Novel Landfill-Gas-to-Biomethane Route Using a Gas–Liquid Membrane Contactor for Decarbonation/Desulfurization and Selexol Absorption for Siloxane Removal

Abstract

A new landfill-gas-to-biomethane process prescribing decarbonation/desulfurization via gas–liquid membrane contactors and siloxane absorption using Selexol are presented in this study. Firstly, an extension for an HYSYS simulator was developed as a steady-state gas–liquid contactor model featuring: (a) a hollow-fiber membrane contactor for countercurrent/parallel contacts; (b) liquid/vapor mass/energy/momentum balances; (c) CO₂/H₂S/CH₄/water fugacity-driven bidirectional transmembrane transfers; (d) temperature changes from transmembrane heat/mass transfers, phase change, and compressibility effects; and (e) external heat transfer. Secondly, contactor batteries using a countercurrent contact and parallel contact were simulated for selective landfill-gas decarbonation/desulfurization with water. Several separation methods were applied in the new process: (a) a water solvent gas–liquid contactor battery for adiabatic landfill-gas decarbonation/desulfurization; (b) water regeneration via high-pressure strippers, reducing the compression power for CO₂ exportation; and (c) siloxane absorption with Selexol. The results show that the usual isothermal/isobaric contactor simplification is unrealistic at industrial scales. The process converts water-saturated landfill-gas (CH₄ = 55.7%mol, CO₂ = 40%mol, H₂S = 150 ppm-mol, and Siloxanes = 2.14 ppm-mol) to biomethane with specifications of CH₄^MIN = 85%mol, CO₂^MAX = 3%mol, H₂S^MAX = 10 mg/Nm³, and Siloxanes^MAX = 0.03 mg/Nm³. This work demonstrates that the new model can be validated with bench-scale literature data and used in industrial-scale batteries with the same hydrodynamics. Once calibrated, the model becomes economically valuable since it can: (i) predict industrial contactor battery performance under scale-up/scale-down conditions; (ii) detect process faults, membrane leakages, and wetting; and (iii) be used for process troubleshooting.

1. Introduction

Landfills spontaneously release gases that entail sanitary, safety, and environmental issues, such as odors, combustion/explosion risks, and greenhouse gas emissions [1]. These gases are generically known as landfill-gas. Landfill-gas is generated by the anaerobic degradation of organic wastes in landfills and typically contains methane (CH₄) (30–65 %mol), carbon dioxide (CO₂) (25–47 %mol), hydrogen sulfide (H₂S) (30–500 ppm-mol), saturation water, and trace silicon compounds (0.3–36 ppm-mass dry basis). Landfill-gas may also contain nitrogen/oxygen from air, ammonia, and hydrogen [2]. The greenhouse warming potential of CH₄ is ≈21 times that of its CO₂ counterpart and landfill-gas releases are responsible for ≈17% of worldwide CH₄ emissions [3]. Moreover, landfill-gas can be converted into biomethane, so that sustainable landfills can be designed to simultaneously avoid CH₄ emissions while exporting biomethane [4] for use as, for example, household fuel-gas, renewable electricity generation [5], vehicular fuel-gas [6], and as a natural gas (NG) substitute [7].

The landfill-gas-to-biomethane process can accelerate energy transition to a low-carbon economy while ensuring energy supply, since solar photovoltaic and wind power energy are naturally intermittent [8] and are in the early development stage [9]. In this case, biomethane is delivered to pipeline networks, as well as associated and non-associated NGs [10], and unconventional NGs from shale-gas [11]. Another possibility is the landfill-gas-to-wire process, which consists of direct electricity generation dismissing purification/transportation [12].

Efficient landfill-gas recovery depends on several factors, such as the coating process, gas drainage, and leachate management [13]. In a capped landfill, landfill-gas is collected and processed to create NGs or biomethane specifications, i.e., via CO₂/H₂S removal, dehydration, and siloxane removal [14]. The landfill-gas-to-biomethane process starts with decarbonation, which increases the heating value, reduces the transportation volume, and minimizes CO₂ emissions from combustion [15]. Desulfurization is the next step aiming at reducing H₂S corrosiveness, toxicity, and acid rain potential [16]. Lastly, dehydration is conducted to avoid gas hydrates and condensation in pipelines [17].

Siloxanes are organosilicon compounds inexistent in nature containing Si-O-Si bonds and methyl groups attached to silicon atoms [18]. Landfill-gas siloxanes result from the decomposition of wastes containing silicon compounds (e.g., paints, coats/waxes, and shampoos/cosmetics) [19]. Siloxanes must be removed from landfill-gas because their oxidation in combustion engines creates SiO₂ deposits on metallic surfaces causing abrasion. In gas turbines, SiO₂ deposits cause blade erosion [20].

Table 1. Landfill-gas silicon compounds with typical contents [18,20].

Name	ID	Formula	Molar Mass (g/mol)	Content (mg/Nm3)
Hexamethyl-disiloxane	L2	C6H18OSi2	162.38	6.07
Octamethyl-trisiloxane	L3	C8H24O2Si3	236.53	-
Decamethyl-tetrasiloxane	L4	C10H30O3Si4	310.69	0.04
Dodecamethyl-pentasiloxane	L5	C12H36O4Si5	384.84	-
Tetradecamethyl-hexasiloxane	L6	C14H42O5Si6	458.99	0.01
Hexamethyl-cyclotrisiloxane	D3	C6H18O3Si3	222.46	0.49
Octamethyl-cyclotetrasiloxane	D4	C8H24O4Si4	296.62	12.53
Decamethyl-cyclopentasiloxane	D5	C10H30O5Si5	370.77	4.73
Dodecamethyl-cyclohexasiloxane	D6	C12H36O6Si6	444.93	0.33
Trimethyl-silanol	TMS	C3H10OSi	90.20	-
	Total (ppm-mol)			2.14

shows typical landfill-gas siloxanes.

After purification, biomethane is compressed for injection in NG grids [21], while the removed CO₂ can be compressed and pipeline transported to oil fields as an enhanced oil recovery (EOR) agent generating revenue that compensates carbon capture and storage (CCS) costs [22]. Since the CO₂ source in the landfill is mostly biomass originated from photosynthesis, landfill-gas processing with the CO₂-to-EOR process creates bioenergy with CCS; i.e., a BECCS system [23]. BECCS systems combine carbon-neutral bioenergy generation with CCS in suitable geological formations [24], yielding negative CO₂ emissions and continuous net CO₂ drainage from the atmosphere [25].

1.1. Landfill-Gas Decarbonation: Advantages of Gas–Liquid Membrane Contactors

Different approaches of decarbonation technologies to convert landfill-gas into biomethane are compared, and gas–liquid membrane contactors’ (GLMCs’) advantages are presented to clarify why they are a feasible landfill-gas decarbonation technology.

Common separation technologies for landfill-gas decarbonation present some disadvantages. Firstly, pressure swing adsorption [26] entails a biomethane purity recovery tradeoff [21], and H₂S/water should be removed beforehand [2]. Moreover, the operation is intermittent due to adsorbent regeneration demanding control and maintenance. Secondly, packed-column chemical absorption [27] presents several hydraulic issues, such as foaming, flooding, entrainment, obligatory gravity alignment, etc. [28]. Thirdly, membrane-permeation [29] dismisses solvent handling and is gravity indifferent, modular, and recommended for high CO₂ fugacity streams [30]. In spite of this, membrane-permeation is limited by permeability-selectivity tradeoff [31] and entails compression costs to generate a driving force [32].

A different concept is the new technology known as gas–liquid membrane contactors (GLMCs), a hybrid of chemical absorption and membrane-permeation that combines the high selectivity of the former with the gravity indifference, modularity, phase segregation, and high transfer area of the latter, without the respective drawbacks [33]. Several positive attributes recommend GLMCs for landfill-gas decarbonation (and desulfurization), such as modularity, linear scale-up, independent control of flowrates [34], gravity indifference, no hydraulic issues, high CO₂/CH₄ selectivity entailing low CH₄ losses [35], and high transfer area per shell [36] due to the high packing density [37]. Compared to packed-column chemical absorption, GLMC size and weight reductions reach 70% and 66%, respectively [38]. GLMC design can prescribe gas (V) and solvent (L) flows in parallel (co-current) contact [39] or in countercurrent contact [40]. V can flow inside hollow-fiber membranes (HFMs) [41] or in the shell [42] and vice versa for L. GLMCs can also work as CO₂ strippers for solvent regeneration, albeit they would require a stripping gas in this case, such as nitrogen [43]. A complete description of GLMC principles can be found elsewhere [44].

1.2. GLMC Modeling for Landfill-Gas/Biogas Decarbonation

The literature presents several GLMC modeling approaches for gas decarbonation with varying simplifications. A common simplification involves zero pressure-drop and isothermal operation with feeds/products at the same temperature, as was shown in the one-dimensional (1D) mass balance of Teplyakov et al. [45]. Belaissaoui et al. [46] also developed a simplified isothermal 1D mass transfer model for CO₂ physical absorption with pressurized water neglecting heat effects and pressure-drop. Belaissaoui and Favre [47] used the same isothermal 1D mass transfer model with HFM-side and shell-side pressure-drop calculations via Hagen–Poiseuille and Happel equations, respectively. The common simplifying assumptions of Teplyakov et al. [45], Belaissaoui et al. [46], and Belaissaoui and Favre [47] correspond to neglect convective solvent-gas heat transfer and absorption/phase-change thermal effects, consequently overestimating CO₂ solubility in the solvent. Gas absorption is always exothermic, i.e., isothermal absorption is unrealistic. Moreover, in Teplyakov et al. [45], Belaissaoui et al. [46], and Belaissaoui and Favre [47], H₂O L → V transfer and phase-change thermal effects were also neglected. In a similar context, Fougerit et al. [48] approached isothermal 1D GLMC modeling using OpenFOAM software to investigate decarbonation of CO₂-CH₄ mixtures.

Li et al. [49] studied GLMC landfill-gas decarbonation with aqueous-Selexol solving the two-dimensional (2D) mass transfer partial differential equations via the finite element method with COMSOL software. However, ideal gas behavior was assumed for P ≥ 12 bar and Selexol has high viscosity and is not recommended for GLMC HFMs. Previously, Li et al. [50] used aqueous K₂CO₃ for GLMC chemical absorption. Tantikhajorngosol et al. [51] investigated simultaneous isothermal transfers of CO₂ (40 %mol) and H₂S (500 ppm-mol) to pressurized water using compressibility factors for real gas behavior. Nakhjiri and Heydarinasab [52] compared GLMC decarbonation performances of CO₂-CH₄ mixtures with aqueous ethylenediamine, aqueous-2-(1-piperazinyl)-ethylamine, and aqueous potassium sarcosinate. All these exothermic absorptions were simulated via COMSOL isothermal modeling assuming ideal gas behavior without pressure drops.

