
The combined reaction and activity coefficient models developed to date provide an accurate representation of CO 2 -MEA-H 2 O systems over a range of conditions. However, they require extensive parameterization and the model parameters obtained are not transferable to other solvents. These models are therefore more suited for their application in step two of the solvent selection process, the detailed quantitative analysis of the performance of the most promising solvents.
An alternative treatment of reactions follows a physical approach. Such approaches have previously been proposed to model the phase and chemical equilibria of mixtures of alkanolamines, CO 2 , and H 2 O 81-84 . This methodology is based on a version of the molecular statistical associating fluid theory (SAFT) EoS 85,86 , in which a square-well intermolecular potential of variable range is usedreferred to as SAFT-VR 87,88 . More recently, 89,90 models for aqueous solutions of CO 2 and primary alkanolamine have been developed within a group contribution re-formulation of the SAFT-VR EoS, SAFT-γ SW 91,92 , where a similar physical treatment of reactions was applied. In these approaches, the reaction products are considered as associated aggregates of the reactants and are not modelled as independent chemical species. Aggregation is driven by short-range association sites incorporated in the molecular models, by means of appropriate site-site interactions. Appropriate association schemes are chosen in order to form the desired reaction products (e.g., carbamate, zwitterion). In the SAFT approach, the concentration of each aggregate species in the mixture can be determined from the fraction of association sites that are not bonded 93 . In cases where chemical equilibrium can be assumed and under some appropriate assumptions (e.g., a constant site-site binding energy), such an approach has been shown to be equivalent to the chemical models used to date in which the aggregated chemical species are modelled explicitly 94 . The equivalence of the chemical and physical approaches means that the parameters within SAFT can be related to the experimental equilibrium constant. The thermodynamic properties of the mixture can thus be determined by specifying only the concentration of the reactants, the pressure and temperature, and the parameters describing the interactions between reactant species (including the types, number, energy and bonding volume of association sites). The physical approach applied to CO 2 and aqueous MEA has been found to provide excellent predictions of the concentration of the key reaction products 84,90 . Physical approaches are particularly well-suited to the exploration of a wide solvent space in that models can be developed entirely from experimental VLE data (e.g., vapour pressure and saturated liquid density) and no data on energetics or speciation are required. In addition, parameters can often be transferred from one compound to another compound on the basis of similarities in molecular chemistry, and the resulting predictions are found to be valid over wide ranges of compositions, temperatures, and pressures. These characteristics make SAFT-VR models based on a physical representation of chemical equilibrium a suitable platform for the exploration of a large space of solvent candidates. While the evaluation of potential solvents can be carried out on the basis of thermodynamics alone, it would be more valuable to obtain an evaluation of each solvent based on performance metrics that relate to a specific absorption process; this possibility is investigated in our current work.
The use of the SAFT-VR thermodynamic approach within process models of CO 2 absorption has previously been explored for high-pressure physical absorption in alkanes 95 , ethers and ether derivatives 96 , and also for chemisorption. In an early study the modelling of the absorption process was also considered and explored in the context of solvent blend design 97 . The absorber model was further developed 98 showing reliable results for a set of pilot-plant data. Mac Dowell et al. 99 have proposed a dynamic model of a CO 2 absorber, based on the SAFT-VR thermodynamic treatment, which has been used in investigations of the control 100 and economics 101,102 of post-combustion CO 2 capture processes. Qualitative agreement with data from two pilot-plant runs from 13 was found in these studies [99][100][101] . The column profiles obtained by Mac Dowell et al. 99 are somewhat difficult to interpret as the location of the plant data points reported in the paper does not match the location of the sensors reported in the physical pilot plant. Nevertheless, the findings of this body of work indicate that a physical approach to the modelling the chemical and physical equilibria allows one to capture the process behaviour accurately with a limited set of parameters.
Overall, the models developed to date can be used to adequately represent the general behaviour of the absorber column. However, only the model of Tontiwachwuthikul et al. 13 provides an accurate description of the temperature of the rich solvent at the outlet, and in all published models, the bulge in the temperature profile, which is a well-known characteristic of this absorption process 103 , is reproduced qualitatively but not quantitatively. Existing models reproduce either its magnitude or its location along the column, but not both.
In view of the foregoing discussion, the potential benefits of a novel absorption model in which a SAFT-VR thermodynamic treatment and a rate-based column model are integrated are explored based on CO 2 absorption in aqueous MEA. The proposed approach is based on a two-film model.
Because the reactions are treated implicitly with the SAFT-VR EoS, only the key molecular species (i.e., MEA, CO 2 , H 2 O and N 2 ) need to be taken into account explicitly at the level of mass and energy balances. The detailed model presented in our current paper includes several modifications over previous work 97,104 , leading to enhanced model reliability. In section 2, we describe the SAFT-VR EoS and the heat-and mass-transfer relations used in the rate-based absorber model. The validation of the model, its predictive capabilities, and a sensitivity analysis are presented in section 3. Two scenarios are investigated, corresponding to different levels of data availability: in the first scenario, it is assumed that no pilot-plant data are available, and the suitability of the model to provide a best-case analysis of process performance is studied; in the second scenario, one pilot-plant run is considered and employed to obtain a more realistic quantification of mass-transfer limitations. The transferability of this analysis is then tested against data at other conditions.

Modelling methodology
The development of a rate-based model of a CO 2 absorber is described in this section. In subsection 2.1, the thermodynamic model used for the vapour liquid equilibrium (VLE) and the chemical equilibrium is presented. The non-equilibrium stage approach is introduced in subsection 2.2.

