Simulation of an Offshore Natural Gas Purification Process for CO2

Article pubs.acs.org/IECR

Simulation of an Offshore Natural Gas Purification Process for CO2 Removal with Gas−Liquid Contactors Employing Aqueous Solutions of Ethanolamines José Luiz de Medeiros,* Andressa Nakao, Wilson M. Grava, Jailton F. Nascimento, and Ofélia de Queiroz F. Araújo Escola de Química, Federal University of Rio de Janeiro, Av. Horácio Macedo, 2030, Centro de Tecnologia, EIlha do Fundão, Rio de Janeiro-RJ, 21941-909, Brazil S Supporting Information *

ABSTRACT: In the scenario of offshore rigs angular indifference, modularity, and compactness strongly influence selection of technology for natural gas purification. Gas−liquid contactors have such attributes and perform gas absorption without the concerns of packing columns like weight, flooding, gravitational alignment, and water saturation. This paper proposes a model for gas−liquid hollow fiber contactors using aqueous alkanolamines for CO2 removal from high pressure natural gas. The model assumes high-pressure compressible flows of both permeate and retentate with full thermodynamics via equations of state. Permeate is approached as a reactive vapor−liquid equilibrium flow of solvent with CO2/CH4 from membrane fluxes. Phase change and reactive heat effects are modeled via mass, energy, and momentum balances written for permeate/retentate as a differential-algebraic system for dependent variables temperature, pressure, and component flows. Profiles are obtained via numerical spatial integration with algebraic resolution imbedded, accounting for permeate reactive vapor−liquid equilibrium with only the molecular species incorporated into a Chemical Theory framework. without typical concerns of packing towers like weight, flooding, tall setups, gravitational alignment, water-saturated products, entrainment, phase distributors, and foaming.1−4 In summary, GLC for CO2/NG cuts offers the advantages of both membrane and absorption technologies, but leaving behind the respective drawbacks. GLC combines membrane separation and chemical absorption, using a physically and chemically active solvent4 for selective CO2 capture, and a membrane standing as a physical barrier against the unnecessary mixing of gas and liquid. Further advantages of GLC5 are independent liquid and gas flow manipulation, larger gas−liquid interfaces, flexibility to scale up or down, no dew-point conditioning, and modularity. These advantages have motivated a rush of papers on hollow fiber membrane (HFM) contactors in the last decades.1−8 In the literature HFM−GLCs are generally configured in parallel flow with gas and liquid on the opposite HFM sides. The gas phase can flow either outside (shell) or inside (lumen) the HFM, but in the majority of studies liquid flows in the inner HFM side and gas in the shell side. An aspect that soon became clear was that membrane pores should remain filled with gas, that is, a nonwetted condition, giving low mass transfer resistance. Otherwise, if HFM pores saturate with a stagnant liquid, that is, a wetted condition, mass transfer resistance increases, lowering fluxes and destroying the economic leverage of GLC. This entails that long-term operation of GLC implies utilization

1. INTRODUCTION Separation unit operations in any kind of process have always to be chosen judiciously and carefully, otherwise they may become burdensome appendices undermining the profitability, controllability, safety, and simplicity of the plant. Nevertheless, even when properly selected, separations usually raise concerns like energy consumption, utilities, chemicals demands, size and weight, construction restraints, operational hazards, etc. In oil and natural gas (NG) deep-water offshore rigs, an issue in the processing of large volumes of NG with high CO2/CH4 ratios vis-à-vis climate concerns, involves large capacity CO2 separators, whose targets are exportation of saleable NG via long pipelines and alimentation of enhanced oil recovery (EOR) systems with hyperpressurized CO2. In this scenario, robustness, angular indifference, modularity, and compactness also influence the selection of technology, so that membrane permeation modules are usually favored against the traditional absorbing towers with alkaline solvents.1,2 On the other hand, large permeation batteries also have their own shortcomings, mostly related to the permselective dense (nonporous) skin over the microporous substrate, that is, low fluxes, low capacity per unit area, CO2/CH4 limited selectivity, feed dew-point conditioned, high ΔP, continuous inspection for membrane bursts, and high consumption of power for permeate recompression and recycles.1 Gas−liquid contactors (GLC) are versatile membrane operations for CO2 removal from NG capable to outperform nonporous permeators in terms of capacity per unit area while sustaining high CO2/CH4 selectivity. High fluxes are possible in GLC because the membrane does not have to be selective, since selectivity is imposed by the solvent in the permeate side cutting the necessity of high ΔP.3,4 At the same time, GLC absorbs gas © 2013 American Chemical Society

Special Issue: PSE-2012 Received: Revised: Accepted: Published: 7074

September 25, 2012 February 12, 2013 February 13, 2013 February 13, 2013 dx.doi.org/10.1021/ie302507n | Ind. Eng. Chem. Res. 2013, 52, 7074−7089

Industrial & Engineering Chemistry Research

Article

of hydrophobic HFM.3,4 Initial studies on absorption with hydrophobic HFM were conducted by Zhang and Cussler.6,7 Karoor and Sirkar8 conducted experiments on CO2 and SO2 absorption through hydrophobic HFM, concluding that mass transfer rates can be 10 times higher than analogous packed towers. On the other hand, problems with hydrophobic HFM derive from partial wetting and lumen gas bubbles. Partial wetting has been reported with polypropylene (PP) and polyvinylidene fluoride (PVDF) hydrophobic HFMs and also due to alkanolamines which decrease surface tension and facilitate wetting.4 Bubbling in the inner HFM is viewed,4 in principle, as a bad feature that creates stagnation and hindrance to mass transfer. But, it is arguable that depending on the design, geometry, and pressures on the HFM sides, inner gas is not necessarily bad since it can accelerate the solvent as a two-phase nonstratified flow, destroying stagnation, increasing turbulence, and enhancing mass transfer coefficients. Concerning the solvent, the advantages of the chemical absorption of acid gases by aqueous alkanolamines are wellknown: they are weak acids and such solvents are weak alkali, such that bonding between them is reversibly broken at high temperatures and low partial pressures, leading to efficient gas stripping and solvent regeneration. That is, GLC is suited to separations with acid gas, water and alkanolamines, or AGWA systems. An AGWA system may include one or more acid gas(es) and one or more alkanolamine(s), but water is mandatory. An AGWA system is normally in a reactive vapor−liquid equilibrium (RVLE), under pressures from a few mmHg up to 200 bar and temperatures up to 180 °C. Primary alkanolamines (e.g., monoethanolamine, MEA), secondary alkanolamines (e.g., diethanolamine, DEA), tertiary alkanolamines (e.g., methyldiethanolamine, MDEA) are common in AGWA systems and their properties and comparative advantages can easily be found in the literature.9 Blends of alkanolamines for conjugating desirable qualities have been tried to obtain solvents with good reactivity and lower costs of regeneration and corrosion as in the case of blends MDEA+MEA, which indicate gains relatively to individual amines.10 The design of high capacity GLC experiences a relative unavailability of pilot data and/or reliable mathematical modeling of CO2 separation from high pressure NG. In the last 10 years there were innumerable papers in the GLC literature. But the great majority of them avoided basic facets of industrial NG purification like high loadings, heat effects, temperature profiles, high pressures, two-phase reactive permeate, and the use of equations of state (EOS) replacing ideal gas. Instead, typical GLC references mostly concentrate on mass transfer idiosyncrasies of HFM.3−5,11−13 Such works normally formulate three O(2) mass transfer boundary value problems (BVP) in cylindrical geometry, respectively, for inner liquid, outer vapor, and HFM, using three sets of diffusivities in axial and radial directions, and nonhomogeneity terms (in the inner BVP) accounting for chemical reactions. When used, partial or full pore wetting is accounted via boundary conditions on the outer HFM and by modifying the HFM BVP. The reactive permeate adopts a set of ionic, ideal solution, chemical equilibrium (ChE) reactions based on Zwitterion,14 leading to rate formulas in concentrations of amine and CO2. All these works adopt similar modeling simplifications; that is, all of them used several (declared or not) assumptions in the following list: (i) negligible external and permeate-retentate heat transfers; (ii) negligible heat effects of phase and chemical changes; (iii) no enthalpy formalism; (iv) isothermal permeate/retentate flows;

