evaporation models for multicomponent...

Pannon Egyetem

Kémia Intézet

Fizikai Kémiai Intézeti Tanszék

Evaporation models for multicomponent mixtures

DOKTORI (PhD) ÉRTEKEZÉS

Járvás Gábor

Témavezető

Dr. Dallos András

Kémiai és Környezettudományi Doktori Iskola

2012.


Értekezés doktori (PhD) fokozat elnyerése érdekében

Írta:

Járvás Gábor

Készült a Pannon Egyetem Kémiai és Környezettudományi

Doktori Iskola keretében

Témavezető: Dr. Dallos András Elfogadásra javaslom (igen / nem) ………………………. (aláírás) A jelölt a doktori szigorlaton ........%-ot ért el, Az értekezést bírálóként elfogadásra javaslom: Bíráló neve: …........................ …................. igen /nem ………………………. (aláírás) Bíráló neve: …........................ …................. igen /nem ………………………. (aláírás) A jelölt az értekezés nyilvános vitáján …..........%-ot ért el. Veszprém, ………………………….

a Bíráló Bizottság elnöke A doktori (PhD) oklevél minősítése…................................. ………………………… Az EDHT elnöke

University of Pannonia

Institute of Chemistry

Department of Physical Chemistry


Ph.D. dissertation

Gábor Járvás

Supervisor

Dr. András Dallos

Doctoral School of Chemistry and Environmental Sciences

2012.

Kivonat Többkomponensű elegyek párolgásának modellezése Doktori értekezésemben a többkomponensű folyadékok egyensúlyi párolgásának

modellezésével kapcsolatos főbb eredményeimet foglalom össze. A kapcsolódó

szakirodalom kritikus feldolgozása során megállapítottam, hogy a többkomponensű

rendszerek párolgásának szerteágazó, gazdag elméleti és kísérleti háttere van,

ugyanakkor a legtöbb közölt munka nem fektet elég hangsúlyt a folyadékfázis reális

tulajdonságainak a leírására, Munkám célja olyan sík- és görbült felületre vonatkozó

folyadék-felületi párolgási modellek fejlesztése volt, melyek a reális folyadékfázis

termodinamikai tulajdonságainak valósághű becslésével képesek a párolgási folyamatok

pontos modellezésére. Ugyancsak fontos szempont volt, hogy a kifejlesztett modellek

egyszerűen és gyorsan használhatóak legyenek - az alkalmazott szoftverek

rendelkezésre állása esetén - változatos és újszerű összetételű, kihívást jelentő elegyek

párolgási jellemzőinek számítására, így gyakorlati jelentőséggel bírhatnak a biomassza

eredetű komponenseket tartalmazó üzemanyagok párolgási jellemzőinek vagy tárolási

veszteségeinek számítása során.

A dolgozatban részletesen ismertetem az ún. COSMO-RS elméletet, mely

újdonságánál fogva kevésbé közismert, ugyanakkor fontos részét képezi a kidolgozott

modellezési eljárásoknak. A COSMO-RS módszerrel becsült aktivitási tényezőket

használtam a folyadékelegyek komponenseinek parciális nyomásának becsléséhez az

egyensúlyi párolgás számítása során. A molekulák gázfázisbeli transzportjának

szimulációjához a Maxwell-Stefan féle diffúziós és konvekciós elméletet alkalmaztam.

A számításokat saját fejlesztésű Matlab programmal végeztem, amely a

részszimulációkhoz a COSMOtherm és a COMSOL Multiphysics kereskedelmi

szoftverek egyes standard eljárásait használta. Mind a csepp-párolgás, mind a

síkfelületű párolgás számítására alkalmas modelljeim eredményeit összevetettem a

szakirodalomból származó kísérleti adatokkal. A szimulációs és kísérleti eredmények

összehasonlítása során megállapítottam, hogy a modellek alkalmasak többkomponensű

elegyek párolgásának becslésére annak ellenére, hogy a szimulációkba bevont elegyek

igen változatos összetételűek voltak.

Disszertációmban bemutatom a Hansen-féle oldási paraméterek (HOP)

becslésére kifejlesztett módszeremet is. A többkomponensű folyadékfázis molekulái

között létrejövő kölcsönhatások és a komponensek aktivitási tényezői becslésének egyik

legelterjedtebb módja az oldási modellek használata, melyek közül a Hansen féle oldási

elmélet a leghasználhatóbb. A HOP becslésére nemlineáris mennyiségi szerkezet-

tulajdonság összefüggés (angolul rövidítve QSPR) modelleket állítottam fel, melyekben

a molekula szerkezetre vonatkozó független változók a COSMO-RS elmélethez

kapcsolódó szigma-momentumok. A szerkezet-tulajdonság összefüggés modellezéséhez

előrecsatolt, felügyelt tanítású mesterséges ideghálót alkalmaztam. A QSPR modelleket

kísérleti adatok felhasználásával validáltam, és megállapítottam, hogy alkalmasak

változatos funkciós csoportokkal és eltérő kémia sajátságokkal rendelkező molekulák és

ionpárok (alkánok, alkének, aromások, halo- és nitro-alkánok, aminok, amidok,

alkoholok, ketonok, éterek, észterek, savak, amin-sav ion-párok és ionos folyadékok)

HOP becslésére is.