Thermodynamic Model
The treatment of the chemical-reaction equilibria relevant to the absorption process is a key aspect of the modelling strategy adopted in this work. In the physical approach followed here, the chemical and physical interactions are treated on an equal footing, within the SAFT-VR SW EoS 87,88 . Background information is provided in this section to help the reader understand the models used and the underlying assumptions.
The SAFT family of equations stems from the first-order thermodynamic perturbation theory (TPT1) of Wertheim 93,[105][106][107][108][109] . In SAFT approaches molecules are modelled as chain of fused spherical segments with embedded short-range association sites incorporated to mediate hydrogen bonding which lead to aggregate formation (speciation). The EoS is developed in terms of the Helmholtz free energy using a perturbation approach, such that the free energy of a chain molecule is obtained with respect to the properties (free energy and radial distribution function) of a reference monomeric (non-bonded) system. The original version of SAFT 85,86 has been revisited and modified by a number of researchers so that several versions are now available 76,87,88,[110][111][112][113][114][115][116] . A historical account of perturbation theories for polar and associating liquids has recently been presented by Gubbins 117 .
In the SAFT-VR formulation 87,88 used here, a square-well (SW) potential is used to describe the interaction between spherical molecular segments of the reference fluid. The more recent versions of the SAFT-VR EoS incorporate the Mie (generalized Lennard-Jones) potential (SAFT-VR Mie 116 ), and allow for more reliable representation of the near-critical region and second derivative properties (e.g., heat capacities) which depend mainly on the specific form of the repulsive part of the potential 114,116,118,119 . Additionally, group contribution (GC) 120 versions of the SAFT-VR equations of state, namely SAFT-γ SW 91,92 and SAFT-γ Mie 121 , have also been proposed. These offer additional predictive capabilities in that the properties of a new solvent that has never been synthesized can be predicted (without data specific to that solvent) provided that parameters for the functional groups appearing in that molecule are available. The assessment of the integration of the molecular-based SAFT-VR SW EoS within an absorption model presented in our current paper can readily be extended to a group contribution formulation.
In the SAFT-VR SW approach, a molecule i is modelled as a chain of m i fused homonuclear spherical segments of diameter σ ii . The interactions between two identical segments are described by a square-well potential of range λ ii and depth ε ii (cf. figure 1). For each molecule i, the number of site types N s,i must be defined, as well as the number of sites of each type a, N s,ia . The sites are characterized by SW site-site energetic ε HB ab,ii and range r c;ab,ii parameters (cf. figure 1). In order to model mixtures, combining rules based on the Lorentz-Berthelot approach (cf. Haslam et al. 122 ) are used to describe the unlike interaction between segments on two different molecules i and j: where k ij is an adjustable parameter that characterizes the unlike dispersion attractive interaction. Parameters to describe association between different molecules are estimated using experimental data for the specific mixtures of interest, or for chemically similar mixtures 82,84 .
In recent work 82,84,89,90 the reactions involved in aqueous amine solutions of CO 2 have been treated implicitly within the SAFT-VR and SAFT-γ frameworks, with the products of the chemical reaction represented as aggregates of the reactant molecules. In the case of MEA the overall set of reactions can be reduced to 28 and association sites that allow the complexation of CO 2 and amine are introduced in the SAFT-VR molecular models. The reaction products can thus be modelled as neutral aggregates of CO 2 and MEA, bonded at association sites as shown in figure 2. An accurate overall representation of the vapour-liquid phase equilibria of MEA + CO 2 + H 2 O can be obtained in this manner by estimating the molecular parameters from experimental fluid-phase equilibria data, for both the pure components and mixtures. One important implication of the physical treatment of chemical equilibrium is that there is no need to explicitly specify a reaction scheme or reaction products. The types of products formed (e.g., carbamates or bicarbonates) are dictated by the association scheme chosen (number of sites and strength of their interactions), and the relative extent of formation of the different products depends on the temperature-independent intermolecular parameters that describe the association energies. The fraction of molecules bonded at a given site is an output of the SAFT-VR model and the distribution of reaction products can be determined from a statistical analysis of the values of these fractions at the thermodynamic state of interest 94 . Thus, although no speciation data are used in deriving the SAFT-VR parameters, the speciation equilibria can nonetheless be predicted successfully 84,90 . This type of physical approach greatly reduces the number of parameters needed to describe the mixture compared to explicit approaches such as eNRTL, in which every species must be described as a separate entity and temperature-dependent equilibrium constants must be derived for all the relevant reactions. The SAFT-VR approach has been shown to be applicable to the absorption of CO 2 in a wide range of aqueous alkanolamine solutions 84 . In many cases, it is possible to transfer parameters from one alkanolamine to another based on molecular similarity, further reducing the need for experimental data.
One key assumption in adopting this type of physical approach is that all reactions are assumed to be at equilibrium, which is only applicable to processes in the physical regime, i.e., where mass transfer is the rate limiting process. In the case where a specific treatment of the charged electrolytic species is required, the SAFT-VRE approach [123][124][125][126][127][128] can be adopted coupled to a chemical approach as appropriate. It is also possible to represent some or all species explicitly where any chemical reactions that are not at equilibrium can then be modelled via a separate kinetic model. We do not follow this route here, which would fall within the class of chemical approaches, but instead assess the adequacy of the physical approach.
The molecules considered in our study are MEA, H 2 O, CO 2 , and N 2 . A schematic of the molecular models used in the SAFT-VR SW EoS is presented in figure 3. The values of all parameters are listed tables 1-4. The MEA molecule is represented as 2 tangent spherical segments with 2 association sites of type e (electron lone pairs on the oxygen atom), 1 site of type e * (corresponding to the lone pair on the nitrogen atom), 1 site of type H (the hydrogen atom on the hydroxyl group), and two sites of type H * (hydrogen atoms on the amine group) 82 . The H 2 O is represented as 1 spherical segment with 2 e sites and 2 H sites 129 . The CO 2 model comprises 2 segments and has 1 α 1 site and 1 α 2 site (acceptor sites) that interact only with the e * sites of MEA 84 . The N 2 molecule is modelled as a fused non-spherical diatomic, with an aspect ratio of m = 1.4, and does not have association sites as it is chemically inert and apolar 104,130 . A site of type e/e * can bond with a site H/H * and vice versa. For example, when an e site from MEA and an H site from water come within the cut-off range r HB c,eH,MEA−H 2 O of each other, there is a site-site hydrogen-bonding associative interaction of energy -ε HB c,eH,MEA−H 2 O . A representative example of the good overall description of the temperature and pressure dependence of the absorption VLE behaviour of CO 2 in aqueous MEA obtained with the SAFT-VR SW models can be seen in figure 4 over several orders of magnitude for the partial pressure of CO 2 . In the standard Wertheim TPT1 treatment at the heart of the SAFT EoS, association into linear-chain, branched-chain, and network aggregates are considered. Association into ring-like structures [131][132][133][134] (and even double bonding 135 and bond cooperativity 136 ) can be taken into account but this is not considered for the systems described in our current work.