(v) ideal gas phase; (vi) Henry’s Law for CO2 VLE at interface; (vii) single-liquid phase permeate; (viii) ideal solution behavior (including ions) in the liquid; (ix) no energy/momentum balances for permeate/retentate; (x) no temperature profiles; (xi) parabolic velocity profile for inner liquid; (xii) no compressible or two-phase flow formalisms; (xiii) no fugacity formalism in ChE, VLE, or driving forces. The reiterated use of such assumptions can be inferred from the comprehensive review on GLC by Mansourizadeh and Ismail,12 which states that GLC models systematically considered (1) isothermal condition; (2) parabolic velocity profile in the lumen; (3) negligible axial diffusion; (4) ideal gas phase; (5) low pressure; and (6) Henry’s Law at interface. Perhaps the first exception to this trend, was the work of Marzouk et al.15 on experiments for CO2 removal from NG (90.5% CH4 + 9.5% CO2) up to 50 bar, using expanded PTFE HFM (non elective microporous) for physical absorption with H2O and physical−chemical absorption with H2O + NaOH, H2O + MEA, H2O + DEA, H2O + TETA in several concentrations at pressures 0.5 bar above gas pressure. Despite its potential of interest, they did not report thermal effects or temperature distributions, probably because they used high solvent/gas volumetric ratios from 1:8 to 1:2 (in typical packed columns this ratio ranges from 1:100 to 1:10). The point here is that bold issues in high capacity GLCs cannot be left without attention, for example, thermal effects, temperature profiles, high density compressible flows, RVLE permeate, whereas certain intricate, but localized, theoretical items can be replaced by short-cut formulas as done in conventional unit operation design, such as radial incompressible velocity profile replaced by axial compressible plug flow, O(2) BVP for flux calculations replaced by equivalent permeancy times fugacity differences between outer and inner sides, and 2D spatiality replaced by 1D axial description. To accomplish this, this work addresses a model for CO2 separation from high pressure NG in HFM−GLC with aqueous MEA/MDEA blends, assuming (i) 1D high-pressure parallel compressible flows of permeate and retentate, without velocity presumptions, but with full thermodynamics via EOS (PR/SRK), (ii) permeate as a RVLE two-phase flow of solvent with CO2 + CH4 from membrane fluxes, (iii) mass, energy, and momentum balances written for permeate/retentate, expressed as a differentialalgebraic system, (iv) spatial integration with algebraic resolution imbedded, accounting for RVLE permeate with only molecular species incorporated into a VLE+ChE framework proposed in de Medeiros et al.16,17 This is a Chemical Theory18 approach where complex species are formed in liquid phase via ChE reactions from real species CO2, H2S, H2O, MEA, MDEA. Complexes are reversibly created in absorption and destroyed in stripping. The limited knowledge to handle the numerous possible ions, is circumvented via such nonionic, nonvolatile complexes, with the advantage that model tuning requires only VLE AGWA data, more available than nonequilibrium counterparts. The nonionic assumption relies on the weakness of AGWA electrolytes, all created by incomplete dissociations. The rest of this work is organized as follows: section 2 describes the RVLE formalism for AGWA systems; section 3 approaches the modeling of HFM−GLC with RVLE permeate flow; section 4 simulates a GLC flowsheet for CO2 removal from NG in offshore rigs including a solvent regeneration stripper also in RVLE mode; section 5 ends the paper with its conclusions. 7075

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2. EQUILIBRIUM MODEL FOR AGWA SYSTEMS WITH H2S/CO2/H2O/MEA/MDEA Following the Chemical Theory,16,18 the RVLE approach to AGWA systems superimposes VLE onto ChE for a complete set of molecular chemical reactions, creating a thermodynamically sound tunable framework reproducing physicochemical changes. Since thermodynamics does not need knowledge of the internal structure of phases, the approach is molecular, with fictitious species in the liquid phase, the complexes, replacing the actual weak AGWA ions. Like ions, complexes are nonvolatile, not acting in the VLE, but influencing liquid properties. This framework is tuned to data, with the following assumptions: (i) species divided into two groups: complex and real; real species comprise solutes CO2 and H2S, amines MEA, MDEA and water; (ii) each complex reaction has as reactants an acid gas, water, and an amine with unitary coefficients; (iii) Only real species are subjected to VLE; (iv) complexes CO2−H2O−MEA, CO2− H2O−MDEA, H2S−H2O−MEA, H2S−H2O−MDEA are created by eq 1, which displays ChE constants and standard states as pure saturated liquid (liq) at T for complexes and pure ideal gas (g) at T and 1 bar for real species:

Calibration of this RVLE model demands VLE data with CO2, H2S, H2O, MEA, MDEA for an implicit parameter estimation of ChE constants K1, K2, K3, K4. The procedure is detailed in de Medeiros et al.16 and is only sketched here. All data used in this procedure belong to the AGWA VLE database gathered by these authors. The various sources of this data can be found in this reference and are not cited here. Labels A, S, C, V, L refer to amines, solutes, complexes, vapor, liquid. An experiment contacts L mols of CO2, H2S, H2O, MEA, MDEA at T. After VLE, pressure P is established with X, Y, XC, LL ,LV, PS, αS, namely, liquid and vapor fractions of real species, liquid fractions of complexes, mols of liquid and vapor, solute partial pressures and loadings. The experiment constraints are: eq 3 as mass balances of real species (n eqs); eqs (4) as normalizations (2 eqs); eqs 5 and 6 for solute partial pressures and loadings (2nS eqs); eq 7 for VLE of real species (n eqs); and eq 8 for ChE (nC eqs). L̲ + Π(L LX̲ C) − L LX̲ − L V Y̲ = 0̲ n

(3)

nC

∑ Xj + ∑ XCk − 1 = 0

CO2 (g ) + H 2O(g) + MEA(g )

j=1

(4a)

k=1

K1

↔ CO2 H 2OMEA(liq)

n

∑ Yj − 1 = 0

CO2 (g ) + H 2O(g) + MDEA(g ) K2

H 2S(g ) + H 2O(g) + MEA(g ) K3

↔ H 2SH 2OMEA(liq) H 2S(g ) + H 2O(g) + MDEA(g ) K4

↔ H 2SH 2OMDEA(liq)

(1)

CO2−H2O−MEA CO2−H2O− MDEA H2S−H2O−MEA H2S−H2O−MDEA

TC (K)

PC (bar)

ω

TB (K)

123.11 181.19

845.75 880.41

61.244 42.579

1.699 1.991

659.27 714.24

113.18 171.26

758.39 794.1

60.183 42.214

1.154 1.630

560.55 628.27

(5)

L V S SY̲ + ( 1T̲ S A L̲ ) α̲ S − S SL̲ = 0̲

(6)

V

L

L

(8)

Experiment variables and constraints (eqs 3 to 8) are put as in eq 9. Parameters (θ̲ = K̲ ) are the ChE constants at the chosen temperature (nθ = nC = 4), where CO2, H2S, H2O, MEA, MDEA the maximum numbers of variables and constraints per experiment are nZ = 3n + 4 + nC + 2nS = 27 and nF = 2n + 2 + nr + 2nS = 20. Thus, at a chosen temperature, one has to estimate nθ = nC = 4 parameters using a set of runs each one with about 27 variables and 20 constraints. Z̲ T = [T

P̲ ST

F̲ (Z̲ , θ̲ ) = 0̲

α̲ ST

L̲ T

X̲ T

Y̲ T

X̲ CT

P

LV

L L] (9a) (9b)

Let an extract of nE isothermal AGWA run. The goal is to estimate its four ChE constants by an implementation of the maximum likelihood criterion for implicit parameter estimation known as the method of estimated deviations,16 which can estimate model parameters ( θ )̂ and all variables (Ẑ i (i = 1, ..., nE)). This implementation was successful and capable to handle AGWA NLPs with outstanding numbers of variables and constraints. Thermodynamic properties of both phases were calculated with PR EOS. The optimization formulation is expressed in eq 10:

method;19 (vi) All species participate in 4 ChE constraints written for eq 1 with stoichiometric matrix (n × nC) for real species [CO2 H2S H2O MEA MDEA ] given by ⎡− 1 − 1 0 0 ⎤ ⎢ ⎥ ⎢ 0 0 −1 −1⎥ Π = ⎢−1 −1 −1 −1⎥ ⎢ ⎥ ⎢− 1 0 − 1 0 ⎥ ⎣ 0 −1 0 −1⎦

(7)

ln X̲ C + ΠT ln f ̂ − ln K̲ (T ) = 0̲

Table 1. Nominal Constants of Complexes via Joback Method19 MW (g/mol)

P̲ S − PS SY̲ = 0̲

ln f ̂ − ln f ̂ = 0̲

(v) No ions are present, phases are molecular and treated via EOS SRK/PR up to high pressures and all compositions, provided critical constants (TC, PC, ω) and normal boiling point (TB) of complexes are available and compatible with negligible vapor pressures as given in Table 1 via the Joback contribution

complex

(4b)

j=1

↔ CO2 H 2OMDEA(liq)

(2) 7076



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n

min Ψ =

⎛1⎞ E E T E ⎜ ⎟ ∑ ( Ẑ − Z̲ ) W ( Ẑ − Z̲ ) i i i i i ⎝2⎠ i

{ θ ̂, Z1̂ , ..., Ẑ nE} s.t. F̲ i( Ẑ i , θ )̂ = 0̲

(i = 1···nE)

(10)

16

It can be proved that the best estimator of fundamental variance (σE2) in this problem is given by σ̂E 2 = SR 2 ⎛ ⎞ nE 1 ⎜ ⎟ ∑ (Z̲ iE − Ẑ i)T W (Z̲ iE − Ẑ i) ≡ ⎜ nE ⎟ i ⎝ ∑i nFi − nθ ⎠ i (11) 16

where W i is a known weighting (diagonal) matrix for variables of run i belonging to an extract with nE isothermal runs. Table 2 Table 2. Summary of Parameter Estimation of ChE Constants in eq 1 T = 40 °C

