Two-Phase Flow in Liquid Chromatography, Part 1: Experimental

Article Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

pubs.acs.org/IECR

Two-Phase Flow in Liquid Chromatography, Part 1: Experimental Investigation and Theoretical Description Franziska Ortner and Marco Mazzotti* Institute of Process Engineering, ETH Zurich, 8092 Zurich, Switzerland S Supporting Information *

ABSTRACT: Two-phase flow is investigated in the context of liquid chromatography. In order to describe the column behavior in the presence of multiple convective phases, an equilibrium theory model is established, accounting for both adsorption and multiphase flow. The validity of this model is studied specifically for the experimental system phenetole−methanol−water on a Zorbax 300SB-C18 column. For this purpose, the experimental system is characterized in terms of thermodynamic liquid−liquid equilibria, adsorption behavior, and hydrodynamic properties by independent experimental campaigns. Mathematical relationships are established, describing the investigated effects. Implementation of these relationships in the equilibrium theory model and application of the model to calculate elution profiles provides a quantitative description of experimental data from dynamic column experiments. The good agreement validates the model, its assumptions, and the established relationships. This study provides a thorough insight on the implications of multiple convective phases in the context of liquid chromatography.

1. INTRODUCTION In liquid chromatography, a phase split and subsequent twophase flow can occur if one or more components become considerably enriched within the column and consequently exceed the solubility limit. This enrichment can be due to different reasons, such as the interaction of different adsorbing components, as was recently observed for the system phenetole (PNT) and 4-tert-butylphenole on the adsorbent Zorbax 300SB-C18.1,2 Other reasons might be the impact of modifiers/sample solvents3,4 or chemical reactions occurring in chromatographic reactors. Since standard chromatographic models do not account for the phenomena of phase split and two-phase flow, model descriptions fail at the conditions where these phenomena occur.1 As these conditions are not understood properly and cannot be described mathematically, they are commonly avoided, and conditions are chosen (usually lower initial and/or feed concentrations) which guarantee a single-phase flow. To evaluate whether this is really necessary or whether efficient processes can also be carried out under two-phase flow conditions, we want to physically understand and mathematically describe the phenomena of phase split and two-phase flow in liquid chromatography. For this purpose, we set up an equilibrium theory model accounting for both adsorption and multiphase flow. While the © XXXX American Chemical Society

standard equilibrium theory model in the context of chromatography accounts for adsorption but only considers a single convective phase,5−7 multiphase flow has been studied thoroughly for applications in natural reservoirs,8 where in turn adsorption effects are most often neglected. A model combining multiphase flow and adsorption has, to the best of our knowledge, only been considered once,9,10 in order to account for adsorption effects in enhanced coalbed methane recovery (ECBM). This previous study was purely theoretical and assumed very simple relationships for the thermodynamic equilibria and the adsorption behavior. In this study, we thoroughly assess through independent experimental measurements the physical behavior due to thermodynamic equilibria and adsorption, as well as twophase flow behavior for the experimental system PNT− methanol−water on the adsorbent Zorbax 300SB-C18. Mathematical relationships, describing the investigated effects, are established based on the experimental data, and implemented into the equilibrium theory model. This model Received: Revised: Accepted: Published: A

December 14, 2017 February 6, 2018 February 7, 2018 February 7, 2018 DOI: 10.1021/acs.iecr.7b05153 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

space coordinates, respectively, normalized through the superficial velocity u (which is constant, as shown below) and the column length Lc:

is used in a fully predictive way to simulate elution profiles (for a detailed derivation of the solutions see the companion paper),11 which are compared to dynamic column experiments, involving both adsorption and two-phase flow. We consider this contribution to be unique with respect to various aspects: To the best of our knowledge, this is the first time that a multiphase flow is considered in the context of liquid chromatography, both concerning the experimental and the theoretical analysis. Also, the depth of the experimental and theoretical characterization of a specific model system, featuring two-phase flow through a porous medium combined with adsorption and involving the complex thermodynamic behavior of a partially miscible ternary system, is unparalleled. The established equilibrium theory model, combined with the mathematical relationships derived during the characterization work, describes the experimental system with a high accuracy, for a broad range of experimental conditions in a fully predictive manner, i.e., without the adjustment of further parameters or relationships. This combined experimental and theoretical approach provides a deep understanding of the interplay of two-phase flow through a porous medium and adsorption, which we consider unprecedented, and which is of interest not only for the chromatographic community but also for several applications in natural oil and gas reservoirs. The paper is structured as follows. In section 2, we establish the equilibrium theory model which accounts for both adsorption and two-phase flow, and we thoroughly discuss the underlying assumptions. The description of two-phase flow in a porous medium is explained in greater detail in section 2.2, being novel in the context of liquid chromatography. Subsequently, we describe the experimental procedures applied in different experimental campaigns for system characterization and model verification (section 3). Experimental results and established mathematical relationships, which characterize the system with respect to thermodynamic equilibria in the convective phases, adsorption behavior, as well as two-phase flow behavior, are presented and discussed in section 4. In a next step (section 5), the relationships established in section 4 are implemented into the equilibrium theory model, and predicted concentration and flow profiles are obtained by solving the model at different initial and feed conditions. Predicted profiles are compared to experimental data obtained from dynamic column experiments, which were performed at the relevant initial and feed conditions. The comparison reveals a quantitative description of the experimental behavior under most conditions when accounting for both adsorption and hydrodynamic effects in the model, and it illustrates shortcomings of the model when neglecting either adsorption effects or two-phase flow. Finally, important findings concerning a two-phase flow in liquid chromatography are summarized and discussed in view of possible chromatographic applications in section 6.

τ=

(2)

The variables Ci and Fi are the overall liquid concentration and the overall fractional flow of component i, respectively, being defined as NP

Ci =

NP

∑ xijρj̅ Sj = ∑ cijSj j=1

j=1

NP

Fi =

∑ xijρj̅ f j

(3a)

NP

=

j=1

∑ cij f j j=1

(3b)

Here, xij and cij are the mass fraction and concentration of component i in phase j, respectively, Sj is the saturation of phase j, i.e., the volume fraction of that phase with respect to the overall liquid volume, and f j is the fractional flow, i.e., the volumetric flow fraction of phase j with respect to the overall volumetric liquid phase flow. Since the different convective phases can move with different velocities, in general, f j ≠ Sj; thus, Fi ≠ Ci. Assumptions concerning the hydrodynamic behavior of the convective phases are described in more detail in section 2.2. The density ρ̅j of phase j can be determined from the phase composition and the component densities ρi assuming additivity of volumes: ⎛ NC xij ⎞−1 ρj̅ = ⎜⎜∑ ⎟⎟ ⎝ i = 1 ρi ⎠

(4)

All the liquid phases are assumed to be in thermodynamic equilibrium, i.e., the chemical potentials μik of component i in all the liquid phases k are identical. Furthermore, thermodynamic equilibrium is assumed between the liquid phases and the adsorbed phase (adsorbed phase concentrations ni in mass of component i per volume of adsorbent in the completely regenerated state). This thermodynamic equilibrium is described by a relationship for ni, being a function of the liquid phase activity ai of component i, which is obviously the same in all liquid phases. Finally, since we are dealing with a nondilute system, volumetric effects due to adsorption or desorption have to be accounted for in order to fulfill the overall mass balance (sum of all component mass balances). The volumetric effects are accounted for by a variable porosity: NC ⎛ ⎞ n ϵ = ϵref − (1 − ϵref ) ∑ ⎜⎜ i ⎟⎟ ρ i=1 ⎝ i ⎠

(5)

where ϵref is the porosity at the reference state, i.e., for the adsorbent being equilibrated with pure solvent. Equation 5 is based on the underlying assumption that the stationary phase is constituted of the adsorbent itself and of the adsorbed phase. Adsorbed components take up the same volume in the liquid as in the adsorbed phase, which is characterized by the component density ρi. It can be shown from the overall mass balance, i.e., the sum of all component mass balances (eq 1), that with the implied relationship for the variable porosity ϵ, the superficial velocity u remains constant.11 For a detailed derivation of the

