Extended abstract summary. Summary

Transcription

1 Extended abstract summary Summary The purpose of ths PhD work was to extend the Cubc-Plus-Assocaton (CPA) equaton of state (EoS) to electrolytes and thus be able to handle smultaneously polar/assocatng compounds, hydrocarbons and a varety of ons from fully dssocated salts. The ultmate goal was to develop a theoretcally-orented model whch can handle a wde varety of mxtures and propertes n an as predctve way as possble. The CPA EoS has durng the past year been appled to thermodynamc modelng of a wde range of ndustrally mportant chemcals, manly n relaton to the ol- and gas sector. One of the strengths of the CPA EoS s that t reduces to the Soave Redlch Kwong (SRK) cubc EoS n the absence of assocatng compounds and s therefore compatble wth exstng tools for ol characterzaton. In a smlar fashon, the electrolyte CPA (e-cpa) EoS should reduce to the CPA EoS n the absence of electrolytes, makng t possble to extend the applcablty of the CPA EoS whle retanng backwards compatblty and resung the parameters for non-electrolyte systems. There are many challenges related to thermodynamc modelng of mxtures contanng electrolytes, and many dfferent approaches to the development of an electrolyte EoS have been suggested by scentsts n the feld. However, most of these approaches are focusng on aqueous solutons and cannot easly be extended to handle mxed solvents. Furthermore, the approaches suggested n current lterature have rarely been appled to all types of thermodynamc equlbrum calculatons relevant to electrolyte solutons. Ths project has amed to determne the best recpe to delver a complete thermodynamc model capable of handlng electrolytes n mxed solvents and at a wde range of temperature and pressure. Dfferent terms descrbng the electrostatc nteractons have been compared and t was concluded that the dfferences between the Debye-Hückel and the "mean sphercal approxmaton" models are neglgble. A term accountng for the Gbbs energy of hydraton (such as the Born term) must be ncluded n order to provde suffcent drvng forces for electrolytes towards the most polar phase. The statc permttvty of the mxture was found to be the most mportant property; yet t was shown that the emprcal models suggested by lterature could lead to unphyscal behavor of the equaton of state. A new theoretcal model was developed to extend the framework for modelng of the statc permttvty to hydrogen-bondng compounds and salts. The model relates the geometrcal confguraton of hydrogenbondng dpolar molecules to the Krkwood g-factor usng the Werthem assocaton model that s ncluded wth modern EoS such as CPA or SAFT (Statstcal Assocatng Flud Theory). Ths new model was shown to gve excellent predctons of the statc permttvty of mxtures over wde ranges of temperature, pressure, and composton and thereby generalzes the handlng of electrolytes n mxed solvents n an electrolyte EoS. The CPA EoS was extended wth a Debye-Hückel and a Born term to account for the electrostatcs along wth the new model for the statc permttvty. Ths new e-cpa EoS was parameterzed aganst osmotc coeffcent, densty, and mean onc actvty coeffcent data of pure salts and valdated aganst salt mxture data. The model was then appled to predct: the solublty of lght gases, hydrocarbons, and aromatcs n aqueous mxtures and mxed solvents sold-lqud equlbrum n aqueous salt mxtures and mxed solvents

2 gas hydrate formaton pressures of methane wth salts n water+methanol lqud-lqud and lqud-lqud-lqud equlbrum wth water-propan--ol-nacl-octane solutons It was demonstrated that the model has a good potental for applcatons n relaton to e.g. flow assurance durng the producton of natural gas. The parameterzaton of electrolyte EoS s of hgh mportance and more work s needed n order to obtan good on-specfc parameters that nclude nteracton parameters wth gases and relevant chemcals. Problem addressed and state of the art The Cubc-Plus-Assocaton (CPA) equaton of state (EoS) has been successfully appled n modelng of the phase equlbrum of natural gas components n mxtures of water, glycols or alcohols []. The purpose of ths work was to extend CPA to account for the presence of ons usng some of the followng contrbutons to the Helmholtz energy: It s clear from ths llustraton that several factors are unclear when extendng the CPA or any equaton of state to mxtures contanng electrolytes: Whch and how many electrolyte terms should be ncluded? How.e. by whch theory should the electrostatc nteractons be descrbed? (Debye-Huckel, MSA, other?). Do we need a Born-type term for the chargng of ons?. Do we need an on-solvent assocaton term? 3. Should we approach the problem from the prmtve pont of vew (solvent s a contnuum) or from the more complex molecular approach where all nteractons should be accounted for? 4. If contnuum, how can we estmate the delectrc constant n a relable way and s ths really requred? 5. Last but not least, how can we estmate the model parameters n a way that makes sense and can render the model sutable for a wde range of practcal applcatons and f possble n a predctve way? These are only some of the key questons when developng electrolyte equatons of state and thus t s no bg surprse that a very wde and dfferently developed e-eos have been proposed. These are summarzed and crtcally compared n the frst chapter of the thess (secton.6, esp. tables. and.3). Ths analyss