T = 60 °C T = 80 °C T = 100 °C

230 no. expt (nE) 6210 no. variables (27 × nE) no. constraints (20 × nE) 4600 estimated ChE constants in eq 1 KCO2MEA (K1) 1313.48

90 2430 1800

87 2349 1740

140 3780 2800

55.712

10.622

1.552

KCO2MDEA (K2)

708299.1

8345.0

57.263

4.18

KH2SMEA (K3)

4999.7

155.53

10.025

1.241

KH2SMDEA (K4)

875532.0

7241.7

245.34

19.77

13.11

6.15

estimator of fundamental variance in eq 11 SR2 4.81 11.37

presents results for the estimated four ChE constants in eq 1 at temperatures of 40, 60, 80, and 100 °C, totaling 16 estimated parameters by means of four isothermal maximum likelihood searches. Figure 1 depicts objects belonging to the parameter estimation of ChE constants in eq 1 at 40 °C using 230 runs from the literature cited in de Medeiros et al.16 Figure 1a depicts a logarithm plot of predicted versus observed loadings and partial pressures of CO2 and H2S, with good coalescence on the diagonal. Figure 1b depicts a 3D projection of the 99% probability confidence hyper-ellipsoid for the correct triad KCO2MEA, KCO2MDEA, KH2SMEA expressed as percent deviations (%KCO2MEA × %KCO2MDEA × %KH2SMEA) from the estimated values, showing that correct ChE constants differ less than 15% from estimated ones. Figure 1 panels c and 1d, respectively, present predicted and observed CO2 loadings and CO2 partial pressures (included limits of ±3 standard deviations and 99% confidence) versus run number, showing also reasonable agreement between predictions and observations. The four estimated ChE constants for the reactions in eq 1, ln KEST (k = 1, ..., 4), depicted in Table 2, were fitted by three k parameter formulas in terms of absolute temperature as shown in Table 3. The strong exothermic absorption of CO2/H2S in aqueous alkanolamines is coherently reproduced by the RVLE model. As shown in Figure 2a for the CO2−MEA reaction and in Figure 2b for CO2−MDEA reaction, the estimated (Table 2) and regressed (Figure 2 and Table 3) ChE constants monotonously decrease with temperature implying negative standard enthalpies

Figure 1. Estimation of ChE constants at T = 40 °C: (a) predicted vs observed loadings (mol/mol), and partial pressures (bar) of CO2 and H2S; (b) 99% confidence ellipsoid for correct values in %KCO2MEA, % KCO2MDEA, %KH2SMEA; (c) predicted and observed CO2 loading (mol/ mol); (d) predicted and observed CO2 partial pressure (bar) [with limits of ±3 standard deviations and 99% confidence]. 7077



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Table 3. Regressed ChE Constants vs T (K) (ln Kk = Ak + Bk/T + Ck ln T) complex

k

Ak

Bk

Ck

CO2−H2O−MEA CO2−H2O−MDEA H2S−H2O−MEA H2S−H2O−MDEA

1 2 3 4

−800.3318 −857.6953 −779.3461 −1511.997

51085.431 63723.504 52941.595 93647.363

112.1179 116.208 107.680 213.448

allow us to correctly address heat effects and energy balances in AGWA operations.

3. MODELING OF PARALLEL FLOW GLC WITH HOLLOW FIBER MEMBRANE Figure 3 brings two sketchesboth aligned with axial coordinate z for GLC modeling with HFM. Figure 3a depicts the spatial

Figure 3. HFM−GLC modeling (z is axial position, θ is inclination): (a) trans-membrane fluxes, RVLE permeate, and retentate flows relative to HFM; (b) HFM−GLC module.

Figure 2. Regressions of estimated ChE constants vs T (K): (a) ln KCO2MEA = −800.332 + 51085.43/T + 112.118 ln T; (b) ln KCO2MDEA = −857.695 + 63723.504/T + 116.208 ln T [est, estimated ChE constant].

allocation of flows with respect to the HFM, showing (i) twophase flow inside HFM for the RVLE permeate (L); (ii) transmembrane fluxes (N); and (iii) parallel compressible gas flow of retentate (V). Figure 3b sketches a parallel GLC (resembling a shell and tube exchanger) with the bundle of HFMs axially aligned in the shell and fixed by perforated plates. The parallel flows greatly facilitate the mathematics making unnecessary BVP strategies (e.g., finite differences) or shooting methods. Indexes k = 1 ..., n and j = 1, ..., nC point to real and complex species. Subscripts L, V designate permeate and retentate; whereas Superscripts LL , LV designate liquid and vapor phases in the permeate L. Since complex species only exist in the permeate after some course of reactions, the concept of equivalent moles

of reaction (ΔH0), since d ln K/dT = ΔH0/(RT2) < 0. This behavior, valid for all reactions in eq 1, is a prerequisite for consistency of the RVLE model. The temperature dependence of ChE constants in Table 3 can be further processed16 for conversion of the standard state of complexes from (liq) to (g) in order to get ideal gas properties of formation at T0 = 298.15 K (e.g., G̅ 0,(g) , H̅ 0,(g) ) and ideal gas heat capacity (C̅ 0,(g) (T) = α + βT f f P 2 + γT ). These properties (not shown here, see de Medeiros et al.16) are vital to access ideal gas enthalpy of complex species, which when complemented by residual enthalpy from EOS, 7078



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3.1.2. RVLE-H Implementation. The vector of n + 2 Duhem RVLE-H specifications is U̲ T = [PL ∑L L̲T]. The 2n + 2nC + 3 RVLE-H variables are Y̲, X̲ , X̲ C, η̲, LL, LV, TL and the set of 2n + 2nC + 3 RVLE-H constraints comprises the same eqs 14a to 14f with the addition of the energy constraint for RVLE permeate (eq 16):

without reaction (EMWR) will be invoked for real species: Let L the vector of EMWR rates (mol/s) of real species that have been introduced into the permeate up to the axial position z assuming that no reaction has occurred (Figure 3a). The concept of EMWR mols was also used in RVLE experiments in section 2, eq 3. EMWR is granted by Duhem’s Theorem, in the sense that if two RVLE problems have the same specification of EMWR rates of real species, with two more equal coordinates (e.g., T, P), they will have the same RVLE state. V and N are also vectors of real species designating, respectively, retentate species rates (mol/s) and HFM species fluxes (mol/s·m2). Vector η designates nC complex forming reaction rates (mol/s). Y, X, XC designate mol fractions of real species in the vapor and liquid permeate phases and of complexes in this liquid. TL, PL, TV, and PV are temperatures (K) and pressures (bar) of L and V, whereas ΣL, ΣV designate rates of enthalpy (kW) carried by L and V. EMWR allows permeate species balances to be written only for real species, with the RVLE resolution done a posteriori. By applying ChE + VLE onto an EMWR specification of real species (L), with two more specifications (e.g., TL, PL), Duhem’s Theorem leads to the entire set of RVLE permeate variables (L, LL, LV, Y, X, XC, TL, PL, ΣL). 3.1. RVLE Model for Reactive Vapor−Liquid Permeate. On the basis of Figure 3a and Duhem’s Theorem, two implementations of a RVLE permeate were developed to deal with the algebraic part of the GLC model; namely, RVLE-T and RVLE-H. Both solve a RVLE permeate: RVLE-T with specifications L, TL, PL, whereas RVLE-H uses specifications L, ΣL, PL; that is, enthalpy rate replaces temperature as a Duhem specification. This is expressed in eqs 12 and 13: RVLE ‐ T

TL , PL , Lk (k = 1, ...,n) ⎯⎯⎯⎯⎯⎯⎯⎯→ L L , L V , Y̲ , X̲ , X̲ C , η̲ , ΣL RVLE ‐ H

Σ L , PL , Lk (k = 1, ...,n) ⎯⎯⎯⎯⎯⎯⎯⎯→ L L , L V , Y̲ , X̲ , X̲ C , η̲ , TL

L LH̅ L L + L V H̅ L V − Σ L = 0

(16)

After the same transformation of mol fractions to unbounded variables above, eqs (17) express the working vectors of 2n + 2nC + 3 RVLE-H state variables and 2n + 2nC + 3 RVLE-H constraints. Equation 17b can be numerically solved if vector U is known. E̲ T = [ Θ̲ X T Θ̲ Y T Θ̲ C T η̲ T L L L V TL ]

(17a)

F̲ (E̲ , U̲ ) = 0̲

(17b)

3.1.3. Numerical Solution of RVLE-H and RVLE-T. Since RVLE-T and RVLE-H are reactive versions of flash problems, only the RVLE-H resolution is detailed here. The Newton− Raphson method (NRM) with analytical jacobians is used. Iterations normally start from a previous solved state (E̲ *, U̲ *) seeking for a new state E̲ at a new specification U̲ . First an initial estimate is created with eq 18a; then NMR iterations proceed until convergence via eq 18b. Jacobians J E/U, J E, calculated at E(n), express the influence of U on E and the influence of E on the RVLE constraints (eqs 17b), and are described in Appendix A for RVLE-H (see Supporting Information).