2. THEORETICAL BACKGROUND 2.1. Equilibrium Theory Model. The investigated system is mathematically described by an equilibrium theory model accounting for adsorption and two-phase flow: ∂(ϵCi) ∂n ∂F + (1 − ϵref ) i + i = 0; i = 1, ..., NC ∂τ ∂τ ∂ξ

tu z ;ξ= Lc Lc

(1)

In this mass balance equation, NC is the number of components, and τ and ξ are the dimensionless time and B

DOI: 10.1021/acs.iecr.7b05153 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research model and for an explanation of the model solution (based on the method of characteristics), the reader is referred to the companion paper in the series.11 2.2. Description of the Hydrodynamic Behavior. For laminar flow (Reynolds numbers Re < 1; for the system considered here Re is about 5 × 10−3) of a convective phase through a porous medium, Darcy’s law establishes a linear relationship between pressure gradient and superficial velocity.12,13 This concept can be extended to multiphase flows, and simplified for the assumption of one-dimensional, horizontal flow as follows: k r, j K ∂pj

uj =

∂z

ηj

Figure 1. Viscosity estimation for the binary methanol−water system at 296 K, according to the estimation procedure described elsewhere.15 Pure component data at different temperatures taken from the literature.16,17

(6)

where uj is the superficial velocity of phase j, describing the flow of that phase through the cross-sectional area of the column, K is the permeability, i.e., a property of the porous medium, ηj is the viscosity of phase j, and ∂pj/∂z is the pressure gradient in phase j. The relative permeability kr,j, with values ranging between 0 and 1, accounts for the fact that not the entire void space of the porous packing is available for phase j, since it has to be shared with other convective phases. As a consequence, the resulting superficial velocity uj is lower than it would be at the same pressure gradient, if phase j were the only convective phase. Neglecting capillary pressures between all convective phases, one can conclude that ∂pj/∂z = ∂pk/∂z = ... = ∂p/∂z; as a consequence ⎛ k r, nK ⎞ ∂p ⎟⎟ ηn ⎠ ∂z n=1 ⎝

NP

n=1

(7)

and k r, j

fj =

uj u

=

ηj N k ∑n =P 1 ηr,n n

(8)

In the investigated system, the viscosities of the phase rich in PNT were assumed to be equal to the viscosity of PNT at 23 °C, ηP = 1.17 mPas, which was interpolated linearly from the data reported for 20 and 25 °C.14 The viscosities of the PNTlean phase were assumed to correspond to the viscosity of the water−methanol mixture at the relevant solvent ratio at 23 °C. Viscosities of water−methanol mixtures were estimated as described in the literature,15 with pure component relationships between viscosity and temperature fitted to data for water16 and that for methanol17 (for the estimation at 296 K, see Figure 1). It was found that experimental breakthrough data for twophase (nonadsorbing) systems in porous media is very often described accurately assuming a power-law relationship between relative permeabilities kr,j and phase saturations Sj.18−20 In this contribution, we consider the following relationship: R Seff < 0: k rR = 0; k rL = 1 R R R λ L R λ 0 ≤ Seff ≤ 1: k rR = k r,max (Seff ) ; k rL = k r,max (1 − Seff ) R Seff > 1: krR = 1; k rL = 0

L

(10)

3. EXPERIMENTAL SECTION 3.1. Materials. All chemicals used in this study, i.e., PNT (purity >99%, Sigma-Aldrich), methanol (HPLC-grade, purity >99.9%, Sigma-Aldrich), and deionized water, are liquids at ambient temperature (23 °C) and pressure. All mixtures were prepared gravimetrically, using analytical balances (DeltaRange AX205 and XP2003S, Mettler Toledo) and precision balances (DeltaRange PM4600, Mettler Toledo). Dynamic column experiments were performed with the stationary phase Zorbax 300SB-C18, using a prepacked 50 mm × 4.6 mm stainless-steel column (Agilent Technologies). The total (inter- and intraparticle) porosity of the packing in equilibrium with a water−methanol mixture was determined by pulse injections of the nonadsorbing component Uracil (purity >99%, Sigma-Aldrich); it is ϵref = 0.615. HPLC analysis of PNT was performed with a prepacked 150 mm × 4.6 mm stainlesssteel column containing the adsorbent Zorbax 300SB-C18 (Agilent Technologies) with a total porosity of 0.61. The dynamic column experiments and quantitative analysis of PNT were carried out on a modular HPLC unit (Agilent

(9a) R

S R − Si 1 − Si − Sr

The superscripts R and L denote the PNT-rich (wetting) and -lean (nonwetting) phase, respectively. Equations 9 and 10 include the following six parameters: the irreducible/residual saturation Si/r, determining the saturation of the wetting/ nonwetting phase, respectively, at which this phase becomes hydraulically disconnected and thus cannot be displaced further by the corresponding (nonwetting/wetting) phase in thermodynamic equilibrium; the maximum relative permeabilities kR/L r,max, which determine the pressure drop at the residual/ irreducible saturation; and the exponents λR/L, which determine the shape of the relative permeability functions. In principle, these parameters are functions of the interfacial tension σ between the wetting and the nonwetting phase, and should thus change for different tie-lines connecting phases in thermodynamic equilibrium. The parameters should behave in such a way that an equal-velocity behavior (f j = Sj) is approached as the phases in thermodynamic equilibrium approach the plait point of the binodal curve (since the two phases merge to one phase at that point). In this study, we simplify the description in such a way that we assume the six parameters to be constant, regardless of the change in interfacial tension.

NP

∑ uj = ∑ ⎜⎜

u=

R Seff =

(9b) (9c)

with C


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Industrial & Engineering Chemistry Research Technologies 1200 Series) equipped with a quaternary dual piston pump with an online degasser. Furthermore, the unit comprises an autosampler, a thermostated column compartment and a UV/DAD detector. The unit is additionally equipped with a manual 5, 9, or 20 mL injection loop (Rheodyne 7725i), which is connected to the data acquisition software (Agilent ChemStation for LC 3D systems). All experiments were carried out at a temperature of 23 °C and a volumetric flow rate Q of 1.2 mL/min. For the HPLC analysis of PNT, a mobile phase of methanol/water 63:37 (v/v) was used. 3.2. Density Measurements. Densities of binary mixtures of PNT and methanol, ranging from pure PNT to pure methanol, were determined at 23 °C with a density meter (DMA 48, Anton Paar). 3.3. Liquid−Liquid Equilibria. Different immiscible compositions of water, methanol, and PNT (of a volume of approximately 10 mL) were prepared and stirred at 800 rpm for 36 h in an EasyMax 102 workstation (Mettler Toledo), thermostated at 23 °C. Subsequently, the two liquid phases in thermodynamic equilibrium were separated by centrifugation (10 000 rpm, Rotina 420, Hettich). For both phases, the content of methanol was determined by gas chromatography (Clarus 480, PerkinElmer), on a 80/100 Porapak Q column (6 feet × 1/8 in. × 2.1 mm, Supelco Analytical). The content of PNT in the PNT-lean phase was determined by HPLC, while the water content of the PNT-rich phase was determined by Karl Fischer titration (titrator: 702 SM Titrino, Metrohm, Karl Fischer reagent: Hydranal-Composite 5, Sigma-Aldrich). The content of the relevant third component (in wt %) was calculated from the concentration of the other two components. 3.4. Adsorption Behavior. The adsorption behavior of PNT was determined by frontal analysis at different solvent ratios rs = xM/(xM + xW) (mass fraction of methanol in the pure solvent) and in the soluble region. In frontal analysis, a column, initially equilibrated with pure mobile phase, is completely saturated with a feed solution of a specific concentration of the investigated adsorbing component. Subsequently, the column is regenerated with the mobile phase. In the performed breakthrough experiments, the solvent ratio rs of both the initial state, being pure solvent, and of the feed solution at a certain concentration of PNT cFP, was identical. Initial and feed states are indicated by the superscripts 0 and F, respectively. The feed solution was injected through the 9 mL injection loop, constraining the injection volume to 8 mL for reasons of accuracy (backmixing effects in the loop). The UV signal at the column outlet at a wavelength of 295 nm was recorded and transformed to concentration profiles via a rational calibration function (established from the relationship between the UV signal at the feed plateau and the feed concentration). As the column becomes saturated, i.e., thermodynamic equilibrium between the adsorbed and the liquid phase is reached, the feed state F breaks through the column outlet. Having reached the concentration cFi in the eluate, the adsorbed phase concentration nFi , which is in thermodynamic equilibrium with the liquid phase concentration cFi , can be determined from the resulting elution profile using the integral mass balance of component i:

∫t

tF 0

uciFAc

dt −

∫t

tF 0

uci(t , Lc)Ac dt

= (ϵFciF + (1 − ϵref )niF − ϵ0ci0 − (1 − ϵref )ni0)AcLc (11) 0

where Ac denotes the cross-sectional area, t denotes the time at which feeding of the feed state at the column inlet starts, and tF = t(ci(Lc) = cFi ) can be any time t at which the eluate has reached the feed concentration. In eq 11, the first and second terms account for the fluxes into and out of the column, respectively, whereas the term on the right-hand side describes the hold-up of component i in the liquid and the adsorbed phase. Substituting eq 5 into eq 11 with c0P = n0P = 0, the adsorbed phase concentration of PNT, nFP, for a specific liquid phase concentration, cFP, and solvent ratio, rs, can be determined as nPF =

⎛ tF ⎞ u⎜∫ 0 (c PF − c P(t , Lc)) dt − c PFtvoid⎟ ⎝ t ⎠

(

Lc(1 − ϵref ) 1 −

c PF ρP

)

(12)

where tvoid =

ϵref Lc u

(13)

3.5. Determination of the Column Permeability K. For the subsequent investigation of the two-phase flow behavior, the permeability of the column packing K had to be determined. For this purpose, pure PNT, as well as different mixtures of methanol−water (with different viscosities in the liquid phase, estimated as described in section 2.2) were pumped through the 50 mm × 4.6 mm Zorbax 300SB-C18 column at a constant flow rate of 1.2 mL/min and a temperature of 23 °C. The stable pressure drop (after equilibration) was measured for each mobile phase. For the latter purpose, the HPLC unit was additionally equipped with two pressure sensors, type 21 PY (Keller), located directly before and after the column, and the pressure signal was recorded online with the Labview software. On the basis of Darcy’s law for single-phase flow,12,13 the permeability K could be determined accurately through a linear fit between pressure drop and viscosity data (compare Figure 2). 3.6. Hydrodynamic Two-Phase Flow Behavior. The hydrodynamic two-phase flow behavior through the porous

Figure 2. Pressure drop over the column for different mobile phases (pure PNT and methanol−water mixtures), at a flow rate of 1.2 mL/ min and a temperature of 23 °C. A linear fit provides the permeability K of the column packing. D


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to the wetting phase during imbibition). The breakthrough of the injected phase at the column outlet commonly occurs as a shock-wave transition (see Buckley and Leverett21 and the Supporting Information of this work), especially if the behavior of fractional flows f j (j = 1, 2) with respect to phase saturations Sj can be described by S-shaped relationships, as is the case for most experimental systems. From the wave parts of the experimentally determined flow and pressure profiles, one can derive relationships for the relative permeabilities kr,1 and kr,2 of both phases 1 and 2, respectively, as a function of the phase saturation S1 (S2 = 1 − S1). The procedure is explained in the following. We start from the component mass balance presented in eq 1. With all (convective and adsorbed) phases being incompressible and in thermodynamic equilibrium, and thus ni = constant and ϵ = constant, the component mass balances, considering our specific case of two convective phases, simplify to

medium was determined by displacement experiments, displacing phases which are in thermodynamic equilibrium with each other, i.e., two compositions on the binodal curve connected by a tie-line. Since the activities ai of each component i are identical for both phases and our assumption of the adsorbed phase concentration ni being a function of the liquid phase activity ai, neither adsorption nor desorption occurs during such displacement. Considering eq 5, the porosity ϵ also remains constant. Thus, all the effects observed in the resulting elution profiles are due to the hydrodynamic behavior of the two phases. The two liquid phases in thermodynamic equilibrium were produced by stirring an immiscible composition of 40.4% (w/ w) PNT, 34.6% (w/w) methanol, and 25.0% (w/w) water for 36 h in the EasyMax 102 workstation and separating the resulting phases in thermodynamic equilibrium as described in section 3.3. According to the established UNIQUAC model, the compositions of the resulting phases are PNT/methanol/ water 2.95:55.73:41.32 and 97.11:2.58:0.31 (w/w/w), respectively. The column was equilibrated with the PNT-lean (nonwetting) phase, until reaching a constant plateau in the UV signal at a wavelength of 295 nm (equilibration time ∼20 min). Subsequently, the PNT-rich (wetting) phase was fed to the column by opening the 20 mL injection loop, constraining the feed volume to 16 mL to avoid loss of accuracy due to backmixing effects in the loop. The displacement of the nonwetting phase by the wetting phase, starting from a column completely equilibrated with the nonwetting phase, is called primary imbibition (PI). At the maximum displacement, i.e., when the eluate reaches the composition of the PNT-rich phase, the saturation of the nonwetting phase within the column is reduced to the residual saturation, where the nonwetting phase becomes hydraulically disconnected and remains trapped in the column. Upon closing the injection loop, the wetting (PNT-rich) phase was redisplaced by the nonwetting (PNT-lean) phase. This second displacement is called drainage, and as it starts from a bed including trapped nonwetting phase, it is a secondary drainage cycle (SD). Maximum drainage is achieved when the volume fraction of the wetting phase in the column reaches the irreducible saturation. Drainage was carried out for 26 min (injection volume 31.2 mL), and was followed by another secondary imbibition (SI) and drainage (SD) cycle (starting from the irreducible or residual saturation, respectively), which were performed in the same manner as the previous two cycles. During the entire procedure, the UV signal at a wavelength of 295 nm, as well as the pressure before and after the column (to determine the overall pressure drop Δp), was recorded (see section 3.5). To determine the elution (flow) profiles of PNT, a second HPLC pump (515 HPLC pump, Waters) was attached after the UV detector to automatically dilute the eluate with methanol at a ratio eluate: diluent 1.2:10 (v/v) during imbibition and 1.2:4.5 (v/v) during drainage. The diluted eluate was sampled in fractions with a fraction collector (FC 203B, Gilson) and analyzed by HPLC (samples collected during imbibition were further diluted manually at a ratio sample: methanol 1:50 (v/v) before analysis). For the remainder of this section, the displaced phase will be indicated as phase 2 (corresponding to the wetting phase during drainage and to the nonwetting phase during imbibition), and the injected phase will be indicated as phase 1 (corresponding to the nonwetting phase during drainage and

ϵ

⎞ ⎛ 2 ⎞ ⎛ 2 ∂⎜ ∂ ⎜ ⎟ x S x f ρ + ρ ∑ ij ̅ j ∑ ij ̅ ⎟ = 0; i = 1, ..., NC ∂τ ⎜⎝ j = 1 j ⎟⎠ ∂ξ ⎜⎝ j = 1 j j ⎟⎠ (14)

Furthermore, as the two convective phases are in thermodynamic equilibrium, their compositions and thus xij and ρ̅j remain constant. The component mass balance can therefore be further simplified: 2

⎛ ∂Sj

∑ xijρj̅ ⎜⎜ϵ j=1

⎝ ∂τ

+

∂f j ⎞ ⎟ = 0; i = 1, ..., NC ∂ξ ⎟⎠

(15)

With S2 = 1 − S1 and f 2 = 1 − f1, the equation can be transformed further to ⎛ ∂S ∂f ⎞ (xi1ρ1̅ − xi2ρ2̅ )⎜ϵ 1 + 1 ⎟ = 0; i = 1, ..., NC ∂ξ ⎠ ⎝ ∂τ

(16) 21

We thus end up with the Buckley−Leverett equation, which, for the sake of the further derivation and of the use for the evaluation of experimental data, is given in the dimensional form: ϵ

∂f ∂S1 +u 1 =0 ∂t ∂z

(17)

Note that the same result can be obtained with a very similar derivation, if multiple (more than two) convective phases in thermodynamic equilibrium are present. Obtaining the Buckley−Leverett equation, which represents a “phase balance”, from the component mass balances under the discussed conditions can physically be explained by the fact that since all phases maintain a constant composition and are incompressible, the overall amount (mass and volume) of each phase remains constant throughout the experiment. For the displacement of phase 2 by phase 1, the following initial and feed conditions apply: S1(t , 0) = 1, thus f1 (t , 0) = 1

for t > 0

S1(0, z) = S10

for 0 ≤ z ≤ Lc (18b)