3 showed that, unfortunately, essentally none of the above questons have been answered n a satsfactory matter and the PhD thess addresses several of them. It has been early n the thess work decded to focus on the prmtve approach as non-prmtve engneerng orented approaches were not felt to be wthn the tme horzon of a sngle PhD, but all the other questons were addressed n ths work. The key dscoveres nnovatons are presented n the next secton, whle detaled results and applcatons are presented next. Key nnovatons. Debye-Huckel vs. MSA The frst nnovaton s the sgnfcant understandng of dfferent models for electrostatc nteractons, especally the MSA (mean sphercal approxmaton) and the Debye-Hückel theores. Ths was an mportant task to undertake, as t s often reported that MSA s a superor model, possbly because of ts theoretcal orgn (dervaton from statstcal thermodynamcs). Thus, the valdty of the conclusons from varous research groups regardng the use of MSA vs. Debye-Hückel should be checked. Bjørn Marbo-Mogensen has both derved fully and mplemented hmself both models (ncludng the full verson of MSA). The major concluson from ths work s that MSA and the Debye-Hückel models were found to perform smlarly (n terms of the reduced Helmholtz energy) f the Debye-Hückel dstance of closest approach parameter was set to 5/6 of the MSA dameter. Ths work s publshed n the open lterature [artcle above, also attached, ref.]. It was shown n the same work [] that the delectrc constant s the most mportant parameter whch nfluences the electrostatc nteractons. It was shown that the delectrc constant and ts concentraton dependency are actually a much more mportant parameter than selectng MSA or DH for the electrostatc nteractons. It was also shown that usng an emprcal model for the delectrc constant may ntroduce possble unphyscal behavor nto the equaton of state, whch can greatly nfluence equlbrum propertes.. Predctve model for the delectrc constant These observatons [under pont, from ref.] were the startng step for Bjørn for developng a theoretcally sound model for predctng the delectrc propertes of complex fluds. Ths was consdered necessary as the exstng models for delectrc constants of mxtures are emprcal and only support aqueous solutons. To mprove the relablty of the thermodynamc model, a new predctve model for the delectrc constant and refractve ndex of mxtures has been developed based on the theory of dpole fluctuatons by Onsager, Krkwood and Fröhlch, usng a geometrcal model for the hydrogen-bond networks. The model s capable of predctng the statc permttvty of pure compounds and complex mxtures over wde temperature and pressure ranges. Bjørn was able wth hs model to predct the statc permttvty for a very wde range of systems, from pure compounds (water, alcohols, glycols) are a functon of temperature, bnary water-alcohol and water-glycol mxtures over the whole concentraton range and an extensve temperature-range as well as

4 both low and hgh pressures, and fnally hs model has be used for predctons for ternary mxtures e.g. of water-ethylene glycol-methanol. In contrast to the emprcal models that are often ncluded n electrolyte models, the new model for predctng the statc permttvty of complex mxtures has the correct physcal behavor, whch s mportant when the model s used n the context of equatons of state. It s also possble to determne the complex (frequency-dependent) permttvty usng a relaxaton model. The results are excellent llustratng that a very successful theoretcally sound model for the statc permttvty has been developed, also for salt contanng mxtures. Ths work has been publshed n two artcles n Journal of Physcal Chemstry B. [3,4]. 3. The mportance of the Born term Bjørn s work has also llustrated the mportance of the Born term for solvaton and that such a term should be ncluded n an equaton of state. 4. Development and parameterzaton of the electrolyte CPA equaton of state Bjørn has hmself programmed a full verson of an electrolyte equaton of state based on CPA. Hs e-cpa model ncludes the CPA model for physcal and assocaton nteractons as well as the DH and Born terms. Usng a smple parameterzaton approach, the developed e-cpa s capable of correlatng the densty, the actvty-, and the osmotc coeffcents of the NaCl-water and other aqueous salt systems over wde temperature ranges, when ncludng on assocaton. He proved that on assocaton effects are mportant. A full parameterzaton of the model has been recently completed and an artcle has been submtted to AIChE J. [5] (specal ssue n honour of John M. Prausntz). Very satsfactory results are obtaned, besdes actvty coeffcents, for calculaton of saltng-out and gas hydrate formaton curves. Applcatons, mplementatons and results Some more detals on the above mentoned nnovatons and some results are shown below. Thermodynamcs of Electrostatc Interactons The theoretcal foundatons of electrolyte thermodynamcs have been lad forth by many of the 8 th, 9 th, and th century s most dstngushed scentsts. C. A. de Coulomb publshed n 785 the fundamental result for the force between two charged spheres as a functon of the nverse-square dstance r. Debye and Hückel (DH) [6] solved the Posson equaton assumng that the charges surroundng a central on followed a Boltzmann dstrbuton. They determned the electrcal potental ψ from Eq. () []: λq λq λκ ψ( r, λ) = r d 4πε ε r 4πε ε + λκd r r The electrostatc energy can then be determned through the partal chargng procedure from Eq. (): ()