Non-equilibrium absorber model
The absorber is a counter-current vapour-liquid multistage separation column, with a liquid feed at the top stage and a vapour feed at the bottom stage. The vapour product comes off the top stage and the liquid product off the bottom stage. The inside of the column is filled with an inert packing material designed for a maximum mass transfer between the vapour and the liquid and for a low pressure drop. The vapour and liquid compositions vary continuously with packing height.
The modelling of such a column can be either discrete or continuous in the vertical direction. In a discrete model, the column is divided in hypothetical stages, each of which represents a section of packing in the packed column. For rate-based models, a greater number of stages provides a better description 36 . Taken to the limit, using an infinite number of stages is equivalent to modelling the column continuously. Although both models could be used, we choose the discrete approach for our study.
The modelling of the stages can be equilibrium or rate-based. In an equilibrium model, it is assumed that vapour-liquid equilibrium is achieved at each stage, everywhere on the stage. A rate-based model accounts for limitations due to transport phenomena. Due to the complexity of the process modelled in this case, a rate-based approach is chosen over an equilibrium one in order to capture some of the key physical phenomena 137,138 .
For the absorber model, the assumption is made that the two phases are distinct, and each phase is perfectly mixed at each stage. A diagram of a non-equilibrium stage is presented in figure 5. Vapour from the stage below is brought into contact with liquid from the stage above and the two phases exchange mass and energy through their common interface represented in the diagram by the wavy line. In a rate-based model, separate mass balances are written for each phase. The two phases are in contact through their interface where material lost by the vapour phase is gained by the liquid phase. The heat transfer is treated in a similar way. There is one energy balance in each phase, and the phases are linked via the rate of energy transferred across the phase interface. The compositions of the two phases at the interface are determined assuming equilibrium conditions at the interface.
The behaviour at the phase interface in the rate-based stages is described with a two-film theory 137 . Each phase is split into two parts: the bulk phase; and the film in which the heat and mass transfer occur. In the bulk phase, the concentration, pressure, and temperature are assumed to be uniform, whereas there could be a gradient in composition and temperature in the film. A liquid-gas interface between the two films is sketched in figure 6, where the profiles for composition of component i, temperature and pressure are represented.
The following assumptions are made: • The model is steady state.
• The bulk phases and films are at chemical equilibrium everywhere (i.e., the reaction rates are faster than the mass transfer rates) 25,28 .
• The interface is at phase and chemical equilibrium.
• The interfacial surface area is the same for both heat and mass transfer.
• The effective area is equal to the wetted area.
• The absorption column is considered to be adiabatic.
• There is no pressure drop along the column.

Model equations
In this subsection, we present the equations used to model a packed column subsection (stage). The equations are grouped into different categories: energy and mass balances, rate equations, equilibrium equations, mass-transfer correlations, heat-transfer correlations, and diffusion correlations. We list explicitly all of the model equations to clarify the radically different approach followed here. One important model parameter τ is introduced in this section to account for a scaling of the CO 2 diffusivity. This parameter will be estimated from experimental data. In the following equations, the subscript i refers to the i th component, j to the j th stage, c is the total number of components, and N s is the number of stages in the column. The number of stages is fixed to 50 which is amply sufficient to assure numerical convergence without significantly impacting the computational time.

Heat and mass balances
The liquid and vapour phase mole balances for component i on stage j are given by where L j and V j (mol s −1 ) are the total liquid and vapour molar flowrate leaving stage j respectively, x i,j and y i,j are the bulk liquid and vapour mole fractions of component i on stage j, N L i,j (mol s −1 ) is the net gain of species i in the liquid phase due to interphase transport for stage j, and N V i,j (mol s −1 ) is the net loss of species i in the vapour phase due to interphase transport for stage j.
The mole fractions in the streams leaving each stage must sum to unity: The energy balances for the liquid and vapour phases respectively are where H L j and H V j (J mol −1 ) are the molar enthalpies of the liquid and gas phases respectively of stage j. E L j (W) is the net gain of energy of the liquid phase through the interface at stage j, and E V j (W) is the net loss of energy from the vapour phase through the interface at stage j. The molar enthalpies H L j and H V j are determined from SAFT-VR as functions of the temperature of the bulk phases, T L j and T V j (K), the molar volumes of the phases, V L j and V V j (m 3 mol −1 ), and the composition (vectors) of the phases, x j and y j .
There is no accumulation of mass or energy at the vapour-liquid interface:

Rate equations: mass transfer
According to the film model for mass transfer, we define the mass-transfer rates as follows: where a T,j (m 2 ) is the total interfacial area on stage j available for heat or mass transfer, The total area for heat and mass transfer is given by the product of the effective specific area and the stage volume: where a ′ j (m 2 m −3 ) is the interfacial area density on stage j, A section (m 2 ) is the cross sectional area of the column, and D z (m) is the stage height given by h packing /N s , with h packing (m) as the total packing height.