(18a)

where S01 = 0 for primary imbibition/drainage, S01 = Si for secondary imbibition, and S01 = Sr for secondary drainage. The solution of the Buckley−Leverett equation by the method of E


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Lc) can be calculated based on eq 3b. From eq 26, one can obtain the average saturation over the column Sav,1(t) at any time t, based on which one can next determine S1(t, Lc), applying eq 24. In order to determine relative permeabilities kr,1(t, Lc) corresponding to the saturations S1(t, Lc) thus obtained, we consider the extended Darcy’s law (eq 6) and its sum over all convective phases (eq 7), which provides a relationship between pressure drop and superficial velocity:

characteristics for specific initial and feed conditions has been multiply discussed in the literature,8,21−23 but for the sake of completeness and understanding, it is illustrated in the Supporting Information. The characteristic propagation velocity for a state S1 in the nondimensionless form is given as ⎛ dz ⎞ ⎜ ⎟ ⎝ dt ⎠

= S1

u df1 u = f1′ (S1) ϵ dS1 ϵ

(19)

with u = constant and ϵ = constant and f′1 being independent of space and time, the propagation velocity is constant for a specific state S1, such that u z = f1′ (S1) t (20) ϵ Derivation of eq 20 with respect to S1 yields ut dz = f ″ (S1) dS1 ϵ 1

⎛ NP k ⎞ ∂p r, n ⎟⎟K u = −⎜⎜∑ η ⎝ n = 1 n ⎠ ∂z

For the sake of simplicity, in the following equations we express the sum in eq 27 as Y: NP

Y (S1) =

(21)

∫0

Lc

(22)

Δp(t ) =

S1(t , z) dz

(23)

=

ut ϵLc

S1(t , Lc)

∫S (t ,0)

S1 f1″ (S1) dS1 S1(t , Lc)

∫S (t ,0) 1

⎞ f1′ (S1) dS1⎟ ⎠

=

ut = S1(t , Lc) + (1 − f1 (t , Lc)) ϵLc

Thus, the saturation at the column outlet S1(t, Lc) can be determined from the average saturation Sav,1(t). In turn, the average saturation can be obtained from the fractional flow profile at the column outlet f1(t, Lc) by integrating eq 17 over space and time:

∫0 −

Lc

=

∫t

t 0

(f1 (t , 0) − f1 (t , Lc)) dt = 0

= S10 +

1 tvoid

∫t

1 tvoid

∫t

(1 − f1 (t , Lc)) dt

t 0

1 dz Y (S1)

(29)

∫0

u2 ϵK

dY −1(S1) ∂S1 dz dS1 ∂t

Lc

S1(t , Lc)

∫S (t ,0) 1

Y ′(S1) Y 2(S1)

f1′ (S1) dS1

uLc 1 u + Kt Y (t , Lc) Kt

∫0

uL Δp(t ) 1 − c t Kt Y (t , Lc)

Lc

1 dz Y (S1) (30)

(31)

Finally, a kr,i−Si relationship can be established between the values for S1(t, Lc), derived from experimental fractional flow profiles through mass balance considerations and the values for kr,1(t, Lc) determined from measured pressure profiles by implying the extended Darcy’s law. Note that the presented approach can only be applied for the wave part of the elution profile, since it is only in that part of the profile that the states S1 travel at their characteristic propagation velocity defined in eq 19. This propagation velocity is used in eqs 24 and 29−31 and thus constrains the validity of the corresponding equations to wave transitions. As a

t 0

Lc

kr ,1(t , Lc) = f1 η1 Y (t , Lc)

(25)

Using eqs 18 and 23 in eq 25 yields Sav,1(t ) = Sav,1(t = t 0) +

(28)

To describe the behavior of the pressure drop mathematically, a rational function Δp(t) = (h1t3 + h2t2 + h3t + h4)/(t + q) was fitted to the experimental pressure data. Equation 30 can then be used to obtain Y(t, Lc). Finally, the relative permeabilities (at a saturation S1(t, Lc)) can be determined from eq 8 as

(S1(t , z) − S1(0, z)) dz

u ϵLc

∫0

=−

(24)

1 Lc

u K

dΔp(t ) u = dt K

1

ut ⎛ S (t , L ) ⎜[S1 f1′ (S1)]S11(t ,0)c − ϵLc ⎝

ηn

Differentiation with respect to t and application of eqs 20−22, as well as the Leibnitz rule and partial integration with respect to S1 yields the following:24

Using eqs 18 and 21 in eq 23 yields the following expression:22 Sav,1(t ) =

k r, n

where Y is a function of S1 only, since kr,n of each phase n is a function of S1, and the dynamic viscosity ηn of phase n is constant as the phase composition of the liquid phases, which are in thermodynamic equilibrium, do not change during the displacement. Equation 27 can be integrated to obtain the overall pressure drop Δp:

In a next step, we define the average saturation over the column: 1 Sav,1(t ) = Lc

∑ n=1

Furthermore, rearrangement of eq 20 and derivation with respect to t yields f ′ (S1) ∂S1 ϵz =− 2 =− 1 ∂t t f1″ (S1) ut f1″ (S1)

(27)

(1 − f1 (t , Lc)) dt (26)

Knowing the phase compositions of both phases, as well as the concentration of PNT FP(t, Lc) in the eluate (measured concentrations of the eluate fractions), the fractional flow f1(t, F


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Industrial & Engineering Chemistry Research consequence, relative permeabilities can only be determined for saturations S1(t, Lc) reached during the wave part of the breakthrough profile. 3.7. Dynamic Column Experiments Involving Adsorption and Two-Phase Flow. Breakthrough experiments with partially miscible initial (water−methanol mixtures) and feed (PNT−methanol mixtures) states, which were not in thermodynamic equilibrium, were carried out to validate the model assumptions and the implemented relationships. For this purpose, the (50 × 4.6 mm) Zorbax 300SB-C18 column was equilibrated with the initial state, and 4 mL of the feed state were injected through the 5 mL injection loop. When closing the injection loop, the feed state in the column was redisplaced by the initial state. With this procedure, the entire chromatographic cycle, i.e., adsorption and desorption step, was considered. As in the previous set of experiments (see section 3.6), an additional pump was connected after the UV detector to automatically dilute the eluate with methanol at a ratio eluate: diluent 1.2:10 (v/v). The diluted eluate was gathered in fractions and the concentrations of PNT in the fractions were analyzed offline by HPLC to obtain the flow profile FP. Fractions with high PNT concentrations were further diluted manually at a sample/methanol ratio of 1:50 (v/v) before analysis.

4. SYSTEM CHARACTERIZATION In this section, we want to evaluate the validity of the model assumptions for the experimental system, and establish relationships describing the thermodynamic equilibria between the liquid phases and between liquid and adsorbed phases. Finally, we assess and describe the hydrodynamic behavior of the two convective phases. 4.1. Additivity of Volumes. To validate the assumption of additivity of volumes, densities of binary mixtures water− methanol and methanol−PNT were analyzed. Binary mixtures of water and PNT were not investigated due to very low solubilities of one component in the other. In order to fulfill the assumption, mixture and pure component densities should behave according to the following relationship (which is obtained from eq 4 upon rearrangement):

Figure 3. Densities of mixtures of the binary systems (a) water− methanol and (b) methanol−PNT. Experimental data was taken from the literature25 for the water−methanol system and was determined for the methanol−PNT system in this study. Model predictions are based on eq 32.