Abstract Evaporation models for multicomponent mixtures

This dissertation summarizes the author’s results on simulations of the

evaporation of multicomponent liquid mixtures having flat or curved liquid surface. The

models are based on the quantum chemical description of non-ideality of liquid phase

properties and take into account the possible transport phenomenas in the gas phase.

The models apply the COSMO-RS theory for the estimation of vapour-liquid

equilibrium of non-ideal solutions and the Maxwell-Stefan diffusion and convection

theory for the calculation of gas phase transport characteristics of the components. The

activity coefficients, the liquid and vapour phase compositions, the cumulative and

components evaporation fluxes have been computed. Calculations for the quasi-

equilibrium evaporation of compounds from surface have been performed by

COSMOtherm and COMSOL Multiphysics programs. The calculation results of both

droplet evaporation and flat surface evaporation models are compared against

experimentally determined values. It can be concluded that the models have reasonable

ability for prediction of the evaporation of multi-component liquid systems.

Solubility parameters, such as Hansen Solubility Parameters, (HSPs) are widely

accepted models for describing the interaction between molecules of multi-component

mixtures and the estimation for activity coefficients of their components. New

quantitative structure-property relationship (QSPR) multivariate nonlinear models based

on artificial neural network (ANN) were developed for the prediction of the components

of the three-dimensional Hansen solubility parameters using COSMO-RS sigma-

moments as molecular descriptors. The models for HSPs were built on a

training/validation data set of compounds having a broad diversity of chemical

characters (alkanes, alkenes, aromatics, haloalkanes, nitroalkanes, amines, amides,

alcohols, ketones, ethers, esters, acids, ion-pairs: amine/acid associates, ionic liquids).

HSP prediction was validated on a test set with various functional groups and polarity,

among them drug-like molecules, organic salts, solvents and ion-pairs. COSMO sigma-

moments can be used as excellent independent variables in nonlinear quantitative

structure-property relationships using ANN approaches. The resulting optimal

multivariate nonlinear QSPR models involve the five basic sigma-moments having

defined physical meaning and possess suitable predictive ability for dispersion, polar

and hydrogen bonding HSPs components.

Abstrakt Modellierung der Verdampfung von Mehrkomponenten-Flüssigkeitsmischungen

Diese Dissertation fast zusammen die Ergebnisse der Modellierung von

Verdampfungen der Mehrkomponenten-Mischungen mit flach oder gekrümmten

Oberflächen. Die entwickelten Modelle benutzen die quantenchemische Theorie für die

Beschreibung der Nichtidealität der flüssigen Phasen, und simulieren die wichtigste

Transportphaenomene in der Gasphase. Die Modelle benutzen die COSMO-RS Theorie

für die Abschätzung vom Dampf - Flüssig - Gleichgewicht von nichtidealen Lösungen

und anderseits die Maxwell - Steffan Diffusion und Konvektion Theorie für die

Berechnung der Transporteigenschaften der Komponenten in der Gasphase. Die

Berechnungsergebnisse für Tropfenverdampfungsmodelle und für Verdampfung von

flachen Oberfläche wurden mit im Experiment ermittelten Daten verglichen. Es wurde

festgestellt, dass die entwickelten Modelle zur Vorhersage der Verdampfung von

Mehrkomponenten Systemen verwendbar sind.

Zur Beschreibung der Wechselwirkungen zwischen den Molekülen der

Mehrkomponenten-Mischungen sind auch Modelle, die auf Löslichkeitstheorien

beruhen, akzeptiert. Die Aktivitätskoeffizienten der Komponente können durch den

Einsatz der Löslichkeitstheorie nach Hansen vorhersagen. Quantitative Struktur-

Eigenschafts-Beziehung (QSPR) Modelle wurden zur Abschätzung der Hansen-

Löslichkeitsparameter mit multivariate und künstlichen neuronalen Netzen entwickelt.

Als unabhängige Varianten der QSPR Modellen werden die Sigma-Momente der

COSMO-Theorie angewendet. Die QSPR Modelle wurden mit experimentellen Daten

validiert. Es wurde bestimmt, dass die entwickelten Modelle für die Vorhersage der

Hansen-Löslichkeitsparametern von Molekülen und Ionenparen mit verschiedenem

chemischen Charakter (Alkane, Alkene, Aromaten, Halogenalkane, Nitroalkane,

Amine, Amide, Ketone, Ether, Ester, Alkohole, Säure, Amine-Säure Ionenparen,

ionische Flüssigkeiten) einsatzfähig sind.