Rate equations: heat transfer
The energy fluxes in the liquid and gas phases at stage j are given by the sum of the conductive and diffusive heat fluxes: where Q L cond,j and Q L dif f,j (W) are the conductive and diffusive heat fluxes at the interface into the liquid phase on stage j, and Q V cond,j and Q V dif f,j (W) are the conductive and diffusive heat fluxes at the interface out of the vapour phase on stage j.
The conductive heat-transfer rate into the liquid phase and out of the vapour phase on stage j are driven by temperature gradients between the bulk phases and the interface, and are given by where h L T,j and h V T,j (W m −2 K −1 ) are the heat-transfer coefficients in the liquid and gas phases on stage j, respectively , and T I j (K) is the temperature at the vapour-liquid interface on stage j. The diffusive heat fluxes account for the variation in enthalpy in the liquid and vapour streams associated with the transfer of mass from one phase to another. This is obtained based on the individual component fluxes and the corresponding enthalpies of the pure components: where where P L j and P V j (MPa) are the pressure of the bulk liquid and vapour phases at stage j respectively and P (T L j , V * L i,j , z i ) and P (T L j , V * L i,j , z i ) represent evaluations of the pressure using the SAFT-VR SW EoS.
When equating the vapour and liquid fluxes, one obtains an expression which includes the difference between the enthalpies in the vapour phase and in the liquid phase for a pure component i; this is the enthalpy of vaporisation of this component. As we will see, the enthalpy of vaporisation of water plays a significant role in the process. We assume that the contribution of N 2 due to its change of enthalpy between the two phases is negligible as its mass transfer is very small, and that the enthalpy of vaporisation of CO 2 is negligible as the operating conditions are close to or above its critical point. In the case where only one phase is stable for a pure component at the conditions of interest, as is typically the case for CO 2 , one can access the hypothetical phase information by providing an adequate initial guess for the molar volume to the SAFT-VR SW EoS.

Equilibrium relations
Both chemical and phase equilibrium are assumed to prevail at the interface so that the conditions of equality of pressure, temperature, and chemical potential must be satisfied: The mole fractions at the interface must sum to unity: Each stage is assumed to be at mechanical equilibrium, so that where Finally, the column pressure drop is assumed to be negligible.
All pressure terms present in the model are thus equated to the pressure of the gas inlet stream, P V Ns+1 .
This assumption can easily be removed at a later stage of the model development. Preliminary results have indicated that the pressure drop has a negligible effect on the process used for the validation of our model. The equations presented in the remainder of this section apply to each stage. The subscript j has been omitted for the purpose of clarity.

Mass-transfer correlations
The interfacial area, the liquid-phase mass-transfer coefficient, and the gas-phase mass-transfer coefficient are obtained using the correlations from Onda et al. 57 , 58 . Other correlations are available, the most commonly used being the ones developed by Rocha et al. 60 and Billet and Schultes 61 . Faramarzi et al. 48 compared the performance of the correlations and found that all three correlations provide a good description; they expressed a slight preference for the correlations of Rocha et al. 60 but could not draw a general conclusion which is applicable to all operating conditions. The correlations of Rocha et al. 60 apply to structured packing and have to be adapted for use in random packing, whereas those developed by Onda et al. 57 , 58 have been developed specifically using random Berl saddle packing, the same as the one employed in the pilot plant by Tontiwachwuthikul et al. 13 . This last set of correlations is sometimes reparameterized when used in commercial software such as Aspen or ProTreat. The correlations developed by Onda et al. 57 , 58 , as reported by Treybal 139 , are implemented in our model of the absorber column. The liquid-phase mass-transfer coefficient for component i is given by 139 where is the liquid velocity, and a ′ is the interfacial area density defined in equation (35).
The vapour-phase mass-transfer coefficient is obtained from where D V i (m 2 s −1 ) is the diffusivity of component i in the vapour-phase, Re V is the vapour-phase Reynolds number, and Sc V i is the Schmidt number of component i in the vapour phase.
The Reynolds number and the Schmidt number for the vapour phase are where ρ V (kg m −3 ) is the density of the vapour phase, η V (kg m −1 s −1 ) is the dynamic viscosity of the vapour phase, and u V (m s −1 ) is the vapour velocity.
The interfacial area density a ′ is obtained from 57,58 where σ c (N m −1 ) is the critical surface tension of the packing material, σ (N m −1 ) is the vapour-liquid surface tension, Re L is the liquid-phase Reynolds number based on the specific surface area, F r L is the liquid-phase Froude number, and W e L is the liquid-phase Weber number. The interfacial area density, together with the three dimensionless numbers used in equation (35), are properties of the mixture, not properties of individual components. The expressions for the dimensionless numbers are The velocities of each phase are wherev L andv V (m 3 s −1 ) are the volumetric flowrate of the liquid and gas phase respectively.