As shown above, typical literature GLMC models for landfill-gas/biogas decarbonation present one or more simplifications, such as (i) isothermal operation; (ii) ideal gas behavior at moderate/high pressures; (iii) negligible H₂O L → V transfer and solvent losses; (iv) negligible CH₄ V → L transfer; and (v) the Henry’s law for CO₂ interfacial vapor–liquid equilibrium (VLE). Such simplifications entail unrealistic results for industrial conditions. Isothermal operations may be valid only for some small-scale low-loading physical absorption. Moreover, bidirectional water transfer (L → V or V → L) cannot be neglected since raw landfill-gas is usually water saturated and the solvent is aqueous [53]. Therefore, complete and thermodynamically rigorous GLMC modeling is rare in the literature. An example is presented by de Medeiros et al. [54] for high-pressure NG decarbonation with aqueous-monoethanolamine-methyldiethanolamine (aqueous-MEA-MDEA) adopting an acid-gas/water/MEA/MDEA reactive VLE [55] and assuming high-pressure V and L compressible flows with full thermodynamics via the Peng–Robinson equation-of-state (PR-EOS) and 1D V and L mass/energy/momentum balances [54].

Machine learning techniques typically use a large amount of data to train black-box models to perform reliable and realistic predictions [56]. In principle, these techniques could also be employed to predict GLMC CO₂ absorption [57] with aqueous solvents [58]. However, these models basically rely on a heavy load of information for extensive training in a statistical context and completely ignore mass/energy/momentum conservation and thermodynamics. Consequently, if not sufficiently trained over prohibitively extensive databases of initial/final temperatures/pressures/compositions, they can generate distorted predictions in deterministic processes totally driven by physical principles like distillation columns, heat exchangers, direct-contact columns, chemical reactors, and GLMC separations.

1.3. GLMC Modeling in Process Simulators

Simulation is an important tool for the analysis, monitoring, fault detection, troubleshooting, and design of chemical processes, as evaluations can be performed analytically with high precision, eliminating unnecessary time-consuming and costly experiments [59]. GLMC modeling for professional process simulators is important, because: (i) the model can take advantage of vast numbers of accurate thermodynamic/transport frameworks available in simulators; and (ii) the integration of a GLMC battery with the process flowsheet is immediately performed in the simulator, accelerating industrial design, process analysis, and economic evaluations. Despite this, process simulator GLMC studies are scarce in the literature and still constitute relevant scientific challenges [60].

Hoff et al. [61] developed a GLMC model for flue gas and high-pressure NG decarbonation with mass/energy/momentum balances. The VLE was modeled via Henry’s law with activity coefficients to account for liquid non-ideality. The model was validated with lab-scale experiments. Hoff and Svendsen [62] improved the previous thermodynamic model [61] to investigate the low-pressure decarbonation of offshore gas-turbine flue gas with the SINTEF/NTNU/CO2SIM simulator, and high-pressure NG decarbonation with the process simulator Protreat.

Quek et al. [63] studied high-pressure NG decarbonation with a 2D adiabatic GLMC CO₂ transfer model in gPROMS. Quek et al. [64] developed a more complete GLMC model for gPROMS admitting CO₂ transfer only, HFM pore-wetting prediction, PR-EOS gas behavior, water evaporation via Raoult’s law, hydrocarbon loading via Henry’s law, simplified energy balance, and no pressure drop. The model was validated against lab-scale and pilot-scale experiments. Quek et al. [65] employed the previous gPROMS model [64] for high-pressure NG decarbonation via the GLMC model evincing heat savings at the expense of inefficient CO₂ abatement.

Kerber and Repke [66] studied biogas purification with pressurized water and a flat-sheet membrane GLMC model, considering solvent evaporation and isothermal operation. Villeneuve et al. [67] developed an Aspen Modeler GLMC 1D adiabatic multicomponent transfer model for the comparison of GLMC and packed columns for NG decarbonation with aqueous ammonia considering ideal gas behavior and e-NRTL for the liquid phase. Villeneuve et al. [68] used an older [67] GLMC model to investigate the impact of water condensation on aqueous MEA NG decarbonation.

Usman et al. [69] studied high-pressure pre-combustion GLMC decarbonation using ionic-liquid 1-butyl-3-methylimidazolium tricyanomethanide with a MATLAB 1D transfer model based on resistance-in-series approaches. Posteriorly Usman et al. [70] integrated the MATLAB code with an Aspen-HYSYS simulator via Cape-Open resources, retaining isothermal behavior and Henry’s law VLEs.

Recently, McQuillan et al. [71] developed a one-dimensional distributed GLMC model for the Aspen Custom Modeler to evaluate potassium glycinate as a solvent for direct air capture, but the model presents simplifications, such as isothermal modeling, Henry’s law VLE, no energy balances, and no pressure-drop calculation.

A more efficient way to develop new unit operation models while maintaining access to Aspen-HYSYS rigorous thermodynamic resources is by creating HYSYS unit operation extensions (UOEs) [72]. However, the literature lacks works on GLMC HYSYS modeling, but there are exceptions for CO₂-rich NG high-pressure CO₂ removal. In da Cunha et al.’s study [73], a GLMC-UOE-1 considered just CO₂ transfer, without energy balances and pressure-drop calculations. Posteriorly, da Cunha et al. [74] developed a more complete GLMC-UOE model with multicomponent bidirectional transfer and mass/energy/momentum balances for gas and solvent flows, provided on an HYSYS thermodynamic basis. This study was conducted in offshore high-pressure conditions for the decarbonation of CO₂-rich NG via GLMC with aqueous-amines. However, since landfill-gas decarbonation occurs under milder conditions, the literature lacks a specific GLMC-UOE model developed for its specific hydrodynamic conditions.

1.4. Siloxane Removal from Landfill-Gas: The Advantages of Selexol Absorption

Several studies on landfill-gas sweetening consider pressurized-water absorption, but it was shown [75] that water absorption is inefficient for siloxane removal. Läntelä et al. [76] studied landfill-gas sweetening in a water-absorption column evincing a 16.6% efficiency of siloxane removal as follows: TMS/D5 was significantly removed, while L2/L3/D3 was enriched, and L4/L5/D4 remained invariant.

The literature indicates Dimethyl-Ether-Polyethylene-Glycol (DEPG)—Selexol—as a promising siloxane absorbent from landfill-gas, with successful applications [75]. Ryckebosch et al. [77] suggest DEPG also for CO₂/H₂S removal from landfill-gas, as is performed for synthesis-gas purification [78]. However, DEPG absorption is an expensive technology, i.e., it is not advisable to use DEPG absorption for full landfill-gas purification. Belaissaoui and Favre [47] suggest siloxane and H₂S DEPG removal prior to landfill-gas decarbonation with water. However, in this case, the higher H₂S content will unnecessarily compete with trace siloxanes. Moreover, Faiz et al. [79] already proved pressurized-water reliability for H₂S removal and Li et al. [80] approved pressurized-water packed columns for biogas decarbonation and suggested its replacement by GLMC.

Given these facts, and since landfill-gas is similar to biogas, the present work proposes a new landfill-gas-to-biomethane route; namely: GLMC CO₂/H₂S removal with pressurized water, followed by siloxane removal via a finishing DEPG-absorption column requiring a low DEPG circulation rate. GLMC CO₂/H₂S water absorption and siloxane DEPG absorption are both modeled via the HYSYS Acid-Gas Physical-Solvents Thermodynamic Package based on the PC-SAFT equation-of-state [81].

1.5. The Present Work

As shown in Section 1.2, the literature presents recent studies on GLMC modeling for landfill-gas decarbonation, but all bear modeling deficiencies or scope limitations. Moreover, process simulation GLMC studies are still scarce (Section 1.3). A remarkable literature gap is the absence of complete GLMC models with multicomponent mass/energy/momentum balances, sustained by an adequate thermodynamic framework for vapor–liquid equilibrium and reliable predictions of thermodynamic/transport properties.

The present work discloses a novel GLMC HYSYS unit operation extension—GLMC-UOE—developed for the steady-state simulation of multicomponent bidirectional (V → L, L → V) mass/heat transmembrane transfers using a fugacity-difference driving force, rigorous VLE, and thermodynamic modeling via the HYSYS Acid-Gas Physical-Solvents Package and rigorous energy balances taking into account chemical/phase-change/compressibility temperature effects, pressure drops, and non-isothermal operations. GLMC-UOE-simulated CO₂/H₂S removal from landfill-gas with pressurized water and can handle GLMC countercurrent and parallel contacts generating composition/temperature/pressure profiles. It is demonstrated that the new HYSYS-based contactor model can be validated with bench-scale literature data and used in industrial-scale batteries with the same module hydrodynamics. Moreover, once calibrated, the model becomes economically valuable since it can: (i) predict the performance of industrial contactor batteries under scale-up/scale-down conditions; (ii) detect process faults, membrane leakages, and wetting; and (iii) be used for process troubleshooting.

The literature presents several incomplete landfill-gas/biogas purification studies, as most of them ignore H₂S and siloxane removal and/or CH₄/H₂O bidirectional transfers and/or temperature/pressure changes. In addition, the proposed landfill-gas-to-biomethane route adopts novel intensified operations to reduce space requirements and to improve energy efficiency. As shown in Section 1.4, the selection of an adequate solvent for siloxane removal is a critical step for biomethane specification. Since Selexol is considered efficient, albeit expensive, this process proposes CO₂ and H₂S removal prior to Selexol absorption for siloxane removal, to reduce the amount of Selexol required. This work proved that Selexol absorption is a promising choice for siloxane removal.

In summary, a novel and complete waste-to-energy landfill-gas-to-biomethane process was solved in Aspen-HYSYS 10.0 considering: (i) an intensified non-isothermal GLMC battery with pressurized water (T = 15 °C, P = 7 bar) for CO₂/H₂S removal with CO₂/H₂S/CH₄/H₂O bidirectional transmembrane transfers and heat effect predictions; (ii) intensified CO₂/H₂S stripping at P = 30 bar, reducing costs for CO₂-to-EOR (P = 300 bar) compression; and (iii) multicomponent siloxane removal using a DEPG absorption (T = 15 °C, P = 7 bar) column.