Contents

1. Introduction ......................................................................................................................... 1

2. Literature overview ............................................................................................................. 3

2.1. Flat surface evaporation ........................................................................................... 3

2.2. Droplet evaporation ................................................................................................. 4

2.3. Modelling of evaporation of multi-component mixtures ......................................... 4

2.4. Equilibrium and non-equilibrium evaporation ........................................................ 5

2.5. Hansen Solubility Parameters .................................................................................. 7

2.6. COSMO-RS theory .................................................................................................. 9

3. Calculation of model elements ......................................................................................... 16

3.1. Quantum chemical and COSMO calculations ....................................................... 16

3.2. The vapour-liquid equilibrium model .................................................................... 19

3.3. Calculation of the vapour pressure of the components .......................................... 19

3.4. Calculation of evaporation flux and transport in gas phase ................................... 21

4. Flat surface evaporation model ......................................................................................... 24

4.1. Model description .................................................................................................. 24

4.2. Test calculations of flat surface evaporation model .............................................. 26

4.2.1. Testing of 1D evaporation model for a binary liquid mixture ........................... 26

4.2.2. Testing of 2D evaporation model for multicomponent liquid mixtures ............ 28

4.3. Summary of flat surface evaporation model .......................................................... 37

5. Droplet evaporation model ............................................................................................... 39

5.1. Model description .................................................................................................. 39

5.2. Tests of the droplet evaporation model .................................................................. 43

5.3. Summary of the droplet evaporation model .......................................................... 51

6. Estimation of Hansen Solubility Parameters .................................................................... 52

6.1. Data and σ-moment sets for modelling .................................................................. 53

6.2. Nonlinear QSPR model ......................................................................................... 54

6.3. Test of HSPs estimation methods .......................................................................... 57

6.4. Summary of the models for HSPs prediction ........................................................ 64

7. References ......................................................................................................................... 65

8. Tézisek .............................................................................................................................. 74

8.1. Síkfelületű párolgásra vonatkozó modell kifejlesztése ............................................. 74

8.2. Csepp-párolgási modell kidolgozása ......................................................................... 74

8.3. QSPR modellek kidolgozása a Hansen-féle oldhatósági paraméterek becslésére .... 74

9. Theses ............................................................................................................................... 76

9.1. Development of flat surface evaporation model .................................................... 76

9.2. Development of droplet evaporation model .......................................................... 76

9.3. Model development for estimation of Hansen solubility parameters ................... 76

10. Kapcsolódó publikációk és közlemények - Related publications ..................................... 78

11. Acknowledgement ............................................................................................................ 80

Abbreviations 1D One Dimensional 2D Two Dimensional 3D Three Dimensional CCD Charge Coupled Device CED Cohesion Energy Density CFD Computational Fluid Dynamics CNN Artificial Neural Network CNN Computational Neural Network COSMO Conductor-like Screening Model COSMO-RS COSMO for Real Solvents CSM Continuum Solvation Model DFT Density Functional Theory ECM Effective Conductivity Model ESC Environmental Stress Cracking FCM Finite Conductivity Model FEM Finite Element Method HB Hydrogen Bonding HSP Hansen Solubility Parameter ICM Infinite Conductivity Model MAE Mean Absolute Error MD Molecular Dynamics MLR Multiple Linear Regression PDE Partial Differential Equation PVT Pressure Volume Temperature QM Quantum Chemical Method QSAR Quantitative Structure-Activity Relationships QSPR Quantitative Structure-Property Relationships RMS Root Mean Square SCD Screening Charge Density SD Standar Deviation T Temperature TRC Thermodynamics Research Center TZVP Triple Zeta Polarized Valence UNIFAC UNiversal Functional Activity Coefficient UNIQUAC Universal Quasi-Chemical VLE Vapour-Liquid Equilibrium

List of Symbols AA Antoine-constant A evaporating surface

aeff effective contact area

ai area of segment i

ATS parameter of Thek-Stiel equation BA Antoine-constant c concentration CA Antoine-constant

cHB adjustable parameter of COSMO-RS theory

Ci ith coefficient

cp isobar heat capacitie d diameter of droplet

Di,j binary diffusional coefficient

ECOSMO total energy of the ideally screened molecule

EHB energy of Hydrogen Bonding

Emisfit interaction energy from misfit SCDs

Ev energy of vaporization

EvdW van der Waals energy

EXiCOSMO total energie of the molecule in the COSMO conductor

EXiGas total energie of the molecule in the gas phase

GC,S combinatorial free energy of system S

hTS parameter of Thek-Stiel equation

jt evaporation flux k thermal conductivity

Mij mean molar mass of compound i and j

Mxi ith σ-moment of compound X

N number of measured points

ni amount of compound i

nXiRing number of ring atoms in the molecule

P arbitrary material characteristic p number of parameters

P*r reduced vapour pressure

Pc critical pressure

Pd vapour pressure at the droplet surface

Pi* vapour pressure of compound i

pS(σ) σ-profile of system S

Pt total pressure

Pu unit of pressure

pX(σ) probability distribution or σ-profile q¯ total molecular area of compound i