(12)

E̲ (0) = E̲ * + J

(13)

E̲ (n + 1) = E̲ (n) − [ J ]−1 F̲ (E̲ (n) , U̲ )

E/U

·(U̲ − U̲ *)

(18a)

E

3.1.1. RVLE-T Implementation. The vector of n + 2 Duhem RVLE-T specifications is U̲ T = [PL TL L̲ T]. The 2n + 2nC + 2 RVLE-T state variables include Y̲, X̲ , X̲ C, η̲, LL, LV and the set of 2n + 2nC + 2 RVLE-T constraints comprises real and complex mass balances (eqs 14a, 14b), normalizations (eqs 14c, 14d), VLE (eqs 14e and ChE 14f): L LX̲ + L V Y̲ − L̲ − Π η̲ = 0̲

(14a)

L LX̲ C − I η̲ = 0

(14b)

1T̲ X̲ + 1T̲ X̲ C − 1 = 0

(14c)

1T̲ Y̲ − 1 = 0

(14d)

ln ϕ ̂

LV

+ ln Y̲ − ln ϕ ̂

LL

− ln X̲ = 0̲

ln X̲ C − ln K̲ + ΠT ln X̲ + ΠT ln PL ϕ ̂

LL

{until || E̲ (n + 1) − E̲ (n) || ≤ ε

3.1.4. Influence Terms via Differential Coefficients from Jacobian Terms. Influence terms, under the form of RVLE differential coefficients, can be extracted from the jacobians at a RVLE solution. These terms are RVLE first order properties that can be used as measures of the influence of specifications (U) on the RVLE state (E). Several examples of such coefficients are derived in Appendix B for RVLE-H and RVLE-T (see Supporting Information). As an informal and clearer representation, symbols of variables replace the respective index ranges of J E/U in the RHS of eqs (19):

(14e)

= 0̲

After replacing mol fractions by unbounded variables (Y̲ = exp(Θ̲ Y), X̲ = exp(Θ̲ X), X̲ C = exp(Θ̲ C)), eqs 15a and 15b express the working vectors of 2n + 2nC + 2 RVLE-T state variables and 2n + 2nC + 2 RVLE-T constraints. Equation 15b can be numerically solved if U is known. (15a)

F̲ (E̲ , U̲ ) = 0̲

(15b)

⎛ ∂T ⎞ = J (TL , Σ L) ⎜ L⎟ E/U ⎝ ∂Σ L ⎠ P , L̲

(19a)

⎛ ∂ η̲ ⎞ = J ( η̲ , Σ L) ⎜ ⎟ E/U ⎝ ∂Σ L ⎠ P , L̲

(19b)

⎛ ∂L L ⎞ = J (L L , Σ L ) ⎜ ⎟ E/U ⎝ ∂Σ L ⎠ P , L̲

(19c)

⎛ ∂X̲ ⎞ = [diag(X̲ )] J (Θ̲ X , Σ L) ⎜ ⎟ E/U ⎝ ∂Σ L ⎠ P , L̲

(19d)

L

(14f)

E̲ T = [ Θ̲ X T Θ̲ Y T Θ̲ C T η̲ T L L L V ]

(18b)

L

L

L

7079



Article

Figure 4. Example 1: RVLE-T and RVLE-H loci: (a) YCO2 vs (T, P); (b) LoadingCO2 (mol/mol) vs (T, P); (c) LL (mol/s) vs (T,P); (d) (∂H∂T)P,L (kJ/ mol·K) vs (T, P), (e) YCO2 vs (ΣL, P); (f) LoadingCO2 (mol/mol) vs (ΣL, P); (g) LL (mol/s) vs (ΣL, P); (h) (∂T/∂ΣL)P,L (K/kW) vs (ΣL, P) [ΣL (kW), P (bar), T (°C)].

⎛ ∂ Y̲ ⎞ = [diag( Y̲ )] J (Θ̲ Y , Σ L) ⎜ ⎟ E/U ⎝ ∂Σ L ⎠ P , L̲ L

C5H12, 10 mol/s CH4, 2 mol/s CO2) so that the corresponding vector L of EMWR species rates is LT = [2 2 × 10−3, 10−3, 0.002, 0.002, 10, 2, 10] (MEA, MDEA, C2H6, C3H8, C4H10, C5H12, CH4, CO2, H2O). Without H2S, there are only two active chemical reactions in this system, namely eqs (1a) and (1b), producing complexes CO2−H2O−MEA and CO2−H2O− MDEA with ChE constants given as functions of temperature in Table 3; that is, nC = 2 and the stoichiometric matrix in eqs

(19e)

3.1.5. Example 1: RVLE-T and RVLE-H Loci for Reactive VLE Permeate. A RVLE system with nine real species was created by mixing 14 mol/s of an AGWA solvent (2 mol/s MEA, 2 mol/s MDEA, 10 mol/s H2O) with 12.006 mol/s of NG (10−3 mol/s C2H6, 10−3 mol/s C3H8, 2.10−3 mol/s C4H10, 2.10−3 mol/s 7080



Article

variables inside the HFM, namely, LL, LV, Y̲, X̲ , X̲ C, η.̲ Although the GLC model uses only EMWR dependent variables, the RVLE-T search is necessary to access thermodynamic properties of RVLE permeate, like fugacities of real and complex species fLk̂ (k = 1, ..., n), fLĵ (j = 1, ..., nC), enthalpy rate ∑L, RVLE heat capacity C̅ LP = (∂H̅ L/∂TL)PL,L̲, density ρL = PL·ML/(ZL·R·TL), RVLE isothermal compressibility ΓPL = (∂ρL/∂PL)TL,L̲, RVLE isobaric expansivity ΓTL = (∂ρL/∂TL)PL,L̲ and RVLE compositional expansivity ΓLk = (∂ρL/∂Lk))TL, PL, Li≠k. An analogous set of properties are calculated for the retentate, but now via a direct application of EOS with TV, PV, V1, ..., Vn, leading to fugacities of real species fVk̂ (k = 1, ..., n), enthalpy rate ∑V, heat capacity C̅ VP = (∂H̅ L/∂TV)PV,V̲ , density ρV = PV·MV/(ZV·R·TV), isothermal compressibility ΓPV = (∂ρV/∂PV)TV,V̲ , isobaric expansivity ΓTV = (∂ρV/∂TV)PV,V̲ and compositional expansivity ΓVk = (∂ρV/ ∂Vk)TV, PV, Vi≠k. The modeling of HFM−GLC uses the following premises: (i) steady state; (ii) 1D axial coordinate z (m) for shell and HFM flows; (iii) 1D parallel compressible plug flows for retentate (gas phase) and permeate (two-phase) including RVLE effects, heat transfer terms, shear terms, and compressibility (monophasic or reactive biphasic) terms; (iv) permeate properties in RVLE context; (v) thermodynamic properties of all phases via cubic EOS (PR/SRK); (vi) high pressure gas viscosity via Chung−Reichenberg−Wilke19 and liquid viscosity via mixing rules;19 (vii) real species −CO2, CH4, etc., allowed to pass through HFM with fluxes given as products of equivalent permeancies (Φk) with differences of fugacities on both sides; and (viii) 1D mass, energy, and momentum balances for permeate and retentate flows. GLC specifications and physical parameters for simulation are (i) initial value of dependent (0) (0) (0) (0) (0) variables z = 0: P(0) V , TV , PL , TL , Lk , Vk (k = 1, ..., n); (ii) permeancies of real species Φk; (iii) heat transfer parameters TE, Ω, ΩE; (iv) battery parameters D, ZC, θ, NM; (v) roughness of surfaces εL, εV; and (vi) HFM parameters d, do, ZHF, NHF. 3.2.1. Spatial Dependence of Densities. A basic step to develop a thermodynamically sound GLC model is to correctly express the spatial dependences of retentate and RVLE permeate densities (ρV, ρL). First, differential forms of ρV, ρL on dependent variables are written in EMWR context:

(14) should be understood with only the two first columns of eq 2. Properties of formation of complexes in the ideal gas state (g) were recovered from data in Table 3 as observed in section 2 (de Medeiros et al.16). Enthalpy is calculated with reference to elemental substances in standard states at 298.15 K and 1 bar. PR EOS was used for thermodynamic properties with common binary parameters.18,19 An initial RVLE state is found as a RVLET solution at TL = 320 K, PL = 1 bar giving 52.71% mol vapor phase. Excursions were conducted from this initial state exploring RVLE-T and RVLE-H in 2D domains of specifications. Figure 4 depicts four views of RVLE-T excursions and another four views of RVLE-H excursions with the same reactive mixture aforementioned with 26.006 mol/s EMWR. Figure 4 panels a− d depict 2D behaviors, generated with RVLE-T excursions in the domain T ∈ [20 °C, 100 °C], P ∈ [1 bar, 20 bar], respectively, for equilibrium mol fraction of CO2 in vapor phase (YCO2), CO2 loading (mol/mol) in liquid phase (LoadingCO2), total mol rate (mol/s) of liquid (LL) and molar isobaric RVLE (two-phase and reactive) heat capacity (∂H̅ /∂T )P , L (kJ/mol·K). The best zones for CO2 absorption are clearly identified in Figure 4a,b as high pressure and low temperature sectors, whereas the best zones for CO2 stripping are identified with the breakage of complexes at low pressures and high temperatures. The zones of high liquid content (LL) are identified in Figure 4c at high pressures and moderately high temperatures, which combine reduced vaporization of liquids with the breakage of complexes creating more liquid (Eqs. 1a,b), whereas, liquid is destroyed at high temperatures and low pressures. Figure 4d interestingly reveals that enthalpy changes more rapidly at the borders of the absorption zones; that is, just where the breakage of complexes and reduction of loading initiates. Figure 4 panels e, f, g, and h depict 2D behaviors, generated with RVLE-H excursions in the domain ΣL ∈ [−5950 kW, −5290 kW], P ∈ [1 bar, 20 bar], respectively for vapor mol fraction of CO2 (YCO2), CO2 loading (mol/mol), total molar rate (mol/s) of liquid (LL) and the isobaric derivative of RVLE temperature with enthalpy rate (∂T/ ∂∑L)P,L̲ (K/kW). The influence of enthalpy rate in the RVLE is similar to the temperature influence, but with distortions. Thus, the best zones for CO2 absorption are identified in Figure 4e,f at high pressure and low enthalpy rates, whereas the best zones for CO2 stripping are identified with the breakage of complexes at low pressures and high enthalpy rates. The zones with highest liquid rate (LL) are identified in Figure.4g at moderate enthalpy rates, whereas liquid disappears at high enthalpy rates and lower pressures. Figure 4h is also interesting, revealing that temperature changes more rapidly with enthalpy rate in the intermediate region between absorption and stripping zones, that is, where the breakage of complexes and loading reductions initiate more vigorously. 3.2. HFM-Contactor Model with Reactive Vapor− Liquid Permeate. Figures 3a and 3b sketch a HFM−GLC with inclination θ and parallel flows of gas retentate and RVLE permeate. RVLE-T and/or RVLE-H solvers are necessary to access the permeate RVLE state corresponding to the dependent variables, which comprise: retentate temperature, pressure, and EMWR rates of real species (TV, PV, V1, ..., Vn); and permeate temperature, pressure, and EMWR rates of real species (TL, PL, L1, ..., Ln). The choice of EMWR rates of real species implies that conservation principles must be written only for real species in retentate and permeate in EMWR mode. In the permeate, the EMWR rates serve, jointly with TL, PL, as specifications U̲ T = [PL, TL, L1, ..., Ln] for a RVLE-T resolution to access the RVLE state

⎛ ∂ρ ⎞ dρV = ⎜ V ⎟ ⎝ ∂TV ⎠ P

V

⎛ ∂ρ ⎞ dTV + ⎜ V ⎟ ⎝ ∂PV ⎠T , V̲

dPV

V , V̲

⎛ ∂ρ ⎞ + ∑⎜ V⎟ k ⎝ ∂Vk ⎠ P n

dVk (20a)

V , TV , Vi ≠ k

⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ dρL = ⎜ L ⎟ dTL + ⎜ L ⎟ dPL ⎝ ∂TL ⎠ P , L̲ ⎝ ∂PL ⎠T , L̲ L

L

⎛ ∂ρ ⎞ + ∑⎜ L⎟ dL k k ⎝ ∂Lk ⎠ P , T , L n

L

L

(20b)

i≠k

With eqs (20) and above definitions of differential coefficients of ρV, ρL, it is straightforward to write total derivatives for ρV, ρL with respect to the independent axial coordinate z according to dρV dz 7081

= ΓTV

dTV dP + ΓPV V + dz dz

n

∑ ΓV

k

k

dVk dz

(21a)


Industrial & Engineering Chemistry Research dρL

= ΓTL

dz

dTL dP + ΓPL L + dz dz

n

∑ ΓL k

dL k k dz

Article

A key concept is partial molar EMWR property (PMPEMWR), an extension of the standard partial molar property (PMP) for RVLE. For retentate, PMP and PMPEMWR are exactly the same; they only differ for RVLE permeate, where chemical changes occur via RVLE-T. Appendix C (see Supporting Information) addresses eq (C8) for PMPEMWR using RVLE-T state E, matrix J E/U and analytical EOS gradients of properties for RVLE phases. Formulas below use strict SI units (pressure, Pa; energy, Pa·m3/ mol; density, kg/m3; molar mass, kg/mol) with index k referring to real species. 3.2.4.1. Retentate Properties. Given by EOS using eqs (29) on dependent variables [PV, TV, V1, ..., Vn]. PMP energy (eq 29c) is the sum of PMP enthalpy with kinetic and potential molar terms.

(21b)

Equations (21) are valid even for RVLE, provided L1, ..., Ln are EMWR interpreted, and are extensively used in retentate and permeate balances. Retentate coefficients in eq 21a are given analytically via EOS. RVLE permeate coefficients in eq 21b use analytical differentiation via EOS and J E/U, the jacobian of E regarding U (Appendices A and B in Supporting Information). 3.2.2. Mass Flow Rates and Mass Flux. Due to invariance of mass in chemical reactions, mass flow rates (kg/s) are invariant whether expressed in EMWR or not. Thus retentate (qV) and permeate (qL) mass flow rates are expressed using only the simpler EMWR formulas in eqs 22: n

qV =

n

n

k

(22)

k

k

⎛ ∂H̅ ⎞ C̅ PV = ⎜ V ⎟ ⎝ ∂TV ⎠ P

whence, dq V dz

n

=

∑ Mk k

dVk dqL , = dz dz

n

∑ Mk k

dL k dz

V , V̲

Ek̅ V = H̅kV + Mk

n

∑ MkNk

SL = NHFπd 2/4

4qL ML μL

9eV =

ML = πdNHF ,

,

4qV MV μ V

,

DL = 4SL /ML

MV = πD + πdoNHF ,

(26a)

ΨV = fV ρV λ v 2 /8

(29e)

H̅L = (L V H̅LV(TL , PL , Y̲ ) + L LH̅LL(TL , PL , X̲ , X̲ C))

Friction factors and shear stresses are given, respectively, by Churchill20 and Darcy formulas:

fV = fV (9eV , εV /D V ),

,

V , V̲

3.2.4.2. Permeate Properties. RVLE-T is first solved at UT = [PL, TL, L1, ..., Ln] with real species EMWR rates, giving X, XC, Y, η, LL, LV. RVLE enthalpy, density, enthalpy rate, and molar masses of RVLE phases are then calculated via eqs 30a−30d. PMPEMWR enthalpies H̅ L,EMWR are given by eq (C8) in eq 30e, k and then used in PMPEMWR energies via eq 30f. The informal representation (Section 3.1.4 and Appendix B in Supporting Information) is used to access subfields of J E/U in eqs 30g−30j.

D V = 4S V /MV

ΨL = fL ρL λL 2 /8

⎛ ∂ρ ⎞ ΓPV = ⎜ V ⎟ ⎝ ∂PV ⎠T

,

V , PV , Vi ≠ k

(25b)

(26b)

fL = fL (9eL , εL /DL ),

(29c)

⎛ ∂ρ ⎞ ΓVk = ⎜ V ⎟ ⎝ ∂Vk ⎠T

Then Reynolds Numbers, flow perimeters and hydraulic diameters follow by eqs (26): 9eL =

(29b)

(29d)

V , V̲

(25a)

S V = S − NHFπdo 2/4

λV 2 + Mk ·g ·h(z) 2

⎛ ∂ρ ⎞ ΓTV = ⎜ V ⎟ ⎝ ∂TV ⎠ P

3.2.3. Velocities and Shear Terms. Hydrodynamic terms in permeate and retentate flows are addressed. Velocities and flow cross sections are first given by eqs (25):

λV = qV /(ρV S V ),

V , PV , Vi ≠ k

ρV = ρV (TV , PV , V̲ )

(24)

k

λL = qL /(ρL SL),

⎛ ∂Σ ⎞ H̅kV = ⎜ V ⎟ ⎝ ∂Vk ⎠T

,

(29a)

(23)

Analogously, the total trans-membrane mass flux is written as

q=

Σ V = (∑ Vk)H̅ V

H̅ V = H̅ V(TV , PV , V̲ ),

∑ MkVk , qL = ∑ MkLk

/(L V + L L)

(30a)

⎛ ⎞ L V MLV L LMLL ⎟⎟ + ρL = qL /⎜⎜ ρLL (TL , PL , X̲ , X̲ C) ⎠ ⎝ ρLV (TL , PL , Y̲ )

(27a) (27b)

(30b)

20

Churchill factor, originally for gas or liquid, is used for RVLE permeate (eq 27a) by a conservative averaged two-phase viscosity with β2 = LV2/(LV + LL)2 the squared RVLE vapor fraction:

Σ L = L V H̅LV(TL , PL , Y̲ ) + L LH̅LL(TL , PL , X̲ , X̲ C) n

MLV =

n

∑ YkMk ,

MLL =

k

μL = (1 − β 2)μLL (TL , PL , X̲ , X̲ C) + β 2μLV (TL , PL , Y̲ )

∑ XkMk + k

nc

∑ XCjMj j

⎛ ∂Σ ⎞ H̅kL,EMWR = ⎜ L ⎟ ⎝ ∂Lk ⎠T , P , L

(28)

3.2.4. Thermodynamic Properties of Retentate and RVLE Permeate. Momentum and energy balances require O(1) and O(2) thermodynamic properties for retentate and RVLE permeate. The RVLE permeate uses EMWR-dependent variables, thus thermodynamics must also operate in EMWR.