2. Methods

GLMC-UOE modeling and the landfill-gas-to-biomethane process are addressed.

2.1. GLMC-UOE Development

GLMC-UOE was created using Visual-Basic 6.0 with the embedded HYSYS-Type Library that offers runtime commands to access HYSYS. An external dynamic link library (DLL) file was generated by code compilation. An extension definition file (EDF) was also developed with the Aspen-HYSYS Extension View Editor for DLL linkage to HYSYS and to create user-interface windows for setting GLMC conditions during the simulations in HYSYS PFD. After registering the extension, GLMC-UOE becomes available in a model palette and can access all HYSYS stream/property calculation resources.

GLMC-UOE considers 1D steady-state L and V axial flows for the simulation of GLMC batteries of N_M paralleled modules. After being entered into the GLMC module, both L and V can become two-phase There are two models for different L/V contacts: (i) the countercurrent-contact distributed GLMC model (GLMC-CCC-D); and (ii) parallel-contact (co-current) distributed GLMC model (GLMC-PC-D). Both models are built for landfill-gas decarbonation/desulfurization with physical-solvent pressurized water and involve axially discretizing a GLMC module as a succession of M elements in Figure 1a,b, where the countercurrent V/L streams (arrows) are valid for GLMC-CCC-D (Figure 1a) and the parallel V/L streams for GLMC-PC-D (Figure 1b).

Besides the aforesaid concepts, GLMC distributed model assumptions comprise: (i) L/V countercurrent or parallel contacts; (ii) V on the HFM side and L on the shell side; (iii) 1D axial two-phase plug flow for L/V mass/energy/momentum balances; (iv) rigorous VLEs and thermodynamic and transport property calculations via the Aspen-HYSYS Acid-Gas Physical-Solvents Package (PC-SAFT equation-of-state); (v) multicomponent system with CO₂/CH₄/H₂S/H₂O and the siloxanes in Table 1 (except L3/L5/TMS; number of components nc = 11), where CO₂/CH₄/H₂S/H₂O are the only species for which bidirectional transfers (V → L, L → V) are considered, since siloxanes are heavy species that practically do not transfer to water; (vi) L/V pressure drop; (vii) L/V outlet composition/temperature/pressure calculations; (viii) adiabatic GLMC modules (external heat transfer coefficient U_E = 0) with convective transmembrane heat transfer (internal heat transfer coefficient U_I ≠ 0); (ix) L/V absorption/compressibility/phase-change heat effects; (x) distributed transmembrane heat and species transfer model using respective driving forces log-mean temperature difference and log-mean species fugacity differences; (xi) direction of positive heat/mass transfers: V → L; (xii) countercurrent GLMC module with discretizationas a cascade of M small countercurrent GLMC elements (Figure 1a) solved with simultaneous corrections [82] Newton–Raphson iterations; (xiii) parallel GLMC module discretized as a cascade of M small parallel GLMC elements (Figure 1b) sequentially solved via element Newton–Raphson iterations; and (xiv) GLMC dependent variables are nc × 1 vectors ${\underset{¯}{L}}_n,\hspace{0.17em}\hspace{0.17em}{\underset{¯}{V}}_n$ of component flowrates (mol/s) that leave element n (n = 1…M) (for the entire battery) as well as the temperatures/pressures ${\underset{¯}{L}}_n,{\underset{¯}{V}}_n,T_{L_n},T_{V_n},P_{L_n},P_{V_n}$ (n = 1…M). Interfacial heat and mass transfer fluxes for each element were eliminated from the phenomenological relationships in terms of driving forces. The specifications were battery size, GLMC module geometry, and the two nc × 1 vectors of component feeds (mol/s) and their temperatures/pressures, ${\underset{¯}{L}}_0,{\underset{¯}{V}}_{M+1},T_{L_0},T_{V_{M+1}},P_{L_0},P_{V_{M+1}}$ for GLMC-CCC-D, and ${\underset{¯}{L}}_0,{\underset{¯}{V}}_0,T_{L_0},T_{V_0},P_{L_0},P_{V_0}$ for GLMC-PC-D.

2.1.1. Element Mass Balances

For each GLMC element, n (n = 1…M), the transmembrane mass transfer of species k (k = 1…nc) was calculated with the log-mean differences of species k fugacities (${\Delta \hat{f}}_{k,n}^{\mathit{LM}}$, bar) as the driving force [73] (asymptotically correct as M increases). Supposed constant, the transmembrane species mass transfer coefficients (${\Pi }_k$, mol/(s.bar.m²), k = 1…nc) were calibrated to adjust transmembrane mass transfer rates for element n ($N_{k,n}$, mol/s) as performed by de Medeiros et al. [54], except for the non-transferable siloxanes, where ${\Pi }_k=0, N_{k,n}=0$. The GLMC module heat/mass transfer area was $A_{GLMC}$, while the element transfer area was $A_{GLMC}/M$. Thus, the single-module separation capacity was pre-defined [44]. In order to determined separation targets, the battery transfer area can be increased (or decreased) with the increase (or decrease) in N_M in the GLMC-UOE parameter window. The calibration of mass transfer coefficients, ${\Pi }_k$, consists of adjusting CO₂/CH₄/H₂S/H₂O transmembrane transfers with GLMC experimental data (Appendix A, Appendix B, Appendix C and Appendix D), as in da Cunha et al. [74]. The set of transferable species is {TS} ≡ {CO₂, CH₄, H₂S, H₂O}. Set $\left\{T{S}^{+}\right\}$ contains species with $N_{k,n}>0$ (V → L transfer), while set $\left\{T{S}^{-}\right\}$ contains those with $N_{k,n}<0$ (L → V transfer), i.e., $\left\{TS\right\}=\left\{T{S}^{+}\right\}\cup \left\{T{S}^{-}\right\}$. The transference direction of a transferable component has to be updated for the elements in the axial direction, so that for $k\in \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left\{TS\right\}$, ${\widehat{f}}_{k,n}^V>{\widehat{f}}_{k,n}^L\Rightarrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k\in \{T{S}^{+}\}$; otherwise, $k\in \left\{T{S}^{-}\right\}$. In the landfill-gas context, normally $\left\{T{S}^{+}\right\}$ = {CO₂, H₂S, CH₄} and $\left\{T{S}^{-}\right\}$= {H₂O}. The model does not take into account membrane pore wetting directly, i.e., pore wetting is superseded by the calibration of ${\Pi }_k,\hspace{0.17em}\hspace{0.17em}k\in \left\{TS\right\}$ by combining GLMC-UOE with experimental GLMC data. Once calibrated, ${\Pi }_k,\hspace{0.17em}\hspace{0.17em}k\in \left\{TS\right\}$ can be supposed as invariant to relatively small changes in design and/or hydrodynamics. Moreover, calibrated ${\Pi }_k,\hspace{0.17em}\hspace{0.17em}k\in \left\{TS\right\}$ is invariant for greater (lower) V values with a constant V/L ratio and the same feed temperature/pressure and module geometry, where the module number, N_M, increases or decreases proportionally to V in order to maintain module hydrodynamics [74].

Equations (1a)–(1d) and (2a)–(2c) represent the multicomponent mass balances in element n (n = 1…M), where Equations (1c) and (2a) apply to GLMC-CCC-D and Equations (1d) and (2b) for GLMC-PC-D, and ${\widehat{f}}_{k,n}^V,{\widehat{f}}_{k,n}^L$ are species k respective fugacities (bar) in V/L streams that leave n. In Equations (1a)–(2c), the species transfer rates ($N_{k,n}$, mol/s) are eliminated using the right-hand sides of Equations (1a)–(1d). Consequently, the mass balances reduce to 2nc element equations presented in Equations (2a) (or (2b)) and (2c). Equation (3) represents the resulting 2nc × 1 vector of mass balances for element n (in GLMC-CCC-D or GLMC-PC-D), whose dependent variables are the element 2nc + 4 outlet variables (${\underset{¯}{L}}_n,{\underset{¯}{V}}_n,T_{L_n},T_{V_n},P_{L_n},P_{V_n}$) [82].

$$N_{k,n}={\Pi }_k\left(N_M{A}_{GLMC}/M\right)\mathit{\Delta }{\widehat{f}}_{k,n}^{LM}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(k\in \hspace{0.17em}\hspace{0.17em}\{TS\}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=1\dots M)$$

(1a)

$$N_{k,n}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(k\notin \hspace{0.17em}\hspace{0.17em}\{TS\}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=1\dots M)$$

(1b)

$$\mathit{\Delta }{\widehat{f}}_{k,n}^{LM}={\displaystyle \frac{({\widehat{f}}_{k,n}^V-{\widehat{f}}_{k,n-1}^L)-({\widehat{f}}_{k,n+1}^V-{\widehat{f}}_{k,n}^L)}{\mathit{ln}({\widehat{f}}_{k,n}^V-{\widehat{f}}_{k,n-1}^L)-\mathit{ln}({\widehat{f}}_{k,n+1}^V-{\widehat{f}}_{k,n}^L)}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(k\in \hspace{0.17em}\hspace{0.17em}\{TS\}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=1\dots M,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}GLMC-CCC-D)$$

(1c)

$$\mathit{\Delta }{\widehat{f}}_{k,n}^{LM}={\displaystyle \frac{({\widehat{f}}_{k,n-1}^V-{\widehat{f}}_{k,n-1}^L)-({\widehat{f}}_{k,n}^V-{\widehat{f}}_{k,n}^L)}{\mathit{ln}({\widehat{f}}_{k,n-1}^V-{\widehat{f}}_{k,n-1}^L)-\mathit{ln}({\widehat{f}}_{k,n}^V-{\widehat{f}}_{k,n}^L)}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(k\in \hspace{0.17em}\hspace{0.17em}\{TS\}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=1\dots M,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}GLMC-PC-D)$$

(1d)

$$V_{k,n}+N_{k,n}-V_{k,n+1}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(k=1\dots nc,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=1\dots M,\hspace{0.17em}\hspace{0.17em}GLMC-CCC-D)$$

(2a)

$$V_{k,n}+N_{k,n}-V_{k,n-1}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(k=1\dots nc,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=1\dots M,\hspace{0.17em}\hspace{0.17em}GLMC-PC-D)$$

(2b)

$$L_{k,n-1}+N_{k,n}-L_{k,n}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(k=1\dots nc,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=1\dots M)$$

(2c)

$${\underset{¯}{F}}_n^{MB}=\underset{¯}{0}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(n=1\dots M)$$

(3)