Qcond conducted heat

qi molecular area of compound i R universal gas constant r¯ total molecular volume

R2 squaer of correlation coefficient

ri molecular volume of compound i

Tb normal boiling point

Tbr reduced normal boiling point

Tc critical temperature

Tr reduced temperature

Vm molar volume

w0 initial mass

Xi ith compound

xi mole fraction of compound i in the liquid phase

yi mole fraction of compound i in the gas phase

∆Hvb molar heat of evaporation at the normal boiling point ∆t time interval

Greek letters

α mass loss fraction α’ adjustable parameter of COSMO-RS theory

αc adjustable parameter of Thek-Stiel method

γi activity coefficient of compound i

δd dispersion force component of HSP

δh hydrogen bonding component of HSP

δp polar component of HSP

δt Hildebrand solubility parameter

εij Lennard-Jones characteristic energy

εT termination criteria

ηGas adjustable parameter of COSMO-RS theory

λ0 adjustable parameter of COSMO-RS theory



µc,SXi combinatorial contribution to the chemical potential

µGasXi chemical potential in the gas phase

µS(σ) measure for the affinity of the system S to a surface of polarity σ

µSXi chemical potential of compound Xi in system S

σHB adjustable parameter of COSMO-RS theory

σi screening charge density of compound i

σi,LJ Lennard-Jones scale parameter

σij characteristic length value

σLG surface tension at the liquid/gas surface

τ’vdW adjustable parameter of COSMO-RS theory

τvdW adjustable parameter of COSMO-RS theory

ΩD diffusion collision integral

ωRing adjustable parameter of COSMO-RS theory

1

1. Introduction

The evaporation of liquids has created great interest in engineering since

decades. Understanding of this process is essential for application and development in

numerous areas, however, augmentation of efficiency of evaporation and combustion of

fuel in Diesel- and Otto-engines and aerosol chemistry are the most important.

Notwithstanding, there are only few theoretical and experimental studies that come

close to the basic governing effect of multicomponent mixture evaporation. Among

many forms of evaporation, droplet and flat surface vaporization are the most important

occurrences. Due to its importance, wide range of studies can be found in this research

fields such as original research articles, review articles [1,2] and textbooks [3,4] too.

The theme of evaporation of droplets is close to another typical way of evaporation

studies, focuses on spray evaporation but it is beyond the scope of this doctoral work;

however, a review can be found in the paper of Singnano [5]. With respect to the

composition of evaporating liquids, most of the studies consider pure solvents such as

n-alkanes or water, and just a few deals with multicomponent non-ideal mixtures. As

the ambient atmosphere of vaporization conditions can be varied in wide range, high

temperature and pressure in internal combustion engine design and atmospheric

pressure and temperature close to the room temperature in the aerosol chemistry.

Numerous computational experiments show, that appropriate real mixture model have

huge effect on the accuracy of the evaporation models.

Consequently, the development of an evaporation model for multi-component

real liquid mixtures, based on activity coefficient calculation from theoretical chemical

structures alone, which is completely independent of any experimental vapour-liquid

equilibrium (VLE) data and of any group interaction parameters of the regarded

compounds, would be of great interest in the chemical industries and also in waste

prevention and environmental protection.

Therefore in my doctoral work I have focused on the simulation of the

evaporation of multicomponent mixtures at normal conditions, especially on the

estimation of non-ideal behaviour of liquid phase. Non-ideal behaviour is essential and

allows me to neglect the secondary flow effect due to the applied different levitation

technics (effect of acoustic streaming in the levitator or suspension) during droplet

2

evaporation, i.e. I had to concentrate on the prediction of the activity coefficients of

components of liquid mixtures.

The most powerful “real-solvent” theories are based on a priori quantum

chemical calculations and can provide direct activity coefficient values for

multicomponent mixtures and additionally purely theoretical molecular descriptors,

which can be used as independent variables in quantitative structure-property

relationships (QSPR). These empirical equations can be applied for estimation of

physico-chemical properties of pure compounds and their mixtures.