L

L

7082

(30d)

(30e)

i≠k

Ek̅ L,EMWR = H̅kL,EMWR + Mk

(30c)

λL 2 + Mk ·g ·h(z) 2

(30f)



Article

⎫ dT ⎧ ρ C̅ L ⎧ ⎫ dP T ⎨ L P − ΓTLλL 2⎬ L + ⎨1 + L ΓTL − ΓPLλL 2⎬ L ρL ⎩ ⎭ dz ⎭ dz ⎩ ML

⎛ ∂H̅ ⎞ ⎛ ∂H̅ ⎞ = ⎜ L⎟ + (∇̲ E H̅L)T J (: , TL) CP̅ L = ⎜ L ⎟ E/U ∂ ∂ T T ⎝ L ⎠ P , L̲ ⎝ L ⎠ P , E̲ L

L

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

n

(30g)

−

⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ ΓTL = ⎜ L ⎟ = ⎜ L⎟ + (∇̲ E ρL )T J (: , TL) E/U T T ∂ ∂ ⎝ L ⎠ P , L̲ ⎝ L ⎠ P , E̲ L

k

(30h)

L

(30i)

L

L

= (∇̲ E ρL )T J

E/U

(: , Lk ) (30j)

i≠k

V

L

− fk̂ )

(31)

3.2.6. HFM−GLC Model Equations. The GLC model comprises real species (k = 1, ..., n) EMWR mass balances and momentum/energy balances along the axial direction, written for retentate and RVLE permeate with the terms discussed from sections 3.2.1 to 3.2.5. 3.2.6.1. Real Species EMWR Mass Balances for Retentate and Permeate. dVk = −SaNk , dz

dL k = SaNk dz

4. SIMULATION OF NG PURIFICATION FLOWSHEET WITH HFM−GLC Figure 5 depicts data pertinent to a GLC process, candidate for offshore deployment, for high pressure NG purification with a battery of 40 HFM−GLC for CO2 removal. The NG feed is labeled GN-CONTACT-1 with (% mol): 1MMNm3/d (516.4 mol/s), T = 300 K, P = 50 bar, 73.22% CH4, 10.19% CO2, 9.09% C2H6, 4.25% C3H8, 1.78% C4H10, 1.47% C5H12, ρV = 55.9 kg/m3, ZV = 0.823, ΣV = −58322 kW, C̅ VP = 0.0523 kJ/(mol·K). Battery is fed with solvent mrSOLV3 with (% mol): 17651 kg/h (104.61 mol/s), T = 300 K, P = 5 bar, 20% MEA, 20% MDEA, 60% H2O, ρL = 943.2 kg/m3, ZL = 0.01014, ΣL = −33279 kW, C̅ LP = 0.146 kJ/(mol·K). PR EOS is used with all phases. Figure 5 panels a, b, and c depict three property diagrams for GN-CONTACT-1, respectively, C̅ P(T, P) (kJ/(mol·K)), Z(T,P) and H̅ P(T, P) (kJ/ mol), with bubble, dew, and critical loci superimposed (T ∈ [−150 °C, 120 °C], P ∈ [0 bar, 200 bar]). CO2 freeze-out is an issue below −80 °C, but was not checked since GLC works at warm conditions. Figure 5 panels a−c depict property surfaces in terms of color, covering subcooled liquid, supercritical fluid and superheated vapor, with the two-phase dome left empty. Despite that first order Z and H̅ (Figure 5b,c) do not denote anything special in the critical neighborhood, the situation is quite different for second order properties like C̅ P, which literally “flames” illuminating the supercritical fluid region in Figure 5a due to a O(2) critical transition. Figure 5d describes the purification flowsheet targeting on the final NG with 3.5% mol CO2, comprising (i) GLC battery (labeled GLC) with 40 HFM vertical modules detailed in Table 4; (ii) RVLE-H stripper (MR1) at 1 bar and 4300 kW of duty; (iii) phase separator (S1) at 1 bar; (iv) RVLE-H makeup mixer, pump, and cooler (MR2) at 5 bar and −460 kW of duty. GLC permeate and retentate are labeled with “L@” and “V@” followed by feed label. Other units label products similarly. 4.1. Simulation of HFM−GLC Unit. Sequential flowsheet calculations initiate with the simulation of the GLC unit (section 3.3) since all its feeds are known. Results for HFM−GLC simulation comprises profiles of retentate and RVLE permeate properties across the axial length ZC = 2 m. Several profiles are

(32)

3.2.6.2. Momentum Balances for Retentate and Permeate. {ΓTV λV 2}

dTV dP + {ΓPV λV 2 − 1} V + dz dz

n

∑ {ΓV λV 2} k

k

dVk dz

⎛λ ⎞ ⎛ πD + Sa ⎞ = ρV g sen θ − ⎜ V ⎟qSa + ψV ⎜ ⎟ ⎝ SV ⎠ ⎝ SV ⎠ {ΓTLλL 2}

dTL dP + {ΓPLλL 2 − 1} L + dz dz

n

∑ {ΓL λL 2} k

k

(33a)

dL k dz

⎛λ ⎞ ⎛ λ − λV ⎞ ⎛ Sa ⎞ = ρL g sen θ + ⎜ L ⎟qSa + ⎜ L ⎟qSa + ψL⎜ ⎟ ⎝ SL ⎠ ⎝ SL ⎠ ⎝ SL ⎠ (33b)

3.2.6.3. Energy Balances for Retentate and Permeate. ⎫ dT ⎧ ρ C̅ V ⎧ ⎫ dP T ⎨ V P − ΓTVλV 2⎬ V + ⎨1 + V ΓTV − ΓPVλV 2⎬ V ρV ⎩ ⎭ dz ⎭ dz ⎩ MV ⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

n

−

∑ {ΓV λV 2} k

k

dVk dz

⎛λ ⎞ Ω EπDρV (TE − TV ) =⎜ V ⎟qSa − ρV g sen θ + qV ⎝ SV ⎠ ⎛ ΩSaρ ⎞ V⎟ − ⎜⎜ ⎟(TV − TL)) ⎝ qV ⎠

(34b)

3.3. Numerical Resolution of HFM−GLC Model. The set of 2n + 4 ODEs in eqs 32 to 34 for 2n + 4 variables [TV, PV, V1, ..., Vn, TL, PL, L1, ..., Ln], is an algebraic-differential system with an embedded algebraic RVLE-T problem for 2n + 2nC + 2 RVLE E variables. It was integrated numerically from z = 0 to z = ZC via the IVP gear method for stiff ODE keeping the algebraic RVLE-T always solved for each proposed [TL, PL, L1, ..., Ln] from the IVP layer. This strategy was able to simulate GLC in all tested cases, generating meaningful profiles for all retentate and permeate variables.

3.2.5. Trans-Membrane Fluxes of Real Species. Real species fluxes are given by species equivalent permeancies times the respective fugacity differences across HFM: Nk = Φk (fk̂

dL k dz

⎛ Saρ ⎞ n V L,EMWR L⎟ + ⎜⎜ ) ⎟ ∑ Nk(Ek̅ − Ek̅ ⎝ qL ⎠ k

⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ ΓPL = ⎜ L ⎟ = ⎜ L⎟ + (∇̲ E ρL )T J (: , PL) E/U ⎝ ∂PL ⎠T , L̲ ⎝ ∂PL ⎠T , E̲ ⎛ ∂ρ ⎞ ΓLk = ⎜ L ⎟ ⎝ ∂Lk ⎠T , P , L

k

⎛ ΩSaρ ⎞ ⎛λ ⎞ L⎟ = −⎜ L ⎟qSa − ρL g sen θ + ⎜⎜ ⎟(TV − TL) ⎝ SL ⎠ ⎝ qL ⎠

L

L

∑ {ΓL λL 2}

(34a) 7083



Article

Table 4. Parameters of HFM−GLC Modules in Figure 5d object

item

value

GLC GLC GLC GLC GLC HFM

θ D ZC volume NM d

π/2 rd 0.8 m 2m 1.005 m3 40 0.5 mm

HFM

do

0.502 mm

HFM HFM HFM

item

value

TE ΩE Ω gas feed solvent ΦCO2

27 °C 5 W/(m2·K) 2 W/(m2·K) GN-Contact-1 mrSOLV3 1.7 × 10−4 mol/s·m2 bar

ΦCH4

5.8 × 10−6 mol/(s·m2·bar)

NHF

6

2.188 × 10

ΦC2H6

10−8 mol/(s·m2·bar)