2.1.2. Element Energy Balances

V → L is the positive transfer direction, as previously stated. For element n (n = 1…M), energy balances are presented for V/L streams—of the entire battery—considering the energy transport of V/L inlet/outlet streams, transmembrane convective heat transfer, and the transmembrane energy transfer coupled to the transfer of species k presented by the species k transfer rate in element n times k partial molar enthalpy at origin. The V/L partial molar enthalpies (kJ/mol) of k at origin in element n, $<{\overline{H}}_{k,n}^V>,\hspace{0.17em}\hspace{0.17em}<{\overline{H}}_{k,n}^L>$, are approximated by the arithmetic means of respective inlet/outlet partial molar enthalpies ($<{\overline{H}}_{k,n}^V>=({\overline{H}}_{k,n+1}^V+{\overline{H}}_{k,n}^V)/2$ for GLMC-CCC-D, $<{\overline{H}}_{k,n}^V>=({\overline{H}}_{k,n-1}^V+{\overline{H}}_{k,n}^V)/2$ for GLMC-PC-D, and $<{\overline{H}}_{k,n}^L>=({\overline{H}}_{k,n-1}^L+{\overline{H}}_{k,n}^L)/2$), which becomes asymptotically correct as M → ∞. V energy balance is expressed via Equation (5a) for GLMC-CCC-D and via Equation (5b) for GLMC-PC-D, while L energy balance is expressed via Equation (5c) for GLMC-CCC-D and via Equation (5d) for GLMC-PC-D, where ${\overline{H}}_{V_n},\hspace{0.17em}\hspace{0.17em}{\overline{H}}_{L_n}$ (kJ/mol) are V/L molar enthalpies leaving element n, and 1 is an nc × 1 vector of ones. Since, partial molar enthalpies are not available in the HYSYS (only molar enthalpies are), they have to be calculated via Equations (4a) and (4b) using mole fraction ($Y_k,X_k$) derivatives of molar enthalpies numerically generated with finite differences, as in da Cunha et al. [74].

$${\overline{H}}_k^V={\overline{H}}_V-{{\displaystyle \sum _{i=1}^{nc}\hspace{0.17em}\hspace{0.17em}\left(\frac{\partial {\overline{H}}_V}{\partial Y_i}\right)}}_{T,P,Y_{j\ne i}}+{\left({\displaystyle \frac{\partial {\overline{H}}_V}{\partial Y_k}}\right)}_{T,P,Y_{j\ne k}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(Y_k=V_k/{\underset{¯}{1}}^T\underset{¯}{V})$$

(4a)

$${\overline{H}}_k^L={\overline{H}}_L-{{\displaystyle \sum _{i=1}^{nc}\hspace{0.17em}\hspace{0.17em}\left(\frac{\partial {\overline{H}}_L}{\partial X_i}\right)}}_{T,P,X_{j\ne i}}+{\left({\displaystyle \frac{\partial {\overline{H}}_L}{\partial X_k}}\right)}_{T,P,X_{j\ne k}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(X_k=L_k/{\underset{¯}{1}}^T\underset{¯}{L})$$

(4b)

Element n transmembrane heat transfer is presented with the transmembrane log-mean temperature difference for element n ($\Delta T_n^{LM}$ in Equation (6a) for GLMC-CCC-D, or in Equation (6b) for GLMC-PC-D, also asymptotically correct as M → ∞), the internal heat transfer coefficient $U_I$(kW/m².K), and $N_M$$A_{GLMC}/M$ the element n battery internal transfer area. For GLMC-CCC-D [74], Equation (6a) is substituted in Equations (5a) and (5c), reducing the energy-balance equations of element n to only Equations (5a) and (5c). Analogously, for GLMC-PC-D [54], Equation (6b) is substituted in Equations (5b) and (5d). In both cases, the resulting 2 × 1 vector of element n energy balances is written as ${\underset{¯}{F}}_n^{EB}=\underset{¯}{0}$ in Equation (7). The 2nc + 4 outlet variables of element n (${\underset{¯}{L}}_n,{\underset{¯}{V}}_n,T_{L_n},T_{V_n},P_{L_n},P_{V_n}$) are the dependent variables of ${\underset{¯}{F}}_n^{EB}=\underset{¯}{0}$.

$$({\underset{¯}{1}}^T{\underset{¯}{V}}_{n+1}){\overline{H}}_{V_{n+1}}-({\underset{¯}{1}}^T{\underset{¯}{V}}_n){\overline{H}}_{V_n}-{\displaystyle \sum _{k\in \left\{T{S}^{+}\right\}}{N}_k\left(\frac{{\overline{H}}_{k,n+1}^V+{\overline{H}}_{k,n}^V}{2}\right)}-{\displaystyle \sum _{k\in \left\{T{S}^{-}\right\}}{N}_k\left(\frac{{\overline{H}}_{k,n-1}^L+{\overline{H}}_{k,n}^L}{2}\right)}-{\displaystyle \frac{U_I{N}_M{A}_{GLMC}\mathit{\Delta }{T}_n^{LM}}{M}}=0$$

(5a)

$$({\underset{¯}{1}}^T{\underset{¯}{V}}_{n-1}){\overline{H}}_{V_{n-1}}-({\underset{¯}{1}}^T{\underset{¯}{V}}_n){\overline{H}}_{V_n}-{\displaystyle \sum _{k\in \left\{T{S}^{+}\right\}}{N}_{k,n}\left(\frac{{\overline{H}}_{k,n-1}^V+{\overline{H}}_{k,n}^V}{2}\right)}-{\displaystyle \sum _{k\in \left\{T{S}^{-}\right\}}{N}_{k,n}\left(\frac{{\overline{H}}_{k,n-1}^L+{\overline{H}}_{k,n}^L}{2}\right)}-{\displaystyle \frac{U_I{N}_M{A}_{GLMC}\mathit{\Delta }{T}_n^{LM}}{M}}=0$$

(5b)

$$({\underset{¯}{1}}^T{\underset{¯}{L}}_{n-1}){\overline{H}}_{L_{n-1}}-({\underset{¯}{1}}^T{\underset{¯}{L}}_n){\overline{H}}_{L_n}+{\displaystyle \sum _{k\in \left\{T{S}^{+}\right\}}{N}_k\left(\frac{{\overline{H}}_{k,n+1}^V+{\overline{H}}_{k,n}^V}{2}\right)}+{\displaystyle \sum _{k\in \left\{T{S}^{-}\right\}}{N}_k\left(\frac{{\overline{H}}_{k,n-1}^L+{\overline{H}}_{k,n}^L}{2}\right)}+{\displaystyle \frac{U_I{N}_M{A}_{GLMC}\mathit{\Delta }{T}_n^{LM}}{M}}=0$$

(5c)

$$({\underset{¯}{1}}^T{\underset{¯}{L}}_{n-1}){\overline{H}}_{L_{n-1}}-({\underset{¯}{1}}^T{\underset{¯}{L}}_n){\overline{H}}_{L_n}+{\displaystyle \sum _{k\in \left\{T{S}^{+}\right\}}{N}_k\left(\frac{{\overline{H}}_{k,n-1}^V+{\overline{H}}_{k,n}^V}{2}\right)}+{\displaystyle \sum _{k\in \left\{T{S}^{-}\right\}}{N}_k\left(\frac{{\overline{H}}_{k,n-1}^L+{\overline{H}}_{k,n}^L}{2}\right)}+{\displaystyle \frac{U_I{N}_M{A}_{GLMC}\mathit{\Delta }{T}_n^{LM}}{M}}=0$$

(5d)

$$\mathit{\Delta }{T}_n^{LM}={\displaystyle \frac{\left(T_{V_n}-T_{L_{n-1}}\right)-\left(T_{V_{n+1}}-T_{L_n}\right)}{\mathit{ln}(T_{V_n}-T_{L_{n-1}})-\mathit{ln}(T_{V_{n+1}}-T_{L_n})}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(GLMC-CCC-D)$$

(6a)

$$\mathit{\Delta }{T}_n^{LM}={\displaystyle \frac{\left(T_{V_n}-T_{L_n}\right)-\left(T_{V_{n-1}}-T_{L_{n-1}}\right)}{\mathit{ln}(T_{V_n}-T_{L_n})-\mathit{ln}(T_{V_{n-1}}-T_{L_{n-1}})}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(GLMC-PC-D)$$

(6b)

$${\underset{¯}{F}}_n^{EB}=\underset{¯}{0}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(n=1\dots M)$$

(7)

2.1.3. Element Pressure Drop

The element n pressure-drop calculation aims at determining its outlet pressures $P_{L_n},P_{V_n}$. To calculate hydraulic diameters and flow-section areas, HFMs are perceived as rigid with external and internal diameters, $d_o,d_i$. The HFM bundle is distributed with HFM centers on an equilateral triangular lattice in a GLMC shell (Figure 2). Hence, the center–center distance, p_HF, of adjacent HFMs is constant, and S_FREE defines in each triangle the free flow area. In Figure 2a, simple reasoning shows that the number of triangles (N_TRI) is asymptotically presented in Equation (8) for $N_{HF}$ HFMs per shell. Since the entire shell transversal section (diameter D) is covered by the triangular lattice without triangle superposition, Equation (9) and Equation (10), respectively, hold for p_HF and S_FREE. With $A_{CSS}$ (m²) as the shell-side flow section in Equation (11)—which asymptotically ($N_{HF}\to \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\infty$) equals N_TRI*S_FREE—and with HP_S as the shell-side hydraulic perimeter, one can see that the shell-side hydraulic diameter, $d_{HS}$ (m), in the second term of Equation (12) is also the hydraulic diameter for S_FREE in the last term of Equation (12).

$$N_{TRI}=2N_{HF}-2\sqrt{N_{HF}}$$

(8)

$$N_{TRI}{\displaystyle \frac{\sqrt{3}}{4}}{p}_{HF}^2={\displaystyle \frac{\pi D^2}{4}}\Rightarrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{p}_{HF}=\sqrt{{\displaystyle \frac{\pi D^2/4}{(\sqrt{3}/2)(N_{HF}-\sqrt{N_{HF}})}}}$$

(9)

$$S_{FREE}={\displaystyle \frac{\sqrt{3}}{4}}{p}_{HF}^2-(1/2){\displaystyle \frac{\pi d_o^2}{4}}$$

(10)

$$A_{CSS}={\displaystyle \frac{\pi D^2}{4}}-N_{HF}{\displaystyle \frac{\pi d_o^2}{4}}$$