Heat-transfer correlations
For the gas and liquid heat-transfer coefficients we use the correlations presented in Treybal's Mass Transfer Operations 139 . Once again these expressions apply to each stage. The expression for the liquid-phase heat-transfer coefficient is where h L t (W m −2 K −1 ) is the liquid-phase heat-transfer coefficient , d s (m) is the diameter of a sphere of the same surface area as a single packing particle (not the same as a p ), λ L T (W m −1 K −1 ) is the liquid thermal conductivity, L spec (kg s −1 ) is the specific liquid flowrate, and P r L is the liquid phase Prandtl number. The latter is obtained from the following expression: where C L p (J kg −1 K −1 ) is the specific isobaric heat capacity of the liquid phase. The vapour-phase heat-transfer coefficient is given by where is the specific isobaric heat capacity of the vapour phase, P r V is the Prandtl number for the gas phase, V spec (kg s −1 ) is the specific vapour flowrate, and ǫ L 0 is the operating void space in the packing. It is assumed to be equal to the void fraction ǫ. The Prandtl number for the gas phase is obtained from where λ V T (W m −1 K −1 ) is the vapour thermal conductivity.

Vapour-phase diffusion coefficient
The vapour-phase diffusion coefficient used in equation (34) is obtained from the expression of Fuller and co-workers 140-142 as reported by Poling et al. 77 . It is assumed that the components are diffusing through air: where Σ v (Å) is the atomic diffusion volume, and M i−air is given by where M i (g mol −1 ) is the molar mass of component i.

Liquid-phase diffusion coefficient
The liquid-phase binary diffusion coefficients are described with different correlations. The mutual diffusion coefficients of CO 2 at very low concentration in pure MEA and pure H 2 O are derived from the Wilke-Chang correlation 77,143 : where φ k is the "association factor" of solvent k, and V m CO 2 (cm 3 mol −1 ) is the molar volume of CO 2 at its normal boiling temperature.
These mutual diffusion coefficients are used in the correlation of Takahashi 144 to determine the diffusivity of CO 2 in a liquid mixture of H 2 O and MEA: where we have introduced the scaling prefactor τ to the correlation. The presence of this scaling factor provides an effective approach to modelling several effects that are not accounted for due to the use of a chemical approach and the consequent implicit treatment of the reaction products: the acceleration of mass-transfer due to the depletion of CO 2 via chemical reactions 40 ; the reduction in mass transfer due to the ion pairs formed and their electrostatic interaction with other species 40 ; and the reduction in the mass transfer due to CO 2 being present in larger aggregated product species (bicarbonate and carbonate species). The latter effect was observed by Han et al. 145 in a molecular dynamics study of the diffusion coefficient of CO 2 in MEA, indicating a significant decrease in the diffusion coefficient with increasing CO 2 loading. The value of the parameter τ and its impact on model predictions are discussed in section 3. Finally, the Perkins and Geankoplis 146 method is used to calculate the diffusivity of H 2 O, MEA, and N 2 in the liquid phase: where V m i (cm 3 mol −1 ) is the molar volume of component i at its normal boiling temperature, and the term (φM ) i (g mol −1 ) represents the "association factor" of component i in the liquid mixture which is derived from The other correlations of experimental data used in our model are listed in table 5.

Results
The model described in section 2.2 is implemented in gPROMS 147 and an in-house implementation of the SAFT-VR SW EoS 148 is accessed via a Foreign Object Interface.
In this section, we investigate the predictive capabilities of the proposed model by thorough comparisons with the the pilot-plant data obtained by Tontiwachwuthikul et al. 13 . The inputs required to model these data and relevant assumptions are discussed in section 3.1. As discussed in section 3.2, we first assume that there are no pilot-plant data available for the solvent in question and analyze the suitability of the model for providing a best-case analysis of the process performance. It is then assumed that only one pilot-plant run is available and the corresponding concentration profiles are used to estimate the single parameter τ , related to mass transfer limitations, as summarized in section 3.3. The transferability of this parameter is assessed by comparing model predictions against data for other pilot-plant runs. Finally, in section 3.4, the sensitivity of the model to several parameters is explored, providing insights into the behaviour of CO 2 absorption columns.

Model inputs
The absorption column studied by Tontiwachwuthikul et al. 13 has an internal diameter of 0.1 m and a total packed height of 6.6 m; care has to be taken not to mistake the total column height of 7.2 m with the packed height, which is the relevant dimension in modelling the absorber. The absorption column internals are randomly packed 12.7 mm ceramic Berl saddles. The gas inlet stream is assumed to be free of MEA and the liquid solvent inlet stream free of N 2 . The input values used are listed in Table 6 and the characteristics of the column and the packing are listed in Table 7. Some of the data needed for the simulation of the process were not explicitly reported by Tontiwachwuthikul et al. 13 : in particular, the temperature of the flue gas, and the concentration of water in the flue gas. Close inspection of an earlier paper detailing the experimental apparatus 149 reveals that the temperature of the flue gas is controlled to be that of the lean solvent with a thermostatic bath. The inlet vapour stream is composed of ambient air and bottled CO 2 . Typical values of relative humidity of air (defined as the mole fraction of water vapour divided by the mole fraction of air saturated with water at the same temperature and pressure ) lie between 20% to 70%, which, at 20