The evaporation of special blends and mixtures, which can not be simulated by

real solvent theories could be described by models using activity coefficients estimated

by the cohesive energy density theory of Hildebrand. The Hildebrand and Hansen

solubility parameters (HSPs) are related to the thermodynamic chemical potential of

ingredients in binary or multi-component systems. Therefore a method, which applies

theoretically well-grounded molecular descriptor set for prediction of Hansen solubility

parameters, could be of great interest in many fields of engineering. Hence, a novel

method, which can be applied for the prediction of Hansen Solubility Parameters using

COSMO-RS sigma-moments as molecular descriptors have been developed.

3

2. Literature overview

2.1. Flat surface evaporation

Studying the flat surface evaporation is the stepbrother of evaporation works,

however, this phenomenon has created great interest at evaporation lost estimation in

oil/fuel industry, in waste prevention, environmental protection and also in the design of

perfumes and of coating systems. Unfortunately, oil spills always give actuality for this

field, the last one was the accident of Deepwater Horizon drilling rig at 20th April 2010,

which is one of the most serious accidents. Stiver and Mackay [6] derived an equation

between the evaporated volume fraction of oil spills and time and they compared the

relationship with the evaporative data of crude oil. Their equation has been modified by

many researchers, one of the most known works was published by Fingas [7]. He also

clarified [8, 9 and 10] empirically that most crude oil and petroleum products evaporate

at logarithmic rate with respect to time and presented a simple model for predicting the

weight loss fraction considering the temperature variations. However, under limited

conditions, the Fingas model cannot be applied to predict the amount of generated

vapour under different evaporation conditions because it is an empirical model with

adjustable system specific parameters. Okamoto et al. [11] also developed a model for

flat surface evaporation of multicomponent mixtures. For the calculation of the

evaporation rate of a multicomponent system individually measured (a priori

information) mass transfer coefficients of solvents were used in their model, which

makes the application of the method harder. Lehr [12] reported a paper with three

different possible models for evaporation of liquid pools, but unfortunately he

investigated the vaporing behaviour of pure benzene. McBain et al. [13] published a

work in which the evaporation from the wetted floor of an open cylinder was studied.

They dealt with pure water also, however, it was reported that beyond diffusion

phenomena, secondary effects can be important such as buoyancy force. It can be

concluded from the above cited works [11-13] and also from [14, 15] that the

evaporation rate of pure solvents is constant with respect to time. However, mass loss

by evaporation is not direct proportional with time for multi-component mixtures

because of the different volatility of compounds. Therefore a model, which can take into

account the non-ideal behaviour of multi-component systems could have great interest

in many fields of engineering.

4

2.2. Droplet evaporation

Droplet evaporation has very rich literature background; hence a literature

survey can be subjective. According to the basic approach of geometry and flow setup

many of these works can be grouped as levitated [16-19] or sessile droplets [20-23] with

or without forced convection. Furthermore, suspended droplets are also in the focus of

investigations [24-26] together with electrostatically levitated single droplets. It is also a

good possibility to order the huge amount of available studies by the composition of the

evaporating liquids such as pure [17-19] or multi-component mixtures [16, 27-29]. Last

but not least, coupled and uncoupled models are traditional ways of ordering the

available scientific literature. The coupling between transfers of species complicates the

solution of differential equations governing the quasi-stationary evolution of

evaporation process, composition and temperature [30]. This coupling means that the

mass flux of a species also depends on the mole fraction gradients of other species, and

the coupling generates diffusional interaction phenomena. One of the most widely

accepted studies on uncoupled theory is given by Kulmala et al. [31] in which the

authors describe the meaning of uncoupling between mass transfer rates: the mass flux

of species is dependent only on its own mole fraction gradient. In the uncoupled model

the mass transfer of another species is ignored when the mass transfer rate of the other

species is calculated.

2.3. Modelling of evaporation of multi-component mixtures

Former works [11-15] focus on the evaporation of pure components.

Understanding such systems is easier because properties are constant in time and there

is no property gradient in the space. The only effect, which can make the evaporation so

complex in this case is the temperature dependency. It is well known, that vapour

pressure of individual compounds - which is one of the key parameters of evaporation -

is strongly temperature dependent. Binary and ternary mixtures are the minor parts of

studies, however, many different approaches, models and simplifications can be found

to account the influence of non-ideality of the liquid phase on partial vapour pressures

of the components. Unfortunately, ideal case when compounds follow the Raoult’s law

is not frequent. Additionally, average thermodynamic properties cannot be used for

multi-component liquid mixtures. For the modelling of binary mixtures the authors

usually applied the van Laar equation [11] or the Wilson equation [27] to describe

5

activity coefficients of organic components in the mixture. However, both activity

coefficient models contain adjustable parameters, which cannot be determined in the

lack of experimental data for vapour-liquid equilibrium. Furthermore, some

questionable simplifications have been proposed to reduce the number of components

and to obtain the activity coefficients during modelling of multi-component liquids

containing more than three components. Okamoto et al. [32] assumed that components

having similar chemical structures behave similarly in liquid phase, consequently the

unique concentrations of similar compounds can replaced by their cumulative

concentration. Another oversimplified process is reported by Kryukov et al. [33] who

replaced a rather complex mixture such as diesel fuel with a hypothetical pure solvent.