(11)

$$d_{HS}={\displaystyle \frac{4A_{CSS}}{H{P}_S}}={\displaystyle \frac{4S_{FREE}}{\pi d_o/2}}=d_o\left({\displaystyle \frac{2\sqrt{3}}{\pi }}{\left({\displaystyle \frac{p_{HF}}{d_o}}\right)}^2-1\right)$$

(12)

Equation (13) presents the HFM-side hydraulic diameter, $d_{HHF}$ (m), while HFM-side flow-section $A_{CSHF}$ (m²) and HFM outlet gas velocity $v_{V_n}$ (m/s) of element n, with ${\overline{\rho }}_{V_n}$ (mol/m³) as the gas molar density, follow in Equation (14) and Equation (15), respectively. The HFM-side pressure-drop (head-loss) for element n, $h_{V_n}^{HF}$ (Pa), is presented by the Hagen–Poiseuille equation in Equation (16a) for GLMC-CCC-D and in Equation (16b) for GLMC-PC-D, where the property inlet/outlet arithmetic means are used and $Z_M$ (m) is the module length. The shell-side pressure-drop across element n, $h_{L_n}^S$ (Pa), is calculated via the Happel equation (Equation (17)) with the support of Equation (18), Equation (19), and Equation (20), respectively determining the Kozeny factor, $\kappa$ [83], the outlet liquid velocity, $v_{L_n}$, and the packing ratio, $\phi$ [83], where ${\overline{\rho }}_{L_n}$ (mol/m³) represents the outlet L mol density. Dynamic viscosities ${\mu }_V$,${\mu }_L$ in Equations (16a), (16b), and (17) are approximated by the viscosity of the predominant phase (gas for V and liquid for L), since V and L streams can become (or not) two phase at an axial position in the GLMC. The GLMC outlet V/L absolute pressures of element n, $P_{V_n},P_{L_n}$(bar), are calculated via Equation (21a) for GLMC-CCC-D, Equation (21b) for GLMC-PC-D, and Equation (22).

$$d_{HHF}=d_i$$

(13)

$$A_{CSHF}=N_{HF}\pi d_i^2/4$$

(14)

$$v_{V_n}={\displaystyle \frac{{\underset{¯}{1}}^T{\underset{¯}{V}}_n}{{\overline{\rho }}_{V_n}{N}_M{A}_{CSHF}}}$$

(15)

$$h_{V_n}^{HF}={\displaystyle \frac{32(Z_M/M)}{d_{HHF}^2}}\left({\displaystyle \frac{v_{V_{n+1}}+v_{V_n}}{2}}\right)\left({\displaystyle \frac{{\mu }_{V_{n+1}}+{\mu }_{V_n}}{2}}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(GLMC-CCC-D)$$

(16a)

$$h_{V_n}^{HF}={\displaystyle \frac{32(Z_M/M)}{d_{HHF}^2}}\left({\displaystyle \frac{v_{V_{n-1}}+v_{V_n}}{2}}\right)\left({\displaystyle \frac{{\mu }_{V_{n-1}}+{\mu }_{V_n}}{2}}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(GLMC-PC-D)$$

(16b)

$$h_{L_n}^S={\displaystyle \frac{16\kappa (Z_M/M)}{d_{HS}^2}}\left({\displaystyle \frac{{\mu }_{L_{n-1}}+{\mu }_{L_n}}{2}}\right)\left({\displaystyle \frac{v_{L_{n-1}}+v_{L_n}}{2}}\right){\displaystyle \frac{{\left(\phi \right)}^2}{{\left(1-\phi \right)}^2}}$$

(17)

$$\kappa =150{\phi }^4-314.44{\phi }^3+241.67{\phi }^2-83.039\phi +15.97$$

(18)

$$v_{L_n}={\displaystyle \frac{{\underset{¯}{1}}^T{\underset{¯}{L}}_n}{{\overline{\rho }}_{L_n}{N}_M{A}_{CSS}}}$$

(19)

$$\phi =N_{HF}{d}_o^2/D^2$$

(20)

$$P_{V_{n+1}}-P_{V_n}-{10}^{-5}{h}_{V_n}^{HF}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(GLMC-CCC-D)$$

(21a)

$$P_{V_{n-1}}-P_{V_n}-{10}^{-5}{h}_{V_n}^{HF}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(GLMC-PC-D)$$

(21b)

$$P_{L_{n-1}}-P_{L_n}-{10}^{-5}{h}_{L_n}^S=0$$

(22)

Equations (8)–(16a) and (16b) can be substituted into Equation (21a) for GLMC-CCC-D or into Equation (21b) for GLMC-PC-D, and Equations (17)–(20) can be substituted into Equation (22), so that the set of pressure-drop equations is reduced to Equations (21a) or (21b) and Equation (22). For element n, the final 2 × 1 vector of pressure-drop equations is expressed by Equation (23). Dependent variables of ${\underset{¯}{F}}_n^{PD}$ are the 2nc + 4 outlet variables ${\underset{¯}{L}}_n,{\underset{¯}{V}}_n,T_{L_n},T_{V_n},P_{L_n},P_{V_n}$ of element n.

$${\underset{¯}{F}}_n^{PD}=\underset{¯}{0}$$

(23)

For element n, the vector ${\underset{¯}{\eta }}_n$ of 2nc + 4 dependent variables and the vector ${\underset{¯}{R}}_n$ of 2nc + 4 independent equation residues—either for GLMC-CCC-D or for GLMC-PC-D—are presented in Equation (24) for n = 1, …, M. Equation (25) represents the complete vector of (2nc + 4)*M GLMC dependent variables ($\underset{¯}{\eta }$) and the complete vector of (2nc + 4)*M GLMC residues ($\underset{¯}{R}$).

$${\underset{¯}{\eta }}_n=\left[\begin{array}{c}{\underset{¯}{L}}_n\\ T_{L_n}\\ P_{L_n}\\ {\underset{¯}{V}}_n\\ T_{V_n}\\ P_{V_n}\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\underset{¯}{R}}_n=\left[\begin{array}{c}{\underset{¯}{F}}_n^{MB}\\ {\underset{¯}{F}}_n^{EB}\\ {\underset{¯}{F}}_n^{PD}\end{array}\right]=\underset{¯}{0}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(n=1\dots M)$$

(24)

$$\underset{¯}{R}(\underset{¯}{\eta })=\left[\begin{array}{c}{\underset{¯}{R}}_1\\ {\underset{¯}{R}}_2\\ ⋮\\ {\underset{¯}{R}}_M\end{array}\right]=\underset{¯}{0},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\underset{¯}{\eta }=\left[\begin{array}{c}{\underset{¯}{\eta }}_1\\ {\underset{¯}{\eta }}_2\\ ⋮\\ {\underset{¯}{\eta }}_M\end{array}\right]$$

(25)

2.1.4. Algorithm to Solve the Countercurrent GLMC Model (GLMC-CCC-D)

The GLMC-CCC-D system of (2nc + 4)*M residues is solved for the (2nc + 4)*M variables in Equation (25) using a cascade Newton–Raphson method known as the simultaneous corrections method [82] originally developed for countercurrent multistage cascades [74]. In this method, iterations occur simultaneously over all elements in order to solve Equations (24) and (25) until convergence. This rather involved and rigorous method will be used to solve GLMC-CCC-D for CO₂/H₂S removal from landfill-gas using partially analytical and partially numerical Jacobian matrices. It was developed elsewhere [74] and will not be further explained here. Figure 3 presents an algorithm flowchart for solving countercurrent-contact GLMCs (GLMC-CCC-D) (Figure 3a).

2.1.5. Algorithm to Solve the Parallel-Contact GLMC (GLMC-PC-D)

Despite the fact that the system in Equation (24) seems to be the same for GLMC-CCC-D and GLMC-PC-D models, the truth is that there are important differences between them because GLMC-CCC-D is a boundary value problem, while GLMC-PC-D is an initial value problem, which means that the GLMC-PC-D elements can be sequentially solved. As shown by da Cunha et al. [74], to solve GLMC-CCC-D, Newton–Raphson iterations throughout the entire cascade are necessary, while to solve GLMC-PC-D, it is only necessary to conduct Newton–Raphson iterations to solve Equation (24) until convergence for each element, n, sequentially starting from element 1 and terminating at element M. The algorithm for GLMC-PC-D is shown in Equation (26). Figure 3 presents a flowchart of the algorithm for solving a parallel-contact GLMC (GLMC-PC-D) (Figure 3b).

$$\left\{\begin{array}{l}Enter\hspace{0.17em}\hspace{0.17em}Feed\hspace{0.17em}\hspace{0.17em}Data:\hspace{0.17em}\hspace{0.17em}{\underset{¯}{L}}_0,{\underset{¯}{V}}_0,T_{L_0},T_{V_0},P_{L_0},P_{V_0}\\ Create\hspace{0.17em}\hspace{0.17em}Initial\hspace{0.17em}\hspace{0.17em}Estimate\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}{\underset{¯}{\eta }}_1:{\underset{¯}{\eta }}_1^{(0)}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(e.g.,\hspace{0.17em}\hspace{0.17em}{\underset{¯}{\eta }}_1^{(0)}={[{\underset{¯}{L}}_0\hspace{0.17em}\hspace{0.17em}{T}_{L_0}\hspace{0.17em}\hspace{0.17em}{P}_{L_0}\hspace{0.17em}\hspace{0.17em}{\underset{¯}{V}}_0\hspace{0.17em}\hspace{0.17em}{T}_{V_0}\hspace{0.17em}\hspace{0.17em}{P}_{V_0}]}^T)\\ For\hspace{0.17em}\hspace{0.17em}n=1\to M\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}With\hspace{0.17em}\hspace{0.17em}{\underset{¯}{\eta }}_n^{(0)}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Solve\hspace{0.17em}\hspace{0.17em}{\underset{¯}{R}}_n=\underset{¯}{0}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}{\underset{¯}{\eta }}_n\hspace{0.17em}\hspace{0.17em}via\hspace{0.17em}\hspace{0.17em}Newton-Raphson\hspace{0.17em}\hspace{0.17em}Method\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}If\hspace{0.17em}\hspace{0.17em}n<M\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Create\hspace{0.17em}\hspace{0.17em}Initial\hspace{0.17em}\hspace{0.17em}Estimate\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}{\underset{¯}{\eta }}_{n+1}:{\underset{¯}{\eta }}_{n+1}^{(0)}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(e.g.,\hspace{0.17em}\hspace{0.17em}{\underset{¯}{\eta }}_{n+1}^{(0)}={\underset{¯}{\eta }}_n)\\ End\end{array}\right.$$