(and from equal specific volumes in liquid and adsorbed phase, which is the basis of eq 5) entail a change in the superficial velocity u. Theoretically, an equilibrium theory solution accounting for varying velocities is more complex but still possible.9,10,26 Experimentally, an accurate determination of the change in porosity (due to adsorption/desorption) and in superficial velocity (due to excess volume upon mixing or different specific volumes in fluid and adsorbed phases) is extremely challenging. In the current system, we expect possible changes in the fluid velocity to be negligible in the face of dominant adsorption and fluiddynamic effects. 4.2. Thermodynamic Liquid−Liquid equilibrium. The thermodynamic liquid−liquid equilibria of the ternary system PNT−methanol−water was established experimentally as described in section 3.3, and the resulting data is presented in Table 1 and in Figure 4 with green data points and tie-lines. With reference to Figure 4, center points located in the immiscible region indicate initial compositions at the start of the experiments, while points at the ends (located on the binodal curve) represent the composition of phases formed at thermodynamic equilibrium, which are connected by (green) tie-lines. LLE experiments at 7 out of 9 different initial compositions were repeated twice or thrice, and the overlapping data points (cannot be discerned from Figure 4), as well as the initial compositions being located on the tie-lines, demonstrate good reproducibility and experimental accuracy. In order to describe the LLE behavior mathematically, a UNIQUAC model27 was used. This model allows the

NC

ρ̅ =

∑ ϕρ i i i=1

(32)

where ϕi is the volume fraction of component i (as a pure species, i.e., before mixing). Note that this rearrangement of eq 4 was done in order to obtain a direct linear relationship between ρ̅ and ϕi, with the pure component densities resulting at ϕi = 0 and 1. In Figure 3, model predictions based on eq 32 are compared to experimental data, determined specifically for this study at 23 °C for the system methanol−PNT and taken from the literature for the system water−methanol at 20 °C.25 While the assumed ideal behavior can be confirmed for the system methanol−PNT, nonidealities are observed for the water− methanol system, which result in prediction errors of up to 3.4% when assuming volume additivity. Although the analysis does not include data of ternary mixtures, judging from the binary mixture data, only minor errors are expected from the assumption of volume additivity. It is worth noting that the relaxation of the assumption of volume additivity involves considerable theoretical and experimental complications. Deviations from this assumption G


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Industrial & Engineering Chemistry Research Table 1. Experimental and Predicted (Italics and Indicated by model) LLE Dataa overall condition 1 1 1 2 2 2 3 3 3 3 4 4 4 5 5 6 6 6 7 7 7 8 8 8 9 9 a

model

model

model

model model

model

model

model model

PNT-lean, L

PNT-rich, R

P

M

W

P

M

W

P

M

W

50.03 49.93 50.03 48.98 48.97 48.98 47.94 47.94 47.90 47.94 46.95 46.91 46.95 47.07 47.07 47.51 47.54 47.51 46.02 46.00 46.02 46.04 46.09 46.04 61.71 61.71

0.00 0.00 0.00 7.99 8.00 7.99 15.99 15.99 15.97 15.99 23.98 23.96 23.98 27.95 27.95 31.02 31.00 31.02 37.97 38.00 37.97 43.95 43.90 43.95 33.83 33.83

49.97 50.07 49.97 43.03 43.03 43.03 36.07 36.07 36.13 36.07 29.07 29.13 29.07 24.98 24.98 21.47 21.46 21.47 16.01 16.00 16.01 10.01 10.01 10.01 4.46 4.46

0.08 0.12 0.03 0.18 0.16 0.11 0.25 0.25 0.26 0.36 0.90 0.88 1.09 1.69 1.94 3.03 2.92 3.15 8.02 7.83 7.78 22.12 21.68 20.73 41.09 37.37

0.00 0.00 0.00 15.05 14.83 15.35 29.73 29.96 29.00 30.09 44.07 44.14 44.04 50.73 51.04 56.10 56.63 56.42 63.78 64.07 64.05 63.03 63.28 63.83 50.91 54.08

99.92 99.88 99.97 84.77 85.02 84.54 70.01 69.79 70.74 69.55 55.03 54.98 54.87 47.58 47.02 40.86 40.45 40.42 28.20 28.10 28.17 14.85 15.04 15.44 7.99 8.55

99.87 99.86 99.87 99.43 99.49 99.44 98.35 99.01 98.79 98.92 98.20 98.19 98.20 97.47 97.65 97.08 96.93 97.02 95.03 95.24 94.97 89.48 89.82 88.72 78.04 77.76

0.00 0.00 0.00 0.42 0.36 0.40 1.45 0.80 1.01 0.89 1.57 1.58 1.56 2.26 2.08 2.62 2.77 2.67 4.53 4.33 4.59 9.65 9.33 10.42 19.99 20.48

0.13 0.14 0.13 0.16 0.15 0.16 0.21 0.20 0.20 0.19 0.23 0.23 0.24 0.26 0.27 0.30 0.30 0.31 0.44 0.43 0.44 0.87 0.85 0.86 1.97 1.76

The PNT-rich and lean phases (R and L) form at thermodynamic equilibrium from the initial compositions denoted as “overall”.

phase split occurs, and liquid phase activities ai can be determined explicitly from the underlying UNIQUAC equations. In the case of phase instability, i.e., location in the immiscible region, phases in thermodynamic equilibrium are determined by a flash calculation based on a successive substitution iteration algorithm and Newton’s method.29 Combinatorial parameters of the UNIQUAC model, namely, the parameters qi, qi′, and ri, are summarized in Table 2 and Table 2. Combinatorial Parameters of PNT (P), Methanol (M) and Water (W) in the UNIQUAC Model comp. i

ri

qi

q′i

P M W

4.8411 1.4311 0.9200

3.7480 1.4320 1.4000

3.7480 0.9600 1.0000

were taken from the literature for water and methanol.30 For PNT, the parameters qP and q′P are identical, and qP and rP were estimated from group volume and area parameters Rk and Qk, as suggested for the UNIFAC model:31

Figure 4. Ternary diagram (mass franctions) of the system PNT− methanol−water. Green data points and tie-lines: experimental data, center green points indicate the initial composition, points at the ends of the tie-lines denote compositions of phases in equilibrium. Black data points and tie-lines: preditions by the UNIQUAC model. Red line and empty circle: Binodal curve and plait point, predicted by the UNIQUAC model.

rP =

∑ νkPR k ; qP = ∑ νkPQ k k

k

(33)

νPk

where is the number of chemical groups of type k in PNT. Group parameters Qk and Rk for PNT were taken from Fredenslund et al.,31 apart from the parameters for group “AC”, which were taken from Larsen et al.32 Residual binary interaction parameters were fitted to the experimental data by minimizing the following objective function J (using the built-in Matlab algorithm lsqnonlin):

determination of liquid phase activities ai, and, for compositions in the immiscible region, the calculation of phase split and compositions of the phases in equilibrium. Phase stability of a composition is tested based on the tangent plane criterion of the Gibbs energy of mixing as described by Michelsen.28 In the case of phase stability (location in the miscible region), no H


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Industrial & Engineering Chemistry Research NC Ncond

J=

∑∑ i=1 j=1

1 Nj ,rep

Nj ,rep

∑ k=1

⎛⎛ R L ⎞2 ⎞ R ⎞2 ⎛ L ⎜⎜ xijk − xij̃ ⎟ + ⎜ xijk − xij̃ ⎟ ⎟ ⎜ xL ⎟ ⎟ ⎜⎜ x R ⎟ ⎠⎠ ⎝ ⎠ ijk ijk ⎝⎝

Predicted compositions of the phases in equilibrium, connected by tie-lines, are plotted in black in Figure 4. The fact that experimental and predicted compositions, as well as the tie-lines, overlap nicely, confirms a good quantitative description of the liquid−liquid equilibria by the established UNIQUAC model. Predicted and experimental values are also compared in detail in Table 1. The red line in Figure 4 represents the binodal curve (determined by 5000 flash calculations for compositions distributed between the base tie-line (xM = 0) and the plait point). The open circle marks the plait point, which is calculated by enforcing the proper condition on the partial derivatives of the Gibbs free energy of mixing.35 From the ternary diagram, it can be noted that both binary systems water−methanol and PNT−methanol are completely miscible, while the binary system PNT−water is almost immiscible, and the miscibility between the two latter components increases with the methanol content. 4.3. Thermodynamic Equilibrium between Liquid Phases and Adsorbed Phase. The adsorption behavior was assessed experimentally by frontal analysis at five different solvent ratios ranging from 0.4 to 0.7, and with feed states located in the soluble region, distributed between xP = 0 and the solubility limit of PNT at the relevant solvent ratio. Operating conditions were constrained to the soluble region in order to exclude any hydrodynamic effects due to two-phase flow, which at this stage could not be assessed. Accordingly, any effects observed in elution profiles could be attributed to the

(34)

where NC, Ncond, and Nj,rep are the number of components, experimental conditions, and experimental repetitions per condition, respectively. Variables xR/L and x̃R/L are experijk ijk imentally determined and predicted mass fractions, respectively, of the PNT-rich phase (superscript R) and the PNT-lean phase (superscript L). The fitted parameters are reported in Table 3, with 95% confidence intervals determined based on the Maximum Likelihood Estimate,33,34 as outlined in the Supporting Information. Table 3. Residual Parameters of the UNIQUAC Model (Fitted) For the Ternary System Phenetole (P)−Methanol (M)−Water (W)a components

Δuij/R [K]

P−M M−P P−W W−P M−W W−M

1.097 (±0.117) × 103 −76.94 (±2.36) 1.165 (±0.025) 270.7 (±30.0) −83.02 (±112.66) 201.0 (±185.6)

a

Confidence intervals estimated as described in the Supporting Information.