Scenario 1
In the absence of pilot-plant data, the scaling factor τ for the diffusion coefficient of CO 2 , cf. equation (48), is set to 1. The predictions with τ =1 are presented in figures 7 and 9-17 as dashed curves. In all but one case (Run T18, figure 14), the rate of absorption of CO 2 throughout the column is found to be over-predicted as the predicted value of the gas-phase CO 2 concentration profiles y CO 2 all fall under the measured values. With the exception of Run T18, the temperature profiles T L and the liquid-phase CO 2 loading profiles θ CO 2 (defined as the number of moles of CO 2 per mole of MEA in the liquid phase) are similarly under-predicted. Complete absorption occurs at approximately stage 30, whilst an analysis of the pilot-plant data suggests that this occurs between stages 10 to 20. The size of the column required to achieve maximum absorption is therefore underestimated based on these predictions alone.
In Run T18 (figure 14), there are two competing effects: the overestimation of the absorption of CO 2 tends to increase the temperature of the liquid phase, which in turn tends to decrease the extent of absorption of CO 2 in the liquid. Since this is the only run in which the CO 2 in the inlet gas is not completely absorbed within the column, the cooling effect of the liquid feed is not as apparent as in other runs. Given the relatively low recovery of CO 2 and the fact that equilibrium is not reached at the bottom of the column, our findings indicate that the model is best used as an indicator of process performance by modelling a sufficiently large absorber to achieve equilibrium.
In general, an optimistic prediction of process performance is thus obtained with the model when no pilot-plant data are used (τ =1). Indeed, although the depletion of CO 2 through chemical reactions has an accelerating effect on mass transfer, the reduced diffusivities of the product species lead to an overall reduction in mass transfer rates. This suggests that the proposed predictive model could be used to obtain a preliminary assessment of novel solvents in the absence of pilot-plant data: if their best-case (τ = 1) performance is found to be significantly less than that of MEA or another suitable benchmark, these solvents would be eliminated from further consideration without undertaking an experimental programme.

Estimation of the scaling factor
In order to investigate whether the proposed model can be used to provide a quantitative agreement with the pilot-plant runs, the value of τ is then estimated by fitting to selected pilot-plant data. In keeping with our requirement to rely on only a limited amount of experimental data in the first step of the solvent selection process (and the exploration of a large space of solvents), the value τ is estimated from only one pilot-plant run (Run T22 of Tontiwachwuthikul et al. 13 ) in our current work. Additionally, this value is determined by minimizing the absolute deviation between the model predictions and the gas and liquid composition profiles, y CO 2 and θ CO 2 . The resulting temperature profile for Run T22 is hence predicted rather than correlated. The value of τ that results in the minimum deviation from the compositional pilot-plant data corresponds to a scaling of the liquidphase diffusivity to 4.1% of its original value (i.e., τ = 0.041). The liquid-phase temperature, gasphase CO 2 concentration, and liquid-phase CO 2 concentration profiles for Run T22, determined with and without the scaling factor for the diffusivity of CO 2 , are represented in figure 7 as continuous and dashed curves, respectively.
As the parameter τ is estimated from pilot-plant data, the value found for τ may be correcting any potential errors in the chosen mass-transfer correlation rather than the diffusivity alone. To the best of our knowledge, there is no published experimental data for the diffusion of the carbamate product in aqueous MEA. In order to assess the impact of the choice of mass-transfer correlation on the value of τ , we implement the mass-transfer correlations of Rocha et al. 151 , 60 in our current model of the absorber with the same scaling factor of τ = 0.041. The liquid-phase temperature, gas-phase CO 2 concentration, and liquid-phase CO 2 concentration profiles for Run T22 obtained with the two different mass transfer correlations are compared in figure 8. It can be seen that the profiles obtained are very similar: with the correlation of Rocha et al. 151 , 60 there is a slight underestimation of the composition of CO 2 in the liquid and gas phases; the temperature profile obtained with the correlation of Rocha et al. 151 , 60 is more faithful to the pilot-plant data than that of Onda et al. 57 , 58 , though one should bear in mind that the diffusivity parameter is estimated from the data using the former correlation. In conclusion, the value of τ is found to be essentially independent of the choice of the mass-transfer correlation (at least for this set of process conditions), confirming our hypothesis that a scaling of the diffusivity is required.

Model predictions
All the runs with the pilot-plant MEA absorber process can now be simulated in a predictive manner using the value of the diffusivity parameter (τ =0.041) obtained based on Run T22. The predictions for the temperature profiles in the liquid phase, and for the gas-phase CO 2 concentration and liquid-phase CO 2 loading profiles for the different Runs T13 to T21 are represented in figures 9 to 17 with continuous curves. These runs represent a variety of operating conditions in terms of amine concentration, inlet liquid-phase CO 2 loading, inlet gas-phase CO 2 concentration, and gas-to-liquid flowrate ratio (see Table 6).
Good agreement is found between the model predictions and the pilot-plant data for Runs T13, T14, T16, T17, T19, and T20 in relation to the liquid-phase temperature and the liquid-and gas-phase composition profiles along the entire length of the column. There is a slight over-prediction of the temperature and the composition profile for Runs T15 and T21, though the compositions at the top and at the bottom of the column are accurately described. The outlet liquid temperature in Run T18 is overestimated by about 12 K; there is a good match between the model predictions and the experimental values for the composition profiles. A good representation of the temperature bulges in Runs T16 and T20 can be seen in terms of its location along the column, and the amplitude is predicted accurately for both of these runs. To the best of our knowledge, no other published models provide a description of the temperature bulge to this level of fidelity 29,33,48,99 .
In all runs except Run T18, flat profiles are observed toward the top of the column. This plateau means that a maximum in the absorption is reached, indicating that equilibrium is achieved not only at the vapour-liquid interface but also between the bulk liquid and the bulk vapour phases. As a result the profiles would not be affected by making the column any higher. This plateau is depicted in figure 18 for Run T22. It can be seen that the temperatures of the liquid phase, the vapour phase, and the vapour-liquid interface are all equal between stages 1 and 15 (i.e., at the top of the column). Similarly, the CO 2 gas-phase composition in the bulk vapour phase and at the vapour interface are equal between stages 1 and 15, as are the CO 2 loadings in the liquid phase and at the interface. Additionally, the profiles for the temperature of the liquid phase and the temperature at the interface are almost identical, suggesting that all the heat-transfer resistance is in the vapour film.
Run T18 stands out from this set of runs as it does not exhibit a plateau in the profiles as found for the other operating conditions, meaning that the whole length of the column is used for absorption. The lack of a plateau region is clearly apparent from figure 19. From Table 6, one can see that Run T18 has the lowest amount of MEA in the lean solvent and the highest amount of CO 2 in the flue gas compared to the other runs. This explains why the whole column is required for absorption. In other runs, the totality of CO 2 is absorbed as the gas travels up the column between stages 50 and 15, so that lean solvent flowing at the top of the absorber (stages 1 to 15) has no CO 2 left to absorb. Run T18 is the only run where there is still CO 2 in the gas stream leaving the absorber, so the lean solvent starts absorbing CO 2 as soon as it enters the column at the top. These differences explain why a larger discrepancy is observed between the predicted and measured profiles for Run T18.
The behaviour of Runs T15, T18 and T21 could be represented more accurately by estimating specific values τ for these runs. However, this would not be in keeping with our objective to develop a predictive modelling platform to support solvent design activities. The good overall quantitative agreement achieved with a unique value of τ indicates that the scaling factor can be applied in a transferable manner at different operating conditions (at least for similar types of column packing).
An analysis of the deviation between the column profiles presented here and those obtained when a different run is chosen to estimate τ is also undertaken. The same method is applied to estimate τ based on pilot plant runs T13 -T21. The values of τ range between 0.027 (for Run T19) and 0.076 (for Run T21). These extreme values are then used to predict the column profiles (T L , y CO 2 and θ CO 2 ). The absolute errors between the values for each variable obtained with τ work, based on choosing any single pilot plant run arbitrarily to estimate the value of τ .