(26)

Since GLMC-PC-D elements are small parallel-contact contactors, the elements’ logarithmic mean driving forces, Equations (1d) and (6b), are asymptotically equal to more palatable arithmetic mean driving forces, Equation (27a) and Equation (27b), respectively. That is, Equations (27a) and (27b) can replace Equations (1d) and (6b), respectively.

$$\mathit{\Delta }{\widehat{f}}_{k,n}^{LM}={\displaystyle \frac{({\widehat{f}}_{k,n-1}^V-{\widehat{f}}_{k,n-1}^L)+({\widehat{f}}_{k,n}^V-{\widehat{f}}_{k,n}^L)}{2}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(k\in \hspace{0.17em}\hspace{0.17em}\{TS\}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=1\dots M,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}GLMC-PC-D)$$

(27a)

$$\mathit{\Delta }{T}_n^{LM}={\displaystyle \frac{\left(T_{V_n}-T_{L_n}\right)+\left(T_{V_{n-1}}-T_{L_{n-1}}\right)}{2}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(n=1\dots M,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}GLMC-PC-D)$$

(27b)

2.1.6. GLMC-UOE Validation

Appendix A, Appendix B, Appendix C and Appendix D report GLMC-UOE validation against the literature data. Appendix A validates GLMC-PC-D via the reproduction of the results of de Medeiros et al. [54] for NG decarbonation with aqueous-MEA-MDEA through a parallel-contact GLMC. Appendix B asymptotically validates an adiabatic GLMC-PC-D for NG decarbonation with aqueous-MEA-MDEA against HYSYS P-H flash. The principle here is that a parallel-contact GLMC-PC-D with a large transfer area asymptotically approaches the response of an adiabatic pressure–enthalpy (P-H) flash with the same parallel feeds. Appendix C asymptotically validates the adiabatic GLMC-CCC-D for NG decarbonation with aqueous-MEA-MDEA against an HYSYS absorption column, because an adiabatic countercurrent-contact GLMC-CCC-D with a large transfer area asymptotically approaches the response of a large adiabatic absorption column with the same countercurrent feed. Appendix D validates the adiabatic GLMC-CCC-D model via the reproduction of the adiabatic countercurrent GLMC results of Belaissaoui and Favre [47] for biogas decarbonation with pressurized water.

In this work, the proposed landfill-gas-to-biomethane route adopts the GLMC modules of Belaissaoui and Favre [47] with a pressurized-water solvent for landfill-gas decarbonation/desulfurization, i.e., the same modules, solvent (T = 15 °C, P = 7 bar), and H₂O/CO₂ capture-ratio were used. The used GLMC mass transfer coefficients were those obtained from Appendix D for GLMC-CCC-D validation. Since Belaissaoui and Favre [47] adopt isothermal GLMC modeling—which precludes obtaining the GLMC internal heat transfer coefficient, U_I, for landfill-gas decarbonation/desulfurization with pressurized water—U_I was estimated using an ad hoc asymptotic procedure in Appendix E.

2.2. Landfill-Gas Composition

Landfill-gas CH₄/CO₂/H₂S/H₂O compositions (

Table 2. Landfill-gas decarbonation/desulfurization siloxane-removal simulation/design assumptions.

Topic	Description
Thermodynamic Modeling	Landfill-Gas Compression, CO2/H2S Separation, Siloxanes Separation: HYSYS Acid-Gas Physical-Solvents Package; CO2-to-EOR: PR-EOS; Cooling-Water(CW)/Chilled-Water(ChW)/LPS,MPS: HYSYS ASME-Steam-Table;
Landfill-Gas	0.5 MMNm3/d; T = 30 °C; P = 1 bar; Mol = 27.73 g/mol; CH4 = 55.7 %mol; CO2 = 40 %mol; H2OSaturation = 4.28 %mol; H2S = 150 ppm-mol; Siloxanes: Table 1.
Biomethane	CH4 ≥ 85 %mol; CO2 ≤ 3 %mol; H2S ≤ 10 mg/Nm3; Siloxanes ≤ 0.03 mg/Nm3 [19,85,86]
GLMC Module [47]	HFM: Polyphenylene-Oxide (Parker P-240); $d_i=370 \mu m$; $d_o=520 \mu m$; HFM-Side:Landfill-Gas; Shell-Side:Water; $D$= 0.36 m; Packing-Ratio: $\phi$= 0.5; $N_{HF}=2.39\times {10}^5 fibers$; GLMC-Absorber: $A_{GLMC}=663.65 m^2/module;\hspace{0.17em}\hspace{0.17em}{Z}_M=2 m$; GLMC-Stripper: $A_{GLMC}=1991.85 m^2/module;\hspace{0.17em}\hspace{0.17em}{Z}_M=6 m$;
GLMC Modeling	$U_I^{GLMC-CCC-D}=U_I^{GLMC-PC-D}=0.09 W{m}^{-2}{K}^{-1}$;  $U_E=0$; ${\Pi }_{C{O}_2}$=${\Pi }_{H_{2}S}$=${\Pi }_{H_{2}O}$=$6.5756\times {10}^{-4}$  mol/(s.bar.m2);${\Pi }_{{\mathit{CH}}_4}$=$3.868\times {10}^{-5}$  mol/(s.bar.m2); ${\Pi }_{Siloxanes}=0$;  Capture-Ratio = 443.41 kgH2O/kgCO2; {TS} = {CO2, H2S, CH4, H2O}; GLMC-CCC-D: Countercurrent-Contact Distributed-Model (Section 2.1); GLMC-PC-D: Parallel-Contact Distributed-Model (Section 2.1).
High-Pressure CO2/H2S Reboilered Stripper	Feed[H2O/CO2/H2S] = 7,275,950 kg/h; $P^{Feed}=30.14 bar;$ $T^{Feed}=223.2 °C$; $Stage{s}^{Theoretical}=10;$ Feed-Stage = 5; $P^{Condenser}=30 bar$; $T^{Condenser}=40 °C$; $P^{Reboiler}=30.2 bar;\hspace{0.17em}\hspace{0.17em}{T}^{Reboiler}=233.8 °C;$ Condenser: Total-Reflux; Reboiler: Kettle (MPS); Reflux-RatioTop = 721.6.
DEPG Absorber	Solvent: 35.06 kmol/h; DEPG = 98.4 %w/w; H2O = 1.6 %w/w;$P=6.9 bar$; $T=15 °C$; $P_V^{in}$ = 6.995 bar; $T_V^{in}$ = 17.29 °C; StagesTheoretical = 20.
DEPG Reboilerd Stripper	Feed: 36.85 kmol/h;$P^{Feed}$ = 1.17 bar; $T^{Feed}$ = 162.7 °C; StagesTheoretical = 10; Feed-Stage = 5; $P^{Condenser}$ = 1.1 bar; $T^{Condenser}$ = 88.72 °C; $P^{Reboiler}$ = 1.2 bar; $T^{Reboiler}$ = 175 °C; Condenser: Total-Reflux; Reboiler: Kettle (LPS); Reflux-RatioTop = 100.
Saturated-Steam [87]	Low-Pressure-Steam (LPS): P = 14.3 bar, T = 196 °C; Medium-Pressure-Steam (MPS): P = 42.5 bar, T = 254 °C.
Compressors	Adiabatic-Efficiency = 75%; Compression-RatioStage = 3 (Landfill-Gas); Compression-RatioStage = 2.25 (CO2-to-EOR).
Pumps	Adiabatic-Efficiency = 75%.
Intercoolers	TGas-Out = 40 °C; ΔPGas = 0.5 bar.
Exchangers	$\Delta T^{Approach}=10 °C;\mathit{\Delta }P=0.5 bar$.
Cooling-Water	$T_{CW}^{in}=30 °C; T_{CW}^{out}=45 °C; P_{CW}^{in}=4 bar; P_{CW}^{out}=3.5 bar$.
Chilled-Water	$T_{ChW}^{in}=10 °C; T_{ChW}^{out}=15 °C; P_{ChW}^{in}=4 bar; P_{ChW}^{out}=3.5 bar$.

) were obtained from Läntelä et al. [76]. Regarding the siloxanes (Table 1), the predominant species in landfill-gas are usually D4/D5 [84], and the lack of L3/L5/TMS compositions is not uncommon [20]. Thus, the present work contemplates seven siloxanes: L2/L4/L6/D3/D4/D5/D6 (all in the HYSYS Library). The landfill-gas from landfill LF-1 [20] was used for defining L2/L4/L6/D3/D4/D5/D6 contents (Table 1).

2.3. Landfill-Gas-to-Biomethane Simulation Assumptions

Table 2 shows the simulation assumptions for the proposed landfill-gas-to-biomethane process.

2.4. Landfill-Gas-to-Biomethane Process

The proposed GLMC-based industrial-scale landfill-gas-to-biomethane process is described in this section. Figure 4 shows a block diagram with the interconnection between the process units, which are detailed in the following subsections.

2.4.1. Landfill-Gas Pre-Processing and Compression

Landfill-gas is collected via sufficiently deep landfill wells avoiding air penetration [1], such that O₂/N₂ contents are negligible. Landfill-gas pre-purification removes solid/liquid particulates.

Water-saturated landfill-gas in atmospheric conditions is compressed to P = 7 bar via 2-staged intercooled compression (Figure 5). In each compression stage, the temperature should not surpass 150 °C, otherwise compressors can be damaged. Intercoolers cool down the gas to 40 °C with cooling-water (CW), whose operating conditions are defined in Table 2. Knock-out vessel aqueous condensates are collected for further treatment (out of scope). Compressed landfill-gas feeds the GLMC battery for decarbonization/desulfurization.

2.4.2. Intensified GLMC Decarbonation/Desulfurization with Pressurized Water

CO₂/H₂S removal from landfill-gas was performed (Figure 6) in an intensified GLMC battery with N_M = 333 modules in countercurrent contact with pressurized water (P = 7 bar, T = 15 °C). The pressure should be higher than atmospheric pressure to increase CO₂/H₂S fugacities in the landfill-gas, improving the mass transfer driving force. The low solvent temperature (15 °C) improves CO₂/H₂S solubility [47]. The decarbonized/desulfurized landfill-gas was subjected to siloxane removal, while rich-water was pumped to another intensified operation, namely, a high-pressure CO₂ stripper. The rich-water was firstly preheated with hot, lean-water from the stripper reboiler in a thermal-integration water–water heat exchangers, where $\mathit{\Delta }{T}^{\mathit{Approach}}=10 °\mathrm{C}$. Unlike ethanolamines, water is insensitive to high temperatures, allowing water regeneration via high pressure (P = 30 bar) CO₂/H₂S stripping. This drastically reduces CO₂ compression costs, but operations above 30 bar are not advisable as column construction costs would be prohibitively higher [82]. The reboiler operated at 233.8 °C and was heated by saturated medium-pressure steam (MPS). The stripper top gas was water-saturated CO₂ with some H₂S (P = 30 bar) at a total reflux condenser temperature of 40 °C. The lean-water returned to the GLMC battery after cooling with chilled-water (ChW) to T = 15 °C. Chilled-water was produced via a propane refrigeration cycle, whose simulation is not in the scope of the present study. Some excess water (from landfill-gas) was collected, i.e., water make-up was unnecessary.