5 Usng ( κd) ( κd ) N el Ne ln A nz + κd A = nq ψ (, ) r λ dλ = + β () 4πε ε r d r = d gves the DH soluton where β =. Usng r = d gves the DH + Born term wth β =. Eq. (3) shows the Born energy of transfer [6] at nfnte dluton from vacuum to lqud calculated from Eq. (). ( ) ( ) ( ) lm Ne z l v A n = µ µ = β = βδaborn 4πε d ε r Fgure shows a comparson of the chemcal potental from a CPA EoS. Eq. (3) provdes the domnatng drvng force for keepng the ons n the polar phase; makng t crucal for ol-water equlbrum [7,8]. (3) Water Decane Vapor -.8 SRK -.8 Assocaton Born Fgure : Chemcal potental (/RT A/ n) of Na + at nfnte dluton from dfferent terms n the ol, water, and vapor phase. Debye-Hückel vs. MSA The DH equaton has often been confused wth the lmtng law arsng from a frst-order Taylor expanson of Eq. () whch s only vald at low concentratons []. Ths has ncorrectly led to the concluson that the DH equaton n Eq. () s not suted for hgh concentratons, and several authors have argued for the use of the mean sphercal approxmaton (MSA) derved from statstcal mechancs by L. Blum [9]. Bjørn showed [] that there are neglgble dfferences between the full (mplct) MSA and the full DH equaton even n concentrated solutons, and that the results were much more senstve to the model used for the statc permttvty than to the use of ether MSA or DH. The man results are summarzed n Fgure : A r /RT [mole] ε r, MSA ε w, MSA - 4 NaCl Molalty [mol/kg] /RT ( A r / V) ε r, DH ε w, DH 4 NaCl Molalty [mol/kg] /RT ( A r / n ) - - Na NaCl Molalty [mol/kg] Fgure : Comparson of the Helmholtz energy when ncludng effect of ons on permttvty (ε r ) and a permttvty equal to water (ε w ). Emprcal model for salt effect may ntroduce non-physcal behavor (e.g. addtonal volume roots).

6 How to calculate the statc permttvty? Despte the mportance of the statc permttvty there s no consensus among researchers; should we use ε r of the pure or mxed solvent [-]?, should we account for on effect through emprcal models [-6]? Or should we use non-prmtve models where the statc permttvty s calculated mplctly wthn the equaton of state tself from dpole-dpole, on-dpole, and on-on nteractons [7,8]? To answer ths mportant queston Bjørn developed a new predctve model for the statc permttvty of flud mxtures contanng both non-polar, polar, hydrogen-bondng compounds [3] and ons [4]. The theory of dpolar fluctuatons [9,3] was extended wth a geometrc model for dpolar correlatons n a hydrogen-bond network to account for dpolar correlatons. The statc permttvty ε r s calculated usng Eq. (4): ( )( ) zp cosγ µ r + r + N A j j j, j = xxgµ, g = + (4) εr 3 εkt B j P cosθj + µ, ε ε ε ε ε ρ Where ε s approxmated by the squared refractve ndex n calculated from the Clausus-Mossott equaton, µ, s the vacuum dpole moment, ρ the molar densty calculated from CPA, x mole fracton, X the fracton of that s not bound to an on (assumng dpoles n the electrcal feld from the on does not contrbute to overall dpole moment densty [4]), g s the Krkwood g-factor for dpolar correlatons calculated from the coordnaton number z j and angles γ j and θ j. The probabltes of assocaton between molecule and j P j as well as probablty of assocaton of component I P = P are both calculated j wthn the CPA equaton of state usng the Werthem assocaton framework []. Based on correlaton of θ for the pure components only, the model gves excellent predctons for bnary and ternary mxtures over wde temperature and pressure ranges as shown n Fgure 3. Statc permttvty ε r 5 Pure component 4 6 Temperature [K] H O MeOH EtOH MEG Statc permttvty ε r 5 5 Water-MeOH.5 Mole fracton MeOH 83.5K 93.5K 363.5K 473.5K Statc permttvty ε r 8 6 Water-MeOH-XX Temperature [K] EtOH MEG Fgure 3: Correlated statc permttvty calc. wth CPA (left) [4]. Predcted statc permttvty of methanol-water (mddle, colored at bar, black at bar) [6]. Predctons for the ternary mxtures (rght) of water (5vol%), methanol (5vol%), ethylene glycol (5vol%) [4], and of water (vol%), methanol (55vol%), ethanol (5vol%) [7]. Data from Landolt-Börnsten [3]. The presence of salts reduces the statc permttvty prmarly through the mechansm of delectrc saturaton, where solvent molecules are drected by the electrcal feld. Bjørn ncluded ths effect through on-solvent assocaton usng the Werthem assocaton framework where each on s assumed to have ϕ j dentcal assocaton stes. Bjørn calculated the on-solvent cross-assocaton strength β j from the solvaton number at nfnte dluton N j, and ether ft the cross-assocaton energy ε j to expermental data or use the mcr- (solvaton) combnng rule ε j =.5ε [4]. Fgure 4 was obtaned usng the solvaton numbers below:

7 Ion Na + K + Ca + Cl - Br - I - - NO 3 Hydraton number SO 4 - Fgure 4: Correspondence of delectrc constant data (left). Dervatves of Debye-Hückel model wth new model (rght). [4] Fgure 4 (left) shows good correspondence wth the statc permttvty of water-salt solutons reported n lterature. Fgure 4 (rght) shows a comparson of compostonal dervatves of Debye-Hückel wth the new model (red) and the emprcal correlatons used n [], wth (dashed) an wthout (sold) account for the effect of ons. It shows that the new model has a dramatc effect on the constant-volume/temperature chemcal potental from Debye-Hückel, and that t now provdes a sgnfcant contrbuton to the water actvty, reflectng the physcal effect that addng more dpolar molecules to a fxed volume contanng water and ons wll ncrease the statc permttvty. The new model also drastcally affects dervatves wrt. T and V. Such effect cannot be captured through the emprcal models commonly adopted n lterature [,4]. In practce results for the statc permttvty of salt-water solutons are scattered, scarce, often nconsstent and are subject to systematc errors [4]. The most severe error arses from the electrophoretc effect dscovered by Hubbard and Onsager [3-35] causng a reducton n the measured permttvty n an electrcal feld due to the movement of ons n the order of 5-75%. It must not be used for thermodynamc modelng as t s a dynamc effect [4,3-37], but as the extent of the effect cannot be determned accurately, the effect has been gnored by prevous groups workng wthn electrolyte thermodynamcs. The fnal phase of the project s summarzed n the most recent publcaton [4] where a complete electrolyte CPA equaton of state s presented. Whle the detals are presented n ths artcle, fgures 5- show some typcal results. Very few adjustable parameters are used and they are ftted to actvty and osmotc coeffcents. It s shown that e-cpa can correlate such actvty coeffcents qute well over extensve temperature ranges, whle t can also predct sold-lqud equlbra (mxed salts), gas solubltes, salt solublty n mxed solvents and gas hydrate curves, among much more. The model has been wdely appled and thus the predctve capabltes of the new EoS wll be assessed for mportant flow assurance propertes, ncludng freezng pont depresson, saltng out, gas hydrate nhbton, and scalng n mxed solvents. The model wll also be made avalable n commercal process smulators, such as Aspen HYSYS through a CAPE-OPEN lbrary n C# developed durng the PhD [38].

8 Osmotc / Actvty Coeffcents Φ (exp.) γ ± m (exp.) Φ (calc.) γ ± m (calc.) Molalty NaCl [mol/kg] Fgure 5: Correlated wth e-cpa osmotc coeffcents and actvty coeffcents for NaCl at 5 o C. Mean onc actvty coeffcent γ ± m o C o C 5 o C o C 5 o C Osmotc coeffcent Φ o C o C 5 o C o C. 4 6 Molalty NaCl [mol/kg] Molalty NaCl [mol/kg] Fgure 6: Results wth e-cpa. Temperature and concentraton dependence of NaCl actvty coeffcents (left) and osmotc coeffcents (rght) compared to expermental data.

9 Fgure 7: Sold-lqud phase dagram wth e-cpa for aqueous mxtures of NaCl+KCl (left) and KCl+K SO 4 (rght) at 5 C. Solublty [mol/kg solvent] NaCl KCl Na SO 4 Na SO 4 H O K SO Water mole fracton Fgure 8: Solublty of dfferent salts n water + methanol wth e-cpa.

10 CH 4 solublty [ppm] mol/kg mol/kg 4 mol/kg Pressure [bar] Fgure 9: Solublty of methane n water wth NaCl at 5.5 C usng e-cpa aganst expermental data from O'Sullvan and Smth (97). 8 No salt mol/kg mol/kg 4 mol/kg.9 mol/kg + wt% MeOH.9 mol/kg + 4wt% MEG 4 Pressure [MPa] Temperature [K] Fgure : Predcted wth e-cpa effect of NaCl on the gas hydrate formaton pressure n pure water and mxtures wth methanol and ethylene glycol. Data was obtaned from the NIST (4) Clathrate Hydrate Database. Note that the pressure scale s logarthmc.

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