Sensitivity analysis
A sensitivity analysis is now employed to assess the relative importance of selected mass-and heattransfer parameters. Different key parameters in the model are altered to observe their impact on the predicted profiles. The values tested are listed in Table 8.

Mass transfer
The mass transfer in the model is mediated via mass-transfer coefficients 57,58 . These coefficients are highly dependent on the viscosity, the diffusivity, and to a certain degree, the surface tension of the fluid considered; these properties are thus considered as key parameters in the model and their influence on the temperature and composition profiles are assessed. The values of the parameters used in the sensitivity analysis are varied within physically realistic ranges. We should note that though marked variations are expected for the diffusivity and viscosity for the reacting system of this type, the change in the vapour-liquid surface tension is expected to be somewhat less dramatic. The values of the vapour and liquid diffusivities are doubled and halved, and the vapour-liquid surface tension is varied within ± 10 mN/m from the nominal value. Varying the diffusivity or viscosity in the gas phase has no visible effect on the liquid temperature profile or the composition profiles. However, varying these parameters in the liquid phase results in a significant variation in the profiles. The effect on the profiles due to the variation of the liquid viscosity and the diffusivity in the liquid phase is represented in figures 20 and 21. Varying the vapour-liquid interfacial tension produces a less significant variation in the profiles as can be seen in figure 22. The data that are originally considered for the surface tension correlation in our model is for a mixture of only H 2 O and MEA 152 . Jayarathna et al. 153 recently published experimental data of the surface tension of liquid mixtures of H 2 O, MEA and CO 2 , finding an increase in the vapour-liquid interfacial tension on absorption of CO 2 . A new correlation from these data is also implemented in our process model. A comparison of the temperature and composition profiles obtained with the two different correlations is shown in figure 22. The difference in the profiles is negligible so changing the surface tension correlation is not considered necessary. This also suggests that the process model is less sensitive to realistic changes in the interfacial tension than the corresponding changes in the the diffusivity and viscosity. We should note that the vapour-liquid surface tension can also be determined from the SAFT-VR EOS within a density functional treatment. 84,[154][155][156] We do not however pursue this approach here as the full DFT treatment of the ternary H 2 O-MEA-CO 2 mixture is computationally challenging, and as has just been discussed the use of the correlations for the surface tension of the H 2 O-MEA binary system provides a good representation of the pilot-plant data for the full mixture. The use of SAFT-VR DFT will be considered in future studies. It should be noted that the values of the mole fractions and temperatures at the outlets are not affected by the variations of the surface tension or the viscosity and diffusivity in the liquid phase. It would be safe to assume that under these operating conditions, where the residence time is large enough for the system to reach equilibrium, the outlet values are determined solely by thermodynamic quantities while the composition profiles are mostly dependent on mass transfer.