Water-saturated CO₂ from the stripper was compressed from P = 30 bar to P = 150.3 bar via 2-staged intercooled compression (Figure 7). Curiously, knock-out vessels were not necessary, since water vapor is stabilized by high-pressure CO₂ at T = 40 °C after cooling with CW. At P = 150.3 bar (T = 40 °C), CO₂ becomes supercritical, requiring the pump to reach P = 300 bar for EOR exportation.

2.4.3. Siloxane Removal via DEPG Absorption

Decarbonated/desulfurized landfill-gas from the GLMC passes through DEPG absorption for siloxane removal at P = 7 bar and T = 15 °C (Figure 8). The low temperature favors siloxanes solubility in DEPG. Landfill-gas feeds through the absorber bottom while lean-DEPG is fed at the top. The top gas is biomethane with less than 0.03 mg^Siloxanes/Nm³, the limit of capstone microturbines [19]. Rich DEPG passes through heat exchangers for preheating with hot lean DEPG from the DEPG stripper reboiler ($\Delta T^{\mathit{Approach}}=10 °C$). Rich-DEPG depressurizes to P = 1.17 bar to feed the atmospheric DEPG stripper, whose reboiler operates at T ≤ 175 °C to avoid DEPG thermal decomposition [17]. The total reflux condenser operates at T = 88.7 °C to favor the release of siloxanes in the top gas, which also contains CO₂/H₂O/H₂S/CH₄. This stream is cooled down to T = 40 °C, condensing some water in a knock-out vessel. The final residual-gas is flared to avoid CH₄ and siloxane emissions. Lean-DEPG is pumped and cooled down to T = 15 °C with chilled-water before returning to the absorber. DEPG make-up is negligible.

3. Landfill-Gas-to-Biomethane Process: Results and Discussion

The process analysis results of the new landfill-gas-to-biomethane process are presented in this section. The design data of all the equipment are shown in Supplement S1 in the Supplementary Materials (reference [88] was used to design heat-exchangers), with TAGs defined in Figure 5, Figure 6, Figure 7 and Figure 8.

3.1. Landfill-Gas Decarbonation/Desulfurization Results

Initially, landfill-gas GLMC decarbonation/desulfurization was simulated separately with a battery of N_M = 333 GLMC-CCC-D modules (discretized with M = 5 elements) and with a battery of N_M = 333 GLMC-PC-D modules (discretized with M = 100 elements) for comparison using the same capture-ratio ($\mathit{CR}={443.41\mathrm{kg}}^{{\mathrm{H}}_2\mathrm{O}}/{\mathrm{kg}}^{{\mathrm{CO}}_2}$) and the same compressed landfill-gas (P = 7 bar, T = 40 °C).

Table 3. Inlet/outlet fugacities: GLMC-CCC-D (M = 5) and GLMC-PC-D (M = 100).

Streams	Fugacity	CO2	CH4	H2S	H2O
Inlets	${\widehat{\mathit{f}}}_{\mathit{k}}^{\mathit{V}}$ (bar)	2.812	3.989	$1.048\times {10}^{-3}$	0.074
	${\widehat{\mathit{f}}}_{\mathit{k}}^{\mathit{L}}$ (bar)	$1.602\times {10}^{-9}$	0	0	0.017
GLMC-CCC-D Outlets	${\widehat{\mathit{f}}}_{\mathit{k}}^{\mathit{V}}$ (bar)	0.203	6.669	$3.318\times {10}^{-5}$	0.019
	${\widehat{\mathit{f}}}_{\mathit{k}}^{\mathit{L}}$ (bar)	1.190	3.899	$1.505\times {10}^{-4}$	0.018
GLMC-PC-D Outlets	${\widehat{\mathit{f}}}_{\mathit{k}}^{\mathit{V}}$ (bar)	0.989	5.876	$1.448\times {10}^{-4}$	0.018
	${\widehat{\mathit{f}}}_{\mathit{k}}^{\mathit{L}}$ (bar)	0.977	5.120	$1.403\times {10}^{-4}$	0.017

presents V/L inlet/outlet component fugacities for GLMC-CCC-D and GLMC-PC-D, while

Table 4. Landfill-gas decarbonation/desulfurization results: GLMC-CCC-D and GLMC-PC-D batteries.

	Landfill-Gas Inlet	Water Inlet	GLMC-CCC-D Landfill-Gas Outlet	GLMC-CCC-D Water Outlet	GLMC-PC-D Landfill-Gas Outlet	GLMC-PC-D Water Outlet
P (bar)	7.0	7.0	6.995	6.213	6.996	6.211
T (°C)	40.00	15.00	17.29	15.37	21.11	15.32
MMNm3/d	0.483961	-	0.261854	-	0.286783	-
kg/h	-	7,259,307	-	7,275,950	-	7,273,439
H2O (%mol)	1.11 *	100	0.29	99.90	0.28	99.91
CH4 (%mol)	57.55	$2.06\times {10}^{-23}$	96.71	0.01	85.14	0.02
CO2 (%mol)	41.32	$1.20\times {10}^{-10}$	3.0	0.09	14.59	0.07
H2S (ppm-mol)	154.94	$2.77\times {10}^{-13}$	4.94	0.34	21.53	0.32
D3 (ppb-mol) #	50.97	0	94.21	0	86.02	0
D4 (ppb-mol) #	977.59	0	1806.79	0	1649.73	0
D5 (ppb-mol) #	1.16	0	2.15	0	1.96	0
D6 (ppb-mol) #	12.46	0	23.03	0	21.03	0
L2 (ppb-mol) #	865.09	0	1598.87	0	1459.89	0
L4 (ppb-mol) #	2.84	0	5.24	0	4.79	0
L6 (ppb-mol) #	0.50	0	0.93	0	0.85	0
Siloxanes (ppb-mol) #	1910.61	0	3531.22	0	3224.27	0
	Final Results: Gas-to-Solvent CO2/H2S %Recoveries and Gas-to-Solvent CH4 %Loss
GLMC Battery	%Recovery CO2		%Recovery H2S		%Loss CH4
GLMC-PC-D	79.08		91.77		12.33
GLMC-CCC-D	96.08		98.28		9.07

* Water saturated; ^# Siloxanes: non-transferable.

shows inlet/outlet streams and CO₂/H₂S %Recoveries evincing that, with the same transfer area and capture-ratio, only the GLMC-CCC-D battery can achieve CH₄/CO₂/H₂S biomethane specifications (Table 2). The reason is the higher fugacity-difference driving force in the countercurrent GLMC-CCC-D. This is the reason for using a GLMC-CCC-D battery (Section 2.4) in the landfill-gas-to-biomethane intensified process. The results of the GLMC-PC-D are depicted via the following axial profiles in Figure 9: ${\widehat{f}}_{C{O}_2}^V,{\widehat{f}}_{C{O}_2}^L$,${\widehat{f}}_{C{H}_4}^V,{\widehat{f}}_{C{H}_4}^L$ (Figure 9a); V %mol CO₂/CH₄ (Figure 9b); ${\widehat{f}}_{H_{2}S}^V,{\widehat{f}}_{H_{2}S}^L$ (Figure 9c); V ppm-mol H₂S (Figure 9d); ${\widehat{f}}_{H_{2}O}^V,{\widehat{f}}_{H_{2}O}^L$ (Figure 9e); V %mol H₂O (Figure 9f); H₂S/CO₂ %Recovery and CH₄ %Loss (Figure 9g); and CO₂/CH₄ Selectivity (Figure 9h). It is interesting to see (Table 3) that, for CO₂/H₂S/CH₄ species, ${\widehat{f}}_k^V>{\widehat{f}}_k^L$ throughout the GLMC-PC-D process due to parallel V → L transfer, while ${\widehat{f}}_k^{V^{OUT}}>{\widehat{f}}_k^{L^{IN}}\hspace{0.17em},\hspace{0.17em}{\widehat{f}}_k^{V^{IN}}>{\widehat{f}}_k^{L^{OUT}}$ in the GLMC-CCC-D process due to countercurrent V → L transfer.

GLMC-PC-D fugacity profiles evince the inefficiency of parallel contact, as transfer driving forces start high and then vanish rapidly, limiting CO₂/H₂S transfer, as seen in Table 3 and Figure 9a–c. In the study of Zhang et al. [39], which tested the increase in the solvent flowrate to decrease CO₂ fugacity in the aqueous phase, it was also observed that it contributes to achieving extra CO₂ mass transfer driving force. On the other hand, in the GLMC-CCC-D process, a higher driving force is maintained because, as V approaches the outlet, it contacts decreasing ${\widehat{f}}_k^L$ values. Moreover, in the GLMC-PC-D process, Figure 9a shows that CO₂ transfer slows down at z = 1.4 m, suggesting that the battery is oversized for CO₂ transfer via parallel contact, benefiting parasitic CH₄ V→L transfers. Meanwhile, at z = 1.95 m, L becomes a two-phase component due to V → L CH₄ transfer, and ${\widehat{f}}_{C{H}_4}^L$ starts decreasing with a tendency to maintain the CH₄ driving force. This effect creates CH₄ bubbles in the L stream generating CO₂/H₂S stripping from the solvent that lowers ${\widehat{f}}_{C{H}_4}^L$. This behavior of CO₂ and CH₄ driving forces decreases CO₂/CH₄ selectivity (Figure 9h) in the 2nd half of $Z_M$. All these aspects of the GLMC-PC-D process were similarly observed by de Medeiros et al. [54]. Figure 9e,f depict strong V → L water transfer driven by the lower L temperature (${\widehat{f}}_{H_{2}O}^L<{\widehat{f}}_{H_{2}O}^V$). This makes landfill-gas decarbonation/desulfurization self-sufficient in water (i.e., water make-up is unnecessary and the process exports water). This behavior is in accordance with Ghasem et al.’s study [53], which previously demonstrated L → V water transfer, and also with Villeneuve et al.’s study [68], which showed bidirectional water transfer, i.e., V → L water transfer is also possible. The higher H₂S %Recovery rate comparative to its CO₂ counterpart (Figure 9g) results from a much higher H₂S capture ratio (kg^H2O/kg^H2S). Table 4 also shows inferior CO₂/H₂S %Recoveries and a greater CH₄ %Loss of GLMC-PC-D compared to GLMC-CCC-D, the former due to poor driving-force utilization and the latter due to the oversized battery for parallel contact benefiting CH₄ V→L transfer.