A widely used method to estimate the non-ideality of liquid mixtures is the UNIFAC

approach [27, 34 and 35]. Unfortunately, the fragmentation methods can be difficult to

apply to complex molecules with diverse functional groups and cannot be used at all for

compounds having atomic groups whose group-contributions are unavailable in the

fragments databases.

2.4. Equilibrium and non-equilibrium evaporation

Many studies assume that the gas phase concentration over the liquid phase is

determined by the vapour-liquid equilibrium [7-11, 16-20, 27]; however it is also

possible to find papers where authors take non-equilibrium evaporation behaviour into

account. V.R. Dushin et al. [36] introduced a new dimensionless parameter I

characterizing the deviation of phase transition from the equilibrium. Accounting for

non-equilibrium effects in evaporation for many types of widely used liquids is crucial

for droplets diameters less than 100 µm. R. S. Miller et al. [1] also demonstrated that in

the case of droplet evaporation there is an important limit for non-equilibrium effects.

Their study reveals that non-equilibrium effects become significant when the initial

droplet diameter is less than 50 µm. In the paper of W.W. Yang et al. [37] it is shown

that the models that invoke a thermodynamic equilibrium assumption between phases

overestimate the mass-transport rates in the case of evaporation of methanol and water

mixture. Although the system quickly evolves to quasi equilibrium state it is necessary

to use non-equilibrium evaporation model in order to calculate accurately evaporation

rates [38]. Non-equilibrium effects have significant importance only in some special

cases [36-38] where conditions are far from normal, for instance in combustion chamber

6

of Diesel-engines where pressure takes place up to 30 bar or even more and temperature

up to 600 K [33].

The evaporation flux - which transports the evaporated molecules from the

evaporating surface towards far away from the surface - is one of the key points of the

evaporation models. Nevertheless, flat surface evaporation is out of focus of recent

studies, which focuses rather on droplet evaporation. There are many different

approaches for the calculation of the evaporation flux (or the concentration gradient) of

droplets. Historically, Fuchs [39] theory is one of the first widely used for the

concentration gradient at the evaporating curved surface as given by eq. (1)

= −

(1)

where c is the concentration, x is the space dimension and d is the diameter of droplet.

Another important method is the so called d2-model first published by Spalding [40] for

evaporation of pure compounds. According to this theory the squared diameter of the

droplet reduces linearly with time during droplet vaporization. As it was pointed out by

Law and Law [41], a multi-component analogue of the classical d2-model exists.

Current works applied the model of Abramzon and Sirignano, which was developed for

pure liquids [42] and the modified version by Brenn et al. [27] for multi-component

cases.

The physical phenomenon of diffusion is omnipresent in every natural as well

industrial process involving mass transfer. In many cases diffusion plays an important

role as the rate limiting mechanism [43]. The almost exclusively employed governing

equation to describe diffusive fluxes within continuum mechanical models is Fick’s

law, which states that the flux of a compound is proportional to the gradient of the

concentration of this species, directed against the gradient. There is no influence of the

other components, i.e. cross-effects are ignored although well-known to appear in

reality. Such cross-effects can completely divert the diffusive fluxes, leading to the so-

called reverse diffusion [44], which is a multicomponent diffusion approach and

required for realistic modelling. Newbold and Amundson [45] established that

Maxwell-Stefan flow plays essential role in the augmentation of the diffusive mass

transport. Finally, the Maxwell-Stefan diffusion matrix is assumed to be symmetric,

which can be obtained from the kinetic theory of gases [44].

7

2.5. Hansen Solubility Parameters

When UNIFAC and COSMO-RS methods cannot be applied for estimation of

activity coefficients, solubility parameters can be alternative possibilities. In case of

theoretically existing molecules, usage of UNIFAC method is hard due to the absence

of interaction parameters. COSMO-RS theory has problems when mixtures contain

polymers. In spite of that ionic liquids have practically zero vapour pressure; they could

have significant effect on evaporation processes as a solvent or co-solvent. In such

complex situations solubility parameters can be used also for taking into account the

non-ideality of mixtures. Solubility parameters, such as the models of Hildebrand or

Hansen were among others perhaps the first attempt to predict interaction of molecules

in the liquid phase. The Hildebrand solubility parameter δt [46], defined with eq. (2) as

the square root of the cohesive energy density, is characteristic for the miscibility

features of solvents.

= .