Figure 5. Breakthrough curves at the solvent ratios (a) rs = 0.40, (b) rs = 0.50, (c) rs = 0.60, and (d) rs = 0.70. Blue curves: Experimental concentration profiles, transformed from the UV profile by calibration functions as described in section 3.4. Red curves: Equilibrium theory predictions based on the model described in section 2.1 with Fi = Ci (single-phase region) and assuming the isotherm determined in section 4.3. I


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When converting the liquid phase concentrations cP to liquid phase activities aP, by applying the UNIQUAC model established above, the adsorption data falls onto one line, regardless of the solvent ratio (see Figure 6b). Horizontal bars through the data points indicate the standard deviations σaP of the predicted activities aP, which were estimated by error propagation of the uncertainties σp of the fitted UNIQUAC parameters p, according to eq 35:36

adsorption behavior. The experimental procedure is described in section 3.4. Recorded breakthrough profiles are provided for four different solvent ratios in Figure 5. The adsorption data derived from the recorded elution profiles is presented as a function of the liquid phase concentrations cP of the feed state in Figure 6a. It can be observed that considering the adsorbed

⎛ ∂a ⎞2 ∑ ⎜⎜ P σpi⎟⎟ ∂pi ⎠ i=1 ⎝ Np

σaP =

(35)

It should be pointed out that experimental uncertainties of the LLE determination were considered to have a minor contribution to the overall uncertainty of the calculated activities, as compared to the parameter uncertainties. From the adsorption data reported in Figure 6, it is concluded that the adsorption behavior can be expressed as a function of the liquid phase activity aP. An anti-Langmuir isotherm of the form nP =

HPaP 1 − KPaP

(36)

was fitted to the experimental data, with fitted isotherm parameters HP = 41.52 g/L and KP = 0.8621. Predicted profiles, based on the equilibrium theory model (eq 1 with Fi = Ci) including the established adsorption isotherm, are compared to the experimental breakthrough curves in Figure 5. Elution profiles at solvent ratios rs = 0.5−0.7 are described very accurately, and major discrepancies can only be observed for the profiles obtained at rs = 0.4. Considering the fact that the description depends on two parameters only and is additionally based on the previously established thermodynamic model which contributes further uncertainties, the agreement between experiments and predictions is judged to be satisfactory. On the basis of the assumption of thermodynamic equilibrium between the liquid phases, i.e., equal liquid phase activities, and between liquid and adsorbed phases, the established adsorption isotherm can also be applied in the presence of multiple convective phases, i.e., in the immiscible region. 4.4. Two-Phase Flow Behavior. The hydrodynamic behavior of the two convective phases was investigated by a displacement experiment with two liquid phases in thermodynamic equilibrium, i.e., the two ends of the same tie-line, located on the binodal curve. Since the two displaced phases have the same liquid phase activities ai of all components i, the adsorbed phase concentration nP, depending on aP, remains constant. As a consequence, no adsorption or desorption, and no changes in porosity ϵ occur throughout the displacement. Thus, all effects observed in the resulting elution profiles can be attributed to the hydrodynamic behavior of the two convective phases. For details on the experimental procedure and the data evaluation, see section 3.6. The experimental pressure and flow profiles over the entire duration of the experiment (equilibration and two imbibition− drainage cycles) are presented in Figure 7. Furthermore, pressure profiles and the evolution of the average saturation of the PNT-rich phase SRav over time are overlaid for both cycles in Figure 8 (imbibition) and in Figure 9 (drainage). Pressure profiles during the secondary drainage cycles (SD(i) and SD(ii)) are reproducible and slowly approach a stable

Figure 6. Adsorption data, determined experimentally by frontal analysis in the soluble region and at different solvent ratios rs. (a) Plotted over the liquid phase concentration cP; (b) plotted over the liquid phase activity aP. The solid black line illustrates the fitted adsorption isotherm (eq 36) with HP = 41.52 g/L and KP = 0.8621. Uncertainties of the adsorption data over activities were estimated by propagation of the confidence intervals of the UNIQUAC parameters.

phase concentration nP as a function of the liquid phase concentrations cP the adsorption behavior changes with the solvent ratio. This is not surprising, as the solubility of PNT also changes considerably with the solvent ratio (compare Figure 4). J


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Figure 7. Flow (blue) and pressure (orange) profile over the entire duration of the displacement experiment, displacing two phases in thermodynamic equilibrium. Different steps are indicated by vertical black lines and the respective abbreviation: EQ: equilibration, PI: primary imbibition, SI: secondary imbibition, SD: secondary drainage ((i) and (ii) denote the first and the second secondary drainage cycles).

Figure 9. Profiles during secondary drainage cycles: (a) average saturation SRav, determined by eq 23; (b) pressure drop Δp.

and secondary cycles (Figure 8b). Furthermore, stable pressure levels are reached neither during primary nor during secondary imbibition, which indicates that the displacement (until reaching the residual saturation Sr of the nonwetting, PNTlean phase) was not completed. An instantaneous decrease in pressure drop can be noted at the start of the imbibition steps, and is ascribed to the opening of the injection loop: the HPLC pump adjusts only gradually to the additional resistance in the system caused by the injection loop, which results in a temporary decrease in flow rate and thus in pressure drop. Thus, this pressure drop is a systematic error and does not result from the investigated hydrodynamic behavior. Neither the pressure drop nor the associated decrease in flow rate is accounted for in the model. Experimentally determined flow profiles (Figure 7) exhibit a greater scattering during imbibition than during drainage. The lower accuracy of the data points during imbibition, determined by HPLC analysis, is due to the additional manual dilution of HPLC samples, which is necessary due to the high concentrations of PNT in the eluate fractions during imbibition. Imbibition profiles exhibit a shock transition reaching directly to high values of FP. A subsequent wave transition is not recognizable, since in the case that a wave transition exists it would be very flat and because the data is scattered during imbibition. In turn, during drainage, the wave part is more distinct and spans a greater range of FP values. The discussed issues in the flow profiles (data scattering, low data accuracy, and short wave part during imbibition) are reflected in the average saturation profiles obtained through eq

Figure 8. Profiles during primary and secondary imbibition cycles: (a) average saturation SRav, determined by eq 23; (b) pressure drop Δp.

pressure level after 15 to 20 min, indicating that the irreducible saturation Si of the wetting, PNT-rich phase is reached (Figure 9b). In contrast, pressure profiles during primary and secondary imbibition differ considerably, suggesting a considerable hysteresis between the displacement behavior during primary K


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estimated relative permeability function, due to the multiple experimental issues and various underlying physical assumptions, only provides a rough description of the actual hydrodynamic behavior. Hysteretic effects, both between primary and secondary cycles and between drainage and imbibition steps, were neglected in the description. Furthermore, a very likely dependence of the hydrodynamic behavior on the interfacial tension between the two phases, i.e., a dependence on the location of the tie-line with respect to the plait point, has not been taken into account. In order to consider the latter effect, similar displacement experiments as discussed in this section, but along different tie-lines, would have been necessary. Considering the lack of accuracy and the considerable effort required by this type of experiments, it was decided to refrain from further experimental investigation while keeping the limitations of the established description in mind.