Heat transfer
The major source of heat in the absorption process is a consequence of the exothermic reaction between MEA and CO 2 , which occurs in the liquid phase. This heat can then be transferred to the vapour phase. Another major element in the energy balance is the heat associated with the transfer of water from one phase to the other. In the model, the heat transfer is mediated via the heat-transfer coefficients 139 . Two notable thermal parameters can be identified in the model: the heat-transfer coefficient for the liquid phase, and the heat-transfer coefficient for the gas phase. In addition, the magnitude of the enthalpy changes as the components are transferred from one phase to another or as the temperature changes in the gas and liquid streams affects the overall temperature profile. For instance, Kvamsdal and Rochelle 103 and Faramarzi et al. 48 have identified the heat capacities as sensitive parameters for the temperature profiles. Here, we consider the impact of the heat of vaporisation of water on model predictions. Although it is a well characterized thermodynamic quantity, its variation provides insights into the behaviour of the column. The value of the heat of vaporisation of water is doubled and halved relative to the value predicted by the SAFT-VR EoS, and the values of the heat transfer coefficients are multiplied by 10 and divided by 10 ( Table 8). The variation of the liquid heat-transfer coefficient has no visible effect on the liquid-phase temperature and composition profile. The effects on the profiles resulting from varying the heat-transfer coefficients and the heat of vaporisation of water are presented in figures 23 and 24, respectively.
As can be seen from figure 23, despite a variation of one order of magnitude in the vapour heattransfer coefficient, the effect on the temperature profile is very limited. It can be concluded that for the operating conditions under consideration, the model is insensitive to the heat-transfer coefficients and there is no need to investigate heat-transfer correlations further.
The variation of the heat of vaporisation of water is found to have a significant impact on the liquid phase temperature profile when its value is doubled, and a lesser impact on the composition profiles; it has a limited effect on the end-point values. An increase in the heat of vaporisation magnifies the amplitude of the temperature bulge, whereas a reduction eliminates the bulge. This suggests that the heat of vaporisation of H 2 O is responsible for the rate at which the liquid heats up and cools down. A similar effect is observed when a sensitivity analysis is carried out on the heat of absorption of CO 2 , however the analysis is not presented here because the adjustment of both properties leads to similar (but opposite) corrections to the energy balance. The key elements to predict the temperature bulge accurately are therefore the enthalpy of absorption of CO 2 and the enthalpy of vaporisation of water. The absorption of CO 2 releases energy that heats both phases. As the gas temperature increases upon entering the column, water is transferred from the liquid phase to the gas phase through evaporation to maintain saturation; this results in an absorption of energy, and a consequent decrease in temperature.
In conclusion, the competition between these two thermal effects (the absorption of CO 2 increases the temperature and the vaporisation of water decreases the temperature) is responsible for the observed and predicted temperature bulge. This interpretation is consistent with that reported in Mac Dowell et al. 99 .

Effect of humidity in the flue gas
The amount of water in the flue gas is considered to be important in determining the position of the temperature bulge in the absorber 103,157 . In the case of the pilot plants studied by Tontiwachwuthikul et al. 13 and Dugas 14 , the inlet flue gas is not saturated in water, while the flue gas is saturated in the pilot-plant studies of Tobiesen et al. 47 and Gabrielsen et al. 16 . In order to assess the impact of humidity in our model of the absorber, we vary the amount of water in the inlet flue gas from ω V H 2 O = 0 to 0.072 (corresponding to a mole fraction of 0.12) (water-rich). The effect on the profiles is represented in figure 25. It is apparent that varying the amount of water in the flue gas does not have a significant impact on the composition profiles. The effect is visible only for the liquid-phase temperature profile, where the outlet temperature value is higher by approximately 5 K for the saturated gas ω V H 2 O = 0.072 than for the dry gas ω V H 2 O = 0. Biliyok et al. 56 also find that increasing the moisture content of the flue gas affects the absorber temperature profile, however the increase in temperature is more significant in their study because a larger change in moisture content is considered (ω V H 2 O varies between 0.015 and 0.15, where the latter value represents a two-phase aqueous system). The negligible effect on the composition profiles is in contrast to the work of Mac Dowell et al. 99 who found a large effect of the humidity of the inlet gas on the flux profiles in the mass transfer zone.

Conclusions
An absorber model for CO 2 capture is developed with the aim of being as predictive as possible, in order to support solvent design activities prior to extensive experimental investigations. The heat-and mass-transfer are described with rate-based equations, in common with many other process models. Unusually, however, a physical approach is taken to model the chemical reactions taking place in the absorber. Both vapour-liquid equilibrium and the chemical equilibrium are treated within the SAFT-VR thermodynamic framework, ensuring a consistent and accurate representation of the physical interactions in the system under the assumption that reaction kinetics are not rate-determining. This approach lends itself to extension to other solvents, as a consequence of the transferable nature of the SAFT molecular models and the relatively small number of parameters and data required to develop them.
Without making use of pilot-plant data in model development, we find that the proposed model can generally be used to obtain a best-case performance of the solvent in question. This modelling approach is valuable for narrowing the solvent search space as solvents may be quickly rejected by comparing their performance in such a test. With very limited pilot-plant data we find that by adjusting a single parameter that corrects for the diffusivity of CO 2 in the liquid phase, the model can be used to predict with quantitative accuracy a variety of different operating conditions. Excellent predictions are obtained for the liquid-phase temperature profiles and the liquid-and gas-phase compositions along the column in most cases, with moderate deviations in a few cases. The comparison of the absorption performance of different solvents via this method may further aid in the narrowing of the solvent search space, and then a more quantitative comparison could be carried out.
Following accurate predictions of the column profiles, a careful sensitivity analysis is conducted. We find that the liquid viscosity and diffusivity, and to a lesser extent the vapour-liquid surface tension, are key properties for the prediction of the composition profiles. The column profiles are also shown to be sensitive to the thermodynamic properties that are major sources of heat generation or dissipation.
The main benefit of the proposed modelling framework, which is based on the physical modelling of the underlying chemical reactions, is the ability to assess new solvents for which there may be limited data available. This can be further enhanced through the adoption of a group-contribution EoS, which makes use of the same physical concepts, such as the SAFT-γ Mie EoS 121 . The modification of the      figure  3). The interaction matrix is symmetrical, i.e., r c;eH,ii = r c;He,ii and so the lower diagonal part has been left blank. The unlike association between sites of the same type is assumed to be symmetric, i.e., r c;eH,ij = r c;He,ij = r c;eH,ji = r c;He,ji 82,84 .      Surface tension +10 mN/m negligible effect -10 mN/m see figure 22 Heat transfer liquid × 10 negligible effect ÷10 Heat transfer vapour × 10 limited effect ÷10 see figure 23 Heat of vaporisation of water doubled significant effect halved see figure 24 Mass fraction of water in inlet flue gas varied between 0 and 0.072 negligible effect see figure 25