Figure 10 presents the GLMC-PC-D temperature/pressure profiles. Regarding the thermal effects, a paramount factor is the high solvent/gas mass ratio (besides ${\overline{C}}_P^L\approx 2{\overline{C}}_P^V$) to provide a high CO₂ capture ratio (443.41 kg^H2O/kg^CO2). As a consequence, T_L slightly increases while T_V quenches rapidly toward T_L (Figure 10a), mainly due to V→L heat transfer and the $V.{\overline{C}}_P^V$ reduction due to CO₂ V → L transfer. Evidently, part of the small T_L value increase comes from exothermic physical absorption. The high solvent/gas mass ratio entails a high shell-side L pressure drop compared to the low HFM-side V pressure drop (Figure 10b). These aspects are clear advantages in comparison with the incomplete isothermal/isobaric GLMC model of Belaissaoui and Favre [47].

GLMC-CCC-D’s and GLMC-PC-D’s performance comparison leads to the selection of the GLMC-CCC-D battery in the landfill-gas-to-biomethane process. Thus, the rich water from GLMC-CCC-D is regenerated via high-pressure (P = 30 bar) stripping (Figure 6).

Table 5. Landfill-gas decarbonation/desulfurization balance and high-pressure stripper products.

	Inlet Landfill-Gas	Outlet Landfill-Gas	Stripper Top Gas	Excess Water
H2O (kmol/h)	10.03	1.43	1.28	7.32
CH4 (kmol/h)	518.05	471.05	47.00	$1.94\times {10}^{-24}$
CO2 (kmol/h)	372.00	14.59	357.41	$1.11\times {10}^{-11}$
H2S (kmol/h)	0.1395	0.0024	0.1371	$5.41\times {10}^{-18}$
	High-Pressure CO2/H2S Stripper—Water Regeneration
		Stripper Top Gas	Lean Water
	P (bar)	30.0	30.2
	T (°C)	40.0	233.8
	kmol/h	405.82	402,960.11
	MMNm3/d	0.218	-
	H2O (%mol)	0.31	100
	CH4 (%mol)	11.58	$2.65\times {10}^{-23}$
	CO2 (%mol)	88.07	$1.52\times {10}^{-10}$
	H2S (ppm-mol)	337.78	$7.39\times {10}^{-13}$

shows the landfill-gas decarbonation/desulfurization balance and the high-pressure stripper outlets. The stripper regenerates water while releasing CO₂ at P = 30 bar, thus lowering CO₂-to-EOR compression investment and power. Some authors naively suggest low-pressure CO₂ stripping for GLMCs with water [89], evidently entailing much higher compression power for CO₂ utilization.

3.2. Siloxane Separation, Process Waste, and Power/Utilities Consumption

GLMC-CCC-D outlet landfill-gas experiences siloxane separation via DEPG absorption (Figure 8), as suggested by Ajhar et al. [75].

Table 6. Siloxane DEPG absorber and DEPG stripper.

	Biomethane Outlet	Absorber Rich DEPG	Stripper Top Gas	Lean DEPG
P (bar)	6.9	6.995	1.1	1.2
T (°C)	15.84	19.35	88.72	175
kmol/h	485.28 (0.26 MMNm3/d)	36.86	1.80	35.06
DEPG (%mol)	$3.72\times {10}^{-6}$	75.98	$1.74\times {10}^{-18}$	79.87
H2O (%mol)	0.31	22.12	61.09	20.13
CH4 (%mol)	96.97	1.24	25.54	$2.65\times {10}^{-25}$
CO2 (%mol)	2.96	0.65	13.26	$7.95\times {10}^{-15}$
H2S (ppm-mol)	4.36 (6.7 mg/Nm3)	7.77	159.44	$5.33\times {10}^{-14}$
D3 (ppm-mol)	$1\times {10}^{-19}$	1.25	25.6	$3\times {10}^{-17}$
D4 (ppm-mol)	$6\times {10}^{-18}$	23.9	490	$8\times {10}^{-15}$
D5 (ppm-mol)	0.00145	12.2	$8\times {10}^{-7}$	12.8
D6 (ppm-mol)	0.00025	57.2	5.09	59.9
L2 (ppm-mol)	$2\times {10}^{-9}$	21.1	434	$9\times {10}^{-19}$
L4 (ppm-mol)	0.00001	0.1935	1.59	0.122
L6 (ppm-mol)	$4\times {10}^{-23}$	0.0123	0.253	$1.5\times {10}^{-17}$
Siloxanes (ppm-mol)	0.0017 (0.029 mg/Nm3)	116	957	72.8

shows the outlets of the DEPG absorber and DEPG stripper, confirming that biomethane CH₄/CO₂/H₂S/Siloxane specifications were attained. But, biomethane still has 3100 ppm-mol H₂O requiring further dehydration.

Table 7. Siloxane separation balance, waste streams, and power/utilities consumption.

	Inlet Landfill-Gas	Outlet Biomethane	Top-Gas Stripper	DEPG Make-Up
DEPG (kmol/h)	0	$1.81\times {10}^{-5}$	$3.13\times {10}^{-20}$	$1.81\times {10}^{-5}$
H2O (kmol/h)	1.43	0.33	1.10	0
CH4 (kmol/h)	471.05	470.59	0.46	-
CO2 (kmol/h)	14.59	14.35	0.24	-
H2S (kmol/h)	0.0024	0.0021	0.0003	-
Siloxanes (kmol/h)	$1.72\times {10}^{-3}$	$8.28\times {10}^{-7}$	$1.72\times {10}^{-3}$	-
	Process Wastes		Power and Utilities Consumption
	Residual Gas	Residual Water		Consumption
kmol/h	0.76	1.04	Power	9.41 MW
DEPG (%mol)	0	0	Chilled Water	16,461 t/h
H2O (%mol)	7.44	99.99	Cooling Water	225,507 t/h
CH4 (%mol)	60.77	0.00124	LPS	4 t/h
CO2 (%mol)	31.53	0.01	MPS	8604 t/h
H2S (ppm-mol)	378.76	0.45
Siloxanes (ppm-mol)	2200	7.71

depicts siloxane separation balance, waste streams with high siloxane/H₂S contents (demanding disposal), and process power/utilities consumption values.

4. Conclusions

A novel GLMC-intensified landfill-gas-to-biomethane process was disclosed, where the decarbonation/desulfurization of 0.5 MMNm³/d of landfill-gas were conducted in a battery of countercurrent GLMC modules using a pressurized-water solvent (P = 7 bar, T = 15 °C). Since water absorption is inefficient for siloxane removal, siloxanes were removed after decarbonation/desulfurization via DEPG absorption (P = 6.9 bar, T = 15 °C) with a small DEPG circulation rate, where DEPG regeneration was conducted via atmospheric stripping requiring a negligible DEPG make-up. A GLMC water solvent is regenerated via intensified high-pressure CO₂ stripping, lowering CO₂-to-EOR compression costs. This process dismisses water make-up and exports water. There are only two small waste streams with high siloxane/H₂S contents. The main revenues comprise clean 0.26 MMNm³/d biomethane and 0.218 MMNm³/d of dense CO₂ traded as an EOR agent. Power consumption reaches 9.41 MW, including CO₂-to-EOR compression/pumping. To the authors’ knowledge, quantitative multicomponent siloxane removal from decarbonated/desulfurized landfill-gas has never been attempted in the literature. The landfill-gas-to-biomethane process proved to be technically viable, attaining CH₄ (>85%mol), CO₂ (<3%mol), H₂S (<10 mg/Nm³), and siloxane (<0.03 mg/Nm³) specifications. Even if biomethane is burned without CCS, due to CO₂ capture from landfill-gas and EOR utilization, the present landfill-gas-to-biomethane results represent BECCS implementation.

To simulate GLMC landfill-gas decarbonation/desulfurization, a thermodynamically rigorous 1D GLMC steady-state model—GLMC-UOE—was developed. GLMC-UOE runs, integrated into Aspen-HYSYS 10.0, as a new HYSYS unit operation using the HYSYS Acid-Gas Physical-Solvents Thermodynamic Package (as well as the HYSYS Chemical Solvents Package). GLMC-UOE can simulate countercurrent-contact 1D-distributed GLMCs with the GLMC-CCC-D model and parallel-contact 1D-distributed GLMCs with the GLMC-PC-D model. Both models were validated via literature comparisons and can handle multicomponent bidirectional transmembrane heat/mass transfers, two-phase V/L flows, and rigorous V/L mass/energy/momentum balances predicting composition/temperature/pressure changes.

After the calibration of component permeances with the literature data, GLMC-UOE can perform rigorous steady-state simulations of landfill-gas decarbonation/desulfurization using a GLMC with pressurized water. This way, GLMC-UOE is useful to design new industrial-scale landfill-gas-to-biomethane projects to perform plant retrofitting, predict process changes during scale-up/scale-down procedures, and to be used for process troubleshooting. The GLMC scale-up was demonstrated in Section 3.1 by using one of the main advantages of the GLMC concept: modularity. This way, the V flowrate can be increased/decreased, as long as the numbers of GLMC modules with the same design are added/removed proportionally in parallel, and the same hydrodynamic conditions for model validation are maintained, i.e., a constant V/L ratio and the same inlet temperatures/pressures.

It is worth noting that GLMC studies on landfill-gas commonly focus only on CO₂ removal, ignoring H₂S removal, water/CH₄ bidirectional transmembrane transfers, temperature effects, and pressure drop, while, in this work, a GLMC was evaluated under multicomponent CO₂/CH₄/H₂S/H₂O bidirectional transmembrane transfers with respective thermal effects and T/P changes. The GLMC-PC-D profiles proved that isothermal/isobaric GLMC modeling is inadequate, despite the GLMC literature’s insistence of this assumption.

As a suggestion for future work, an economic analysis can be performed for novel landfill-gas-to-biomethane processes using GLMC decarbonation/desulfurization and siloxane removal via Selexol absorption.