23 and illustrated in Figures 8a and 9a. Saturation profiles during imbibition quickly reach a rather constant level close to SRav = 1, and due to the scattering and the great span of the shock transition, a wave part is not recognizable. From the profiles, a residual saturation of the nonwetting phase Sr ≈ 0 was inferred. During drainage, a wave transition can be clearly observed, starting at an average saturation SRav = 0.5 down to the plateau value, which for the two cycles ranges between SRav = 0.3−0.4. From the plateau values, an irreducible saturation of Si ≈ 0.32 was estimated. From the saturation and pressure profiles presented, relative permeability data was estimated as described in section 3.6. The resulting relative permeability over saturation data is presented in Figure 10. Data estimated from the drainage curves are

5. COMPARISON OF MODEL PREDICTIONS AND DYNAMIC COLUMN EXPERIMENTS The independently established relationships describing the liquid−liquid equilibrium, the adsorption behavior (thermodynamic equilibrium between liquid and adsorbed phases) and the hydrodynamic behavior were implemented in the model presented in section 2.1, and the model was solved for specific initial and feed conditions, applying the method of characteristics.5,6 It should be stressed that in this context the model was used in a fully predictive way, i.e., no adjustment of neither model parameters nor implemented relationships. The derived elution (flow) profiles were then compared to experimental flow profiles of PNT, determined during dynamic experiments at the same initial and feed conditions (as described in section 3.7). We investigate 8 different conditions, arising from the combination of 2 different initial states (A1 and A2, both mixtures of methanol and water) with 4 different feed states (B1, B2, B3, and B4, all mixtures of PNT and methanol), which are visualized in the ternary phase diagram shown in Figure 11. The resulting concentration (predicted) and flow (experimental and predicted) profiles are presented in Figure 12 for all conditions with initial state A1 and in Figure 13 for all conditions with initial state A2. In contrast to the previous displacement experiments discussed in section 4.4, the initial and feed states of the

Figure 10. Relative permeability data determined from the drainage (circles) and imbibition (crosses) data. Darker blue and red circles were determined from experimental data of the first drainage cycle (SD(i)), lighter symbols were obtained by data analysis of the second drainage cycle (SD(ii)). Model descriptions based on eqs 9a9 and 10 are illustrated by continuous lines.

presented as open circles, with the maximum relative permeability kLr,max at SR = 0.32 (Si = 0.32) determined from the pressure plateau reached during both secondary drainage steps, and applying eq 6 with f L = 1. Due to the multiple experimental issues during imbibition (very short wave part, combined with data scattering, as well as the discussed systematic error in the pressure profile), no reasonable relative permeability data was expected applying the method described in section 3.6 to the imbibition data. A rough estimate was only obtained for the maximum relative permeability kRr,max at SR = 1 (Sr = 0), considering the final pressure drop during the secondary imbibition cycle in eq 6 with f R = 1. This estimated value is indicated as a cross in Figure 10. Finally, the parameters λR/L were adjusted in such a way as to achieve a reasonable description of the experimental data, i.e., λR = 3 and λL = 2. The estimated parameter values, required in eqs 9 and 10, are summarized in Table 4. We want to underline that the Table 4. Parameters of the Relative Permeability Functions for the Experimental System parameters values

kLr,max 0.94

kRr,max 0.77

Sr 0.00

Si 0.32

λL 2.00

Figure 11. Initial (Ai) and feed (Bi) states (mass fractions), investigated in the validation experiments. Combinations of initial and feed states are connected by a continuous line.

λR 3.00 L


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Figure 12. Concentration (predicted, left column) and flow (experimental and predicted, right column) profiles of PNT for conditions with initial state A1 and feed state (a) B1, (b) B2, (c) B3, and (d) B4. Entire chromatographic cycles, i.e., adsorption and desorption steps (separated by a dashed line), are considered. Data points in different shades of blue (darker and lighter blue) indicate experimental replicates. Red, green, and black lines present equilibrium theory predictions, based on the three different models ET, ET (hydro), and ET (ads), which are explained in the text.

validation experiments are not in thermodynamic equilibrium, and they are all located in the soluble region, far from the

binodal curve. As a consequence, the liquid phase activities of PNT aP of initial and feed states are not identical, resulting in M


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Figure 13. As in Figure 12, but for initial state A2.

adsorption and desorption, which brings along also a change in porosity. Apart from conditions with feed state B4, all other conditions lead to the occurrence of states within the immiscible region,11 thus resulting in two-phase flow. Owing to the fact that initial and feed states are not in thermodynamic

equilibrium and are not located on the binodal curve, no phase becomes permanently trapped within the column, as it was the case in section 4.4. In turn, in the case where a state becomes temporarily trapped, the trapped phase is slowly consumed by a N


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adsorption, no phase becomes temporarily trapped when considering different velocities, since the residual saturation Sr = 0. In contrast, during desorption, i.e., displacing a PNT-rich (wetting) phase by a PNT-lean (nonwetting) phase, the ratio of wetting phase temporarily trapped is very high, since Si = 0.32. This portion of trapped phase is removed very slowly by constant adjustment of the thermodynamic equilibria between the liquid phases; this explains the long “tailing” in the desorption steps when accounting for different velocities. In turn, when assuming equal velocities, both liquid phases always remain convective, and the PNT-rich phase can thus be eluted much faster. Finally, let us compare the model predictions with the experimental profiles. We have no possibilities to experimentally determine compositions at a location inside the column (concentration profiles Ci at this location), but we can only measure the composition of the eluate at the column outlet (flow profiles Fi at position Lc). Here, we focused on the determination of the flow profiles of PNT. Experimental replicates of several conditions show a good reproducibility (see overlaid elution profiles in darker and lighter blue in Figures 12 and 13). Models accounting for adsorption accurately describe the conditions with feed state B4, thus providing a good confidence in the established relationship describing the adsorption behavior (no two-phase flow under these conditions). Conditions with feed states B1 and B2 are described quantitatively when accounting for both adsorption and different velocities of the convective phases. The last part of the desorption profile (wave-shock transition) is identical for both feed states with the same initial state A1 or A2, since it maps on the same path in the hodograph plane.11 Elution profiles for conditions with feed state B3 clearly differ from the profiles for feed states B1 and B2: The last part of the desorption profile is considerably shorter and does not overlap with the elution profiles for the other two feed states, both experimentally and in simulations using the “ET” model. According to the equilibrium theory solution, this last part corresponds to an intermediate state (plateau), followed by a shock transition (which differs from the wave-shock transition for feed states B1 and B2). However, these conditions can only be described qualitatively by the “ET” model. In fact, the experimental profiles exhibit a shorter desorption profile, located between the predictions assuming different velocities, and the one assuming equal velocities. This quantitative mismatch is due to the established relative permeability function, featuring constant parameters and thus neglecting an impact of the interfacial tension between the liquid phases on the hydrodynamic behavior. The solution path for conditions with feed state B3 in the hodograph plane passes through the immiscible region, very close to the plait point,11 where the two convective phases have similar compositions (low interfacial tension) and the flow behavior should thus approach the single phase flow behavior (equal velocities). This is not taken into account by the established relative permeability function. The experimental profile is therefore located between the predictions assuming different velocities (but independent of the interfacial tension) and the prediction assuming equal velocities. From the comparison of the three model predictions with the experimental profiles, one can conclude that in the immiscible region and far from the plait point the impact of the hydrodynamic two-phase flow behavior on the elution