NMAX HF

2.303 × 106

ΦC3H8

10−9 mol/(s·m2·bar)

total area

2

6901.4 m

ΦC4H10

retentate and permeate, and in Figure 6 panels c and d for % mol of CO2 and CH4 in retentate and in the two permeate phases. Figure 6e depicts profiles of % recovery of CO2 and CH4. Figure 6f shows profiles of % mol of all species in vapor (Yk) and liquid (Xk, XCj) permeate phases, where liquid mol fractions (XCj) of complexes CO2−H2O−MEA and CO2−H2O−MDEA play central roles. Figure 6f should be analyzed accompanied by the profiles of reaction rates (mol/s) (η1,η2) forming CO2−H2O− MEA and CO2−H2O−MDEA shown in Figure 6h. By last, Figure 6g reports profiles of molar rates of permeate liquid (LL) and vapor (LV), showing that the flow rate of liquid stabilizes near the middle of the unit, whereas the flow of gas increases monotonously. Figure 6 panels a to h all reveal an effluorescent idiosyncrasy of GLC’s which was intentionally seeded in this flowsheet and is also a syndrome of NG purification with membrane permeators. The point is that separation can be spoiled by excess of equipment size, that is, by excess of contact area of HFM. In other words, overdesigned GLC batteries gain capacity, but severely lose selectivity comparatively to tight designs. This is more evident in Figure 6b,d, f−h, which subtly report that the majority of favorable separation has occurred by the middle of the second GLC quarter; that is, the rest of the unit only serves to spoil separation by allowing CH4 to trespass into the permeate with a practically constant flux accompanied by a decreasing faint flux of CO2. This is corroborated by Figure 6b,e, and g, showing that CH4, which has a practically constant and large driving force (see Figure 7c,d), accesses the permeate under constant flux, that is, giving linear profiles of permeate mol/s rate (LCH4, Figure 6b), % recovery (Figure 6e), and permeate gas rate (LV, Figure 6g), while CO2 rapidly loses driving force and is surpassed by CH4 in the permeate by the end of the third quarter. Even worse, the excess of area allows excessive bubbling of CH4 into the permeate, producing more bubbling via CO2 stripping as shown in Figure 6f,h, that is, chemical reactions extensions and liquid phase complex compositions retrocede. In other words, CO2 is converted in the first GLC quarter to the liquid phase under the form of complexes; and, after the middle of the second quarter, is gradually released to the permeate vapor as the breakage of complexes proceeds via stripping action caused by CH4. Figure 7 panels a−e address spatial profiles not directly related to material balances, but also of paramount significance. Figure 7a reports the GLC temperature profiles, showing that the heating of permeate and retentate is under progress as occurs in AGWA absorbers. The RVLE AGWA model calibrated in section 2 with only VLE data (i.e., no thermal data used) was capable, thanks to the inexorable consistence of thermodynamics, to reproduce a reasonable increase of 12 °C in the temperature of GLC products. This heating is created by the exothermal CO2

Figure 5. Natural gas GN-Contact-1: diagrams with bubble and dew loci (a) CP (kJ/(mol·K)) vs (T, P), (b) Z vs (T, P), (c) H (kJ/mol) vs (T, P), (d) flowsheet for CO2 removal with HFM−GLC.

shown in Figures 6 and 7, where the GLC length was divided into 0.5 m segments under distinct background colors for easy recognition of details. Profiles are respectively depicted in Figure 6 panels a and b for EMWR rates (mol/s) of CO2 and CH4 in the 7084



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Figure 6. GLC profiles (k = 1, ..., n, j = 1, ..., nC, L ≡ permeate, V ≡ retentate): (a) Vk (mol/s), (b) Lk (mol/s), (c) % mol Vk, (d) % mol Lk, (e) % recovery, (f) % mol Yk, Xk, XCj, (g) LL, LV (mol/s), (h) η1, η2 (mol/s).

absorption into the permeate, and then, via heat transfer across HFM, it is propagated to the retentate also heating it. Figure 7a corroborates the oversizing of the GLC battery because the heating effect ceases by the middle of the second quarter with maxima in the permeate and retentate temperatures, which then go slowly down due to reverse absorption caused by CH4 stripping (endothermic) and cooling to the environment via external heat transfer. As a consequence of parallel flow and high contact area, retentate and permeate emerge on the other side practically at the same temperature. Figure 7a also discards, as

totally unnecessary, dew-point adjustment of the gas feed since, contrary to membrane permeators, GLC promotes much more chemical heating than Joule-Thomson cooling via gas expansion through HFM. Figure 7b tested the consistency of the GLC model showing that, despite the reversed curvatures of the profiles of retentate and permeate energy rates, their sum is almost constant, with a very subtle fall due to small loss of energy via cooling through the external heat transfer. The profile of retentate energy rate goes up due to the removal of CO2 which has a very negative enthalpy of formation, while the permeate 7085



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Figure 7. GLC profiles (k = 1, ..., n, j = 1, ..., nC, L ≡ permeate, V ≡ retentate): (a) TL,TV (°C), (b) energy rate (kW), (c) V fugacities (bar), (d) L fugacities (bar), (e) PL,PV (bar), (f) Xk, XCj, (g) Yk, (h) CO2/CH4 selectivity.

the CO2/CH4 selectivity defined as (RECCO2/RECCH4)/(YCO2V/ YCH4V) where RECCO2 and RECCH4 are absolute recoveries in mol/s. The retentate effluent leaves the HFM−GLC at 38.8 °C, 49.85 bar, with 0.83 MMN m3/d and with 76.5% CH4, 3.5% CO2, 11% C2H6, 5.1% C3H8, 2.2% C4H10, and 1.8% C5H12. 4.2. Simulation of MR1, S1 and MR2 Units. Separator S1 is merely a physical splitter of liquid and vapor phases already in RVLE as imposed by MR1. MR1 and MR2 are reactive flashes with specified pressure and heat duty, both solved via RVLE-H algorithm (section 3.1) after the resolution of GLC. The liquid

profile goes down by the opposed reason. Figure 7 panels c and d portrait profiles of fugacities (bar) in retentate and RVLE permeate, showing that, as said before, the driving force of CO2 reaches something near zero by the middle of the unit, whereas the CH4 driving force remains almost constant above 30 bar. Figure 7e presents pressure profiles of retentate and permeate, showing that the correspondent head losses are less than 0.2 bar both for retentate and permeate. Figure 7 panels f and g are similar to Figure 6 panels d and f. Finally, Figure 7h corroborates the previous discussion showing the fall, after the first quarter, of 7086



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absorption heat effects in the RVLE permeate flow as consequence of the CH4 intrusion; (v) accurate prediction of head losses associated with the compressible flows of retentate and permeate. The HFM−GLC model was used in the simulation of a small flowsheet for CO2 withdrawal from 1 MMNm3/d of a CO2-rich NG with an approximately 100% oversized battery of 40 GLC modules with 6901.4 m2 each. This analysis confirmed the expected HFM−GLC effects like temperature increase, inertial dominance of the CH4 driving force entailing losses of CH4 to the permeate, creating stripping of CO2 in the permeate due to CH4 intrusion, and higher losses of MEA and water during solvent regeneration at 1 bar due to the high content of light gases in the permeate effluent.

product of MR2 was considered recycle converged with mrSOLV3 in the first trial. MR1 is a reactive, low pressure, one stage stripper for solvent recovery at 1 bar with a heat duty of 4300 kW, resulting in T = 100.51 °C and 78.13% vapor. The vapor phase of S1 has 3.9% MEA, 33.6% CH4, 24.9% CO2, and 37.6% H2O, whereas its liquid product has 35.4% MEA, 49.5% MDEA, 14.4% H2O, 0.1% CO2, 0.02% CH4, 0.56% complex CO2−H2O−MEA, and 0.02% complex CO2−H2O−MDEA. MR2 is a makeup mixer and cooler, pumping recomposed solvent at 5 bar back to the GLC. It is fed with 42.08 mol/s of liquid from S1 at 100.51 °C, makeup streams at 27 °C of MEA and H2O, respectively, with 5.84 and 56.67 mol/s, and a cooling of −460 kW, resulting in a liquid very similar to mrSOLV3 at 27.4 °C. Appendix D (see Supporting Information) presents complementary stream data for units GLC, MR1, S1 and MR2, and also conveys selected figures in this analysis of GLC flowsheet for NG purification.

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ASSOCIATED CONTENT

* Supporting Information S

Appendix A: Jacobian matrices for Newton−Raphson method on RVLE-H, with eqs (A1), (A2), (A3), (A4), (A5), (A6), (A7), (A8), (A9). Appendix B: Differential influence terms from Jacobian Terms, with eqs (B1), (B2), (B3), (B4), (B5), (B6), (B7), (B8), (B9), (B10), (B11), (B12), (B13), (B14), (B15) and (B16). Appendix C: Derivation of partial molar EMWR properties, PMPEMWR, with eqs (C1), (C2), (C3), (C4), (C5), (C6), (C7) and (C8). Appendix D: Selected figures, simulation of GLC Flowsheet for NG Purification [GN-Contact-1]. This material is available free of charge via the Internet at http://pubs. acs.org.