(2)

where Vm is the molar volume of the pure solvent, and Ev is the measurable energy of

vaporization [47]. Hansen [48] proposed an extension of the solubility parameter to a

three-dimensional system. Based on the assumption that the cohesive energy is a sum of

the contributions from non-polar, polar and hydrogen bonding molecular interactions,

he divided the overall solubility parameter into three parts: a dispersion force

component δd, a hydrogen bonding component δh and a polar component δp. These so-

called Hansen solubility parameters are additive:

δ = δ + δ + δ (3)

The three-dimensional Hansen solubility scale gives information about the relative

strengths of solvents and allows determining solvents, which can be used to dissolve a

specific solute. This approach has significantly increased the power and usefulness of

the solubility parameter in screening and selection of the appropriate solvents in

industry and in laboratory applications. HSPs belong to the key parameters for selecting

solvents in chemical, paint and coatings industries, and for selecting suitable solvents

for polymeric resins. They are widely used for characterizing surfaces, for predicting

solubilities, degree of rubber swelling, polymer compatibility, chemical resistance,

suitable chemical protective clothing, environmental stress cracking (ESC), permeation

rates, for explaining different properties of the components forming a formulation in

pharmacy, and in solvent replacement and substitution programs [49]. The solubility

8

parameter and its components can be applied for complete description and selection of

the best excipient materials to form stable pharmaceutical liquid mixtures or stable

coating formulations [50]. Furthermore, both Hildebrand and Hansen solubility

parameters are related to the thermodynamic chemical potential of the ingredients in

binary or multi-component systems [51] which reinforce the physical soundness of this

model.

Although the definition of the solubility parameters is simple, their experimental

determination is not always easy, especially for non-volatile compounds. Several

different methods for the determination of solubility parameter of materials exist:

swelling measurements [52], inverse gas chromatography [53], mechanical

measurements [54], solubility/miscibility measurements in liquids with known cohesive

energy [55] and viscosity measurement [56]. The partial solubility parameters can also

be calculated from experimental PVT data of the systems using equation-of-state

models [57, 58]. In all cases, the experimental determination of the HSPs requires pure

materials and is generally expensive.

In absence of reliable experimental data, the HSPs components can be estimated

based on the molecular structure by cohesive energy density method, using molecular

dynamics computer simulation [59], or by group contribution method [47, 60, 61].

However, the group contribution methods require the knowledge of all chemical group

contributions, which is difficult for ionic liquids or acid/base mixtures (organic salts)

involving molecular association.

Alternatively, multivariate, linear or non-linear regression methods, such as

quantitative structure-property relationships, based on purely theoretical molecular

descriptors have been proposed [62, 63]. The development of such predictive QSPR

models for the HSPs components, based on theoretical chemical structure alone, is of

great interest, because they would allow to obtain valuable information in the early

phase of the development of new molecules, i.e. even before the synthesis of these

molecules is started. Additionally, QSPR seems to be the only way to obtain the HSPs

components of ionic liquids, which are of growing interest in the industry, owing to

their unique properties as sustainable solvents.

However, the molecular descriptors generally used in QSPR are often abstract

quantities related to topological, structural, electrostatic, and quantum chemical features

of the molecules and the models obtained do not always have a straightforward physical

meaning. For example, one of the most widely used software products for calculation of

9

molecular descriptors is DRAGON, which can calculate 4885 descriptors for each

molecule. Variable selection on a huge number of descriptors is not trivial, and random

correlation can occurs. In particular, the link between molecular descriptors and the

thermodynamic properties of materials is generally not obvious.

2.6. COSMO-RS theory [64]

The COSMOtherm program is based on COSMO-RS theory of interacting

molecular surface charges [65, 66]. COSMO-RS is a theory of interacting molecular

surfaces as computed by quantum chemical methods (QM). COSMO-RS combines an

electrostatic theory of locally interacting molecular surface descriptors - which are

available from QM calculations - with a statistical thermodynamics methodology.

The quantum chemical basis of COSMO-RS is COSMO [67], the ”Conductor-

like Screening Model”, which belongs to the class of QM continuum solvation models

(CSMs). In general, basic quantum chemical methodology describes isolated molecules

at a temperature of T=0 K, allowing a realistic description only for molecules in vacuum

or in the gas phase. CSMs are an extension of the basic QM methods towards the

description of liquid phases. CSMs describe a molecule in solution through a quantum

chemical calculation of the solute molecule with an approximate representation of the

surrounding solvent as a continuum. Either by solution of the dielectric boundary

condition or by solution of the Poisson-Boltzmann equation, the solute is treated as if

embedded in a dielectric medium via a molecular surface or “cavity” that is constructed

around the molecule. Hereby, normally the macroscopic relative permittivity of the

solvent is used. COSMO is a quite popular model based on a slight approximation,

which in comparison to other CSMs achieves superior efficiency and robustness of the

computational methodology [68]. The COSMO model is available in several quantum

chemistry program packages. First what I have to mention is PQS [69] because it has

Hungarian origin by Prof. Pulay. Others, such as Turbomole [70], Gaussian [71] and

GAMESS-US [72] are also important. If combined with accurate QM CSMs have been

proven to produce reasonable results for properties like Henry law constants or partition

coefficients. However, as has been shown [73] the continuum description of CSMs is

based on an erroneous physical concept. In addition, concepts of temperature

dependency and mixing are missing in CSMs.