continuous adjustment of the thermodynamic equilibrium between the liquid phases. Let us first comment on the predicted profiles presented in Figures 12 and 13, which are based on three different models. The first (red lines, designated as “ET”) is the model discussed in this contribution, accounting for both adsorption and different velocities of the multiple convective phases and implementing all the relationships established in the previous sections. The other two models, designated as “ET (ads)” and “ET (hydro)”, are simplifications of the first model. The simplification “ET (ads)” (black lines) accounts for adsorption effects (via the established adsorption isotherm) but assumes equal interstitial velocities for all convective phases, i.e., it neglects hydrodynamic effects ( f j = Sj and Fi = Ci). This model represents the equilibrium theory model as it is commonly used to describe chromatographic processes without liquid−liquid phase separation. The second simplification “ET (hydro)” (green lines) accounts for different velocities (via the established relative permeability functions), but it neglects adsorption effects (i.e., nP = 0). This type of model is commonly used to describe two-phase flows in porous natural reservoirs, where interactions with the rock matrix are ignored.8 The equilibrium theory solution for model “ET (hydro)” is derived by Orr.8 For the models “ET (ads)” and “ET”, a detailed derivation of the solutions (specifically for all conditions with initial state A1) is provided in the companion paper of this series.11 Solutions for conditions with initial state A2 can be derived likewise as for the initial state A1. When assuming equal velocities (model “ET (ads)”), concentration (Ci) and flow (Fi) profiles are identical, whereas they differ considerably in the case of different velocities (models “ET (hydro)” and “ET”). While the overall liquid concentration Ci corresponds to the amount of component i per overall void volume at a certain position in the column, the overall fractional flow Fi describes the amount of component i moving per overall volume of flow (over all the fluid phases). Hence, the concentration profiles calculated at location Lc in Figures 12 and 13, left column, describe the concentration of PNT in the void space at the end of the column, whereas the flow profiles at position Lc (right column in Figures 12 and 13) correspond to the concentration of PNT in the (single- or twophase) eluate. Ci and Fi; hence, concentration and flow profiles, are only identical in the case of a single-phase flow, or if multiple fluid phases move with identical interstitial velocities. Since conditions with feed state B4 map on paths in the hodograph plane which do not pass through the immiscible region,11 there is no two-phase flow occurring in the corresponding solution. As a consequence, the two models “ET (ads)” and “ET” yield identical elution profiles. In contrast, the model “ET (hydro)”, neglecting adsorption effects, predicts a simple plug flow displacement of the initial state by the feed state, occurring at the dead time both during the displacement and during the redisplacement step. At all other conditions (feed states B1−B3), predictions of all three models differ. However, similar elution profiles are produced with the two models accounting for different velocities, while profiles based on the model assuming equal velocities exhibit considerable differences. This indicates that at least under the established adsorption and flow conditions hydrodynamic effects dominate over adsorption effects in the immiscible region. Discrepancies are more distinct during desorption than during adsorption. When displacing a PNTlean phase (nonwetting) by a PNT-rich phase (wetting) during O


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immiscible region located far from the plait point. As the plait point is approached during a dynamic column experiment, the convective phases feature similar properties, e.g., similar compositions, small interfacial tension between the convective phases; hence, the dominance of hydrodynamic effects decreases. Strong hydrodynamic effects, i.e., big differences in the velocities of the convective phases, correspond to components being strongly retained in the column and being eluted very slowly, as it is the case for the conditions A1−B1 and A1−B2 in section 5. This effect is in general undesired in liquid chromatography. It can thus be concluded that strong hydrodynamic effects have a negative impact on the chromatographic separation. Operation under two-phase flow conditions hence involves a trade-off between high productivities resulting from high liquid phase concentrations (which are no longer limited to the miscible region) and between the discussed two-phase flow behavior, resulting in slow elution and possibly high dilution of the target product. Accepting a two-phase flow can be the preferred option if the increase in productivity due to higher feed concentrations outweighs the disadvantages correlated to the two-phase flow behavior. This might be the case if hydrodynamic effects are low to moderate, i.e., low interfacial tension between the convective phases, or if the solubility of the different components is extremely low, implying severe constraints to productivity under single-phase conditions. The dynamic column experiments considered in this study (section 5) create a two-phase flow by injecting a feed state, which is immiscible with the initial state. Another interesting phenomenon to be studied is a spontaneous phase split within the column, due to a considerable enrichment of one or more components, as it was observed in a previous study.1,2 Such enrichment can only occur due to an interaction of multiple (two or more) adsorbing components or due to a chemical reaction with immiscible products occurring in the column. In principle, the presented model could also account for a spontaneous phase split, with the underlying assumption that the phase split is not kinetically hindered. As phase separations are prone to kinetic limitations, the physical behavior and mathematical description of this phase split requires further investigation. Finally, the current investigation is limited to a ternary system with only one adsorbing component (which does not feature the formation of a spontaneous phase split, as discussed above). Since chromatographic processes commonly aim at the separation of different adsorbing components, it is crucial to extend this study to binary and multicomponent systems. The equilibrium theory model established in this work is able to account for a random number of components. Further, the established relationships describing physical properties, such as isotherms based on liquid phase activities or fractional flow functions based on Brooks−Corey correlations, form a solid basis, which however needs to be extended to account for a higher number of components. The presence of multiple adsorbing components results in competition or cooperation for adsorption sites, which has to be taken into account by binary and multicomponent adsorption concepts.39 Indeed, the adsorption behavior of a multicomponent system under thermodynamically nonideal conditions has been studied and described very recently.40 Simultaneously, the thermodynamic behavior of the fluid phases increases in complexity, allowing the presence of more than two phases, which also complicates

profiles dominates over the impact of the adsorption behavior. Approaching the plait point, properties of the two convective phases become similar, the interfacial tension between the phases decreases, and an equal velocity behavior is approached. At the same time, the impact of the adsorption effects on the elution profile increases. In the miscible region, with only one convective phase, hydrodynamic effects are absent and the adsorption behavior is the dominating factor.

6. CONCLUSIONS In this study, we have investigated two-phase flow in liquid chromatography, both experimentally and theoretically. In a first step, an equilibrium theory model, combining adsorption and two-phase flow, was established and solved.11 Underlying assumptions of this model (such as additivity of volumes) were verified for the experimental system PNT−methanol−water on the adsorbent Zorbax 300SB-C18, and relationships describing the thermodynamic equilibrium between the liquid phases, the adsorption equilibrium, and the two-phase flow in the column were established through independent experiments. The relationships were implemented in the equilibrium theory model, which in a next step was used in a fully predictive way to simulate concentration and flow profiles for specific initial and feed conditions. The chosen initial and feed conditions were not in thermodynamic equilibrium, and thus resulted in adsorption or desorption during the elution, and 6 out of 8 conditions led to two-phase flow. The resulting simulated profiles were compared to profiles obtained from dynamic column experiments at the same initial and feed conditions, and a quantitative agreement was achieved for a majority of the conditions. For the sake of completeness, we would like to point out that one key assumption was not investigated through an independent experimental campaign, namely, that of negligible kinetic limitations (dispersive effects and mass transfer limitations). Those effects would result in band broadening in the experimental elution profiles.37,38 With experimental profiles in section 5 exhibiting very sharp transitions and being in quantitative agreement with the established equilibrium theory model, we conclude that kinetic effects play a negligible role for the investigated system. This finding is nongeneric, and kinetic effects should indeed be considered when characterizing different systems. The good agreement between simulation results and experimental data confirms the validity of the mathematical model developed, in terms of both physical assumptions and simplifications made and of specific mathematical relationships utilized. This demonstrates a good understanding of both the experimental system and the implications of multiple convective phases in liquid chromatography. Following, we summarize the most important (experimental and theoretical) findings and evaluate them against the background of possible applications in the context of liquid chromatography. The comparative analysis of experimental results and model predictions in section 5 clearly demonstrates a transition from a dominance of hydrodynamic effects in the immiscible region to an adsorption dominated behavior in the miscible region (where only one convective phase is present). The dominance of hydrodynamic effects in the immiscible region is particularly strong (indicated by fractional flows differing considerably from saturations, i.e., where f j ≠ Sj) if one convective phase is considerably more wetting than the other and when the interfacial tension between the convective phases is large. In the considered system, this is the case for conditions in the P


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Industrial & Engineering Chemistry Research the fluiddynamic behavior. Finally, the system of PDEs to solve increases linearly with the number of components, making the derivation of an analytical solution more difficult and elaborate. The presented work provides, through both experimental observations and theoretical considerations, a thorough understanding of two-phase flow in liquid chromatography and creates a theoretical basis, which can be extended to describe a spontaneous phase split within the column and to account for multiple adsorbing components. Combining two-phase flow behavior with adsorption effects, the investigated model is of interest not only for the chromatographic community but also for applications in natural reservoirs involving adsorption effects, such as CO2 storage or enhanced coalbed methane recovery.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b05153. Detailed description of the uncertainty quantification of UNIQUAC parameters fitted in section 4.2 and the solution of the Buckley−Leverett equation by the method of characteristics (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Marco Mazzotti: 0000-0002-4948-6705 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by ETH Research Grant ETH-44 141. REFERENCES

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Two-Phase Flow in Liquid Chromatography, Part 1: Experimental

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