5. CONCLUDING REMARKS This work presents a complete modeling approach focusing on a gas−liquid contactor (GLC) with hollow fiber membranes (HFM) for CO2 removal from high pressure CO2-rich natural gas (NG). This model comprises (i) an entirely novel16,17 AGWA equilibrium model, based on the Chemical Theory, using nonionic complex forming ChE chemical reactions, capable of reproducing chemical absorption and desorption of CO2 (and H2S) in aqueous solutions of alkanolamines, from low to high pressures, which was calibrated in terms of ChE constants with VLE AGWA data from the literature; (ii) two RVLE numerical solvers, namely, RVLE-T and RVLE-H, which are appropriate for RVLE AGWA problems at high pressure using state specifications, respectively, centered in temperature and enthalpy rate; (iii) a 1D parallel flows, HFM−GLC model assuming two-phase compressible plug flow of RVLE permeate and compressible plug flow of retentate gas, built with 1D mass, energy, and momentum balances for retentate and permeate, allowing heat and mass interfacial transfers, shear and compressibility terms, with all properties of all phases involved rigorously calculated via conventional EOS like PR and SRK. The HFM−GLC model has as dependent variables temperature, pressure, and EMWR rates of real species of retentate and permeate. It is an algebraicdifferential system which was numerically integrated by an IVP gear method with numerical resolution of an RVLE-T problem embedded. This HFM−GLC model was able to soundly reproduce several interesting behaviors inside high capacity, high pressure contactors, such as (i) strong reactive and phase change thermal effects, resulting that permeate and retentate temperatures increase along the GLC length, making totally unnecessary any kind of pretreatment of the NG feed for dewpoint adjustment or water removal; (ii) intense, fugacity driven, trans-membrane flux of CO2 pulled by the ChE interaction with the solvent and the fugacity driving force between retentate and permeate, showing high concentration of heat release at the more intense absorbing zones; (iii) almost invariant trans-membrane flux of CH4 due to an extremely inertial, relatively high, driving force (since CH4 strongly dominates the NG retentate) with the consequence that, if GLC is oversized concerning the CO2 capture target, CH4 will linearly bubble into the permeate, despite the vanishing presence of CO2 in the trans-membrane flux, entailing loss of valuable CH4, selectivity degradation of CO2 and CO2 stripping from the liquid permeate, which increases the gas fraction in the permeate flow and hinders the performance of the operation; (iv) equilibrium shifts and reverse

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +5521 2562-7535. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS J.L. de Medeiros and O.Q.F. Araujo gratefully acknowledge CNPq-Brazil for financial support.

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ABBREVIATIONS AGWA = acid gas, water, alkanolamine BVP = boundary value problem ChE = chemical equilibrium EMWR = equivalent moles without reaction EOR = enhanced oil recovery EOS = equation of state GLC = gas−liquid contactor HFM = hollow fiber membrane IVP = initial value problem NG = natural gas NLP = nonlinear programming NRM = Newton−Raphson method ODE = ordinary differential equation PMP = partial molar property PMPEMWR = partial molar EMWR property PR = Peng−Robinson RHS = right hand side RVLE = reactive vapor−liquid equilibrium SRK = Soave−Redlich−Kwong VLE = vapor−liquid equilibrium 1D = one dimensional 2D = two dimensional dx.doi.org/10.1021/ie302507n | Ind. Eng. Chem. Res. 2013, 52, 7074−7089


Article

S̲A, S̲S = selection matrices of amines (nA × n) and solutes (nS × n) S2R = statistic weighted sum of squares of residues of variables T, TE, TV, TL = temperature, external temperature, retentate and permeate temperatures (K) TC, PC, ω, TB = critical temperature and pressure, acentric factor, normal boiling point (K) X̲ , Y̲ = mol fraction vectors (n × 1) of real species in liquid and vapor RVLE phases X̲ C = vector (nc × 1) of mol fractions of complex species in liquid RVLE phase U̲ = specification vector (n + 2 × 1) of RVLE permeate in RVLE-T and RVLE-H V̲ = vector (n × 1) of retentate molar rates of real species (mol/s) Z, ZC, ZHF = axial GLC coordinate and lengths of GLC and HFM (ZC = ZHF) (m) Z̲ , Z̲ E, Ẑ = vector of correct, observed and estimated AGWA experiment variables Z, ZL, ZV = compressibility factor, permeate and retentate compressibility factors W i = diagonal weight matrix of experiment i

Nomenclature

a = interfacial area per unit of volume (m−1) of GLC shell C̅ LP, C̅ VP = molar heat capacities of permeate (RVLE) and retentate (gas) (J/mol.K) diag (.) = diagonal matrix creator operator d, dO = HFM internal and external diameters (m) D, DL, DV = shell diameter and hydraulic diameters of permeate and retentate flows (m) E̲ = vector of state variables in RVLE-T and RVLE-H E̲ V,EMWR , E̲ L,EMWR = PMPEMWR energies of k in retentate and k k permeate (J/mol) f L, f V = Darcy friction factors in permeate and retentate flows f ̂LL, f ̂LV = vectors of real species fugacities in liquid and vapor RVLE phases (bar) F̲(Z̲ , θ̲) = 0̲ = vector of experiment constraints with correct variables and parameters F̲(E̲, U̲ ) = 0̲ = vector of constraints for reactive VLE permeate in RVLE-T and RVLE-H g = gravity acceleration (9.81 m/s2) h(z) = elevation at axial position z in the flow (m) H̅ L, H̅ V = molar enthalpies of permeate and retentate (J/mol) H̅ LL, H̅ LV = molar enthalpies of liquid and vapor RVLE phases (J/mol) H̅ V,EMWR , H̅ L,EMWR = PMPEMWR enthalpies of k in retentate and k k permeate (J/mol) I = identity matrix J E, J U = Jacobians of RVLE-T or RVLE-H constraints regarding E and U J E/U = Jacobian of E̲ with respect to U̲ in RVLE-T or RVLE-H Kk , K̲ = ChE constant of reaction k and the vector (nC × 1) of ChE constants L = vector (n × 1) of EMWR rates of real species in RVLE permeate (mol/s) LL , LV = total mol rate of liquid and vapor phases belonging to permeate (mol/s) Mk = molar mass of real species k (kg/mol) ML, MV = EMWR molar masses of permeate and retentate (kg/mol) MLV, MLL = molar masses of liquid and vapor phases in the permeate (kg/mol) n, nC, nA, nS, nE = no. of real and complex species, amines, solutes, and AGWA experiments nθ, nZ, nF = no. of parameters, experiment variables, and experiment constraints NHF, NMAX HF , NM = no. of HFMs and its maximal value; no. of GLC modules in the battery Nk = trans-membrane flux of real species k (mol/(s·m2)) P, PV, PL = absolute pressure, retentate and permeate absolute pressures (Pa) P̲S = vector (ns × 1) of partial pressures of solutes (bar) qV, qL, q = mass flow rates of retentate and permeate (kg/s); HFM mass flux (kg/(m2·s)) Q̅ , Q̅ LV, Q̅ LL = molar property Q and molar property Q of liquid and vapor phases in permeate = PMP and PMPEMWR Q property for real species k Q̅ k, Q̅ EMWR k M L, M V, 9 eL, 9 eV = flow perimeters and Reynolds numbers of permeate and retentate (m) R = ideal gas constant (8.314 Pa·m3/(mol·K)) S = πD2/4 = section of GLC shell (m2) SL, SV, Sa = NHFπdO = sections of permeate and retentate flows (m2); interfacial area per length (m)

Greek Symbols

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α̲S = vector (ns × 1) of solute loadings (mol/mol amine) in liquid phase β = LV/(LV + LL) = vapor mol fraction of RVLE permeate εV, εL = contact surface roughness for retentate and permeate flows (m) ξ, ξMIN = porosity of HFM bed and its minimal value η̲ = rates of reaction extensions in RVLE permeate (mol/s) Φk = equivalent permeancy of species k (mol/(s·m2·bar)) ϕL̂ L, ϕL̂ V = fugacity coefficients of k in liquid and vapor RVLE phases ΓPL, ΓPV = isothermal compressibility of RVLE permeate and retentate (kg/(Pa·m3)) ΓTL, ΓTV = isobaric expansivity of RVLE permeate and retentate (kg/(K·m3)) ΓLk, ΓVk = Species k mol expansivity of RVLE permeate and retentate (kg.s/(mol·m3)) λV, λL, ρV, ρL = velocities (m/s) and densities (kg/m3) of retentate and permeate flows μV, μL, μLV, μLL = dynamic viscosity of retentate, permeate, vapor, liquid RVLE phases (kg/(m·s)) Ω, ΩE = internal and external heat transfer coefficients (W/ (m2·K)) Π̲ = real species stoichiometric matrix (n × nc) for complex forming reactions ΨV, ΨL = shear stress on contact surfaces in retentate and permeate flows (Pa) σE2 = fundamental variance in the estimation process ∑L, ∑V = enthalpy rates in permeate and retentate flows (kW) θ, θ̲, θ ̂ = inclination of GLC (rd); correct and estimated model parameters (nθ × 1) Θ̲ x, Θ̲ y, Θ̲ c = unbounded mol fraction variables in RVLE-T and RVLE-H

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Simulation of an Offshore Natural Gas Purification Process for CO2

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