COSMO-RS, the COSMO theory for “real solvents” goes far beyond simple

CSMs in that it integrates concepts from quantum chemistry, dielectric continuum

10

models, electrostatic surface interactions and statistical thermodynamics. Still,

COSMO-RS is based upon the information that is evaluated by QM-COSMO

calculations. Basically QM-COSMO calculations provide a discrete surface around a

molecule embedded in a virtual conductor [67]. Of this surface each segment i is

characterized by its area ai and the screening charge density (SCD) σi - illustrated on

Figure 1. - on this segment which takes into account the electrostatic screening of the

solute molecule by its surrounding (which in a virtual conductor is perfect screening)

and the back-polarization of the solute molecule.

Figure 1 Visualization of COSMO screening charges on molecular surfaces of n-hexane and p-xylene.

In addition, the total energy of the ideally screened molecule ECOSMO is provided. Within

COSMO-RS theory a liquid is now considered an ensemble of closely packed ideally

screened molecules. In order to achieve this close packing the system has to be

compressed and thus the cavities of the molecules get slightly deformed (although the

volume of the individual cavities does not change significantly). Each piece of the

molecular surface is in close contact with another one. Assuming that there still is a

conducting surface between the molecules, i.e. that each molecule still is enclosed by a

virtual conductor, in a contact area the surface segments of both molecules have net

SCDs σ and σ’. In reality there is no conductor between the surface contact areas. Thus

an electrostatic interaction arises from the contact of two different SCDs. The specific

interaction energy per unit area resulting from this “misfit” of SCDs is given by

= !" + # (4)

where aeff is the effective contact area between two surface segments and α’ is an

adjustable parameter. The basic assumption of eq. (4) - which is the same as in other

surface pair models like UNIQUAC [74] - is that residual non-steric interactions can be

described by pairs of geometrically independent surface segments. Thus, the size of the

11

surface segments aeff has to be chosen in a way that it effectively corresponds to a

thermodynamically independent entity. There is no simple way to define aeff from first

principles and it must be considered to be an adjustable parameter. Obviously, if σ

equals -σ’ the misfit energy of a surface contact will vanish. Hydrogen bonding (HB)

can also be described by the two adjacent SCDs. HB donors have a strongly negative

SCD whereas HB acceptors have strongly positive SCDs. Generally, a HB interaction

can be expected if two sufficiently polar pieces of surface of opposite polarity are in

contact. Such behaviour can be described by a functional of the form

$% = &$%'() 0;min/0; 0102 + $%'340; 5 02 + $%6 (5)

wherein cHB and σHB are adjustable parameters. In addition to electrostatic misfit and HB

interaction COSMO-RS also takes into account van der Waals (vdW) interactions

between surface segments via

78 = 978 + 978# (6)

wherein τvdW and τ’vdW are element-specific adjustable parameters. The van der Waals

energy is dependent only on the element type of the atoms that are involved in surface

contact. It is spatially non-specific. EvdW is an additional term to the energy of the

reference state in solution. Currently nine of the vdW parameters (for elements H, C, N,

O, F, S, Cl, Br and I) have been optimized. For the majority of the remaining elements

reasonable guesses are available. Figure 2 shows the “misfit” and hydrogen bonding

types interactions incorporated in COSMO-RS theory, however, vdW interaction cannot

be visualized.

12

Figure 2 Visualization of incorporated molecular interactions in COSMO-RS theory (EvdW cannot be

visualized)

The link between the microscopic surface interaction energies and the macroscopic

thermodynamic properties of a liquid is provided by statistical thermodynamics. Since

in the COSMO-RS view all molecular interactions consist of local pair wise interactions

of surface segments, the statistical averaging can be done in the ensemble of interacting

surface pieces. Such an ensemble averaging is computationally efficient - especially in

comparison to the computationally very demanding molecular dynamics or Monte Carlo

approaches which require averaging over an ensemble of all possible different

arrangements of all molecules in a liquid. To describe the composition of the surface

segment ensemble with respect to the interactions (which depend on σ only), only the

probability distribution of σ has to be known for all compounds Xi. Such probability

distributions pX(σ) are called “σ-profiles” . The σ-profile of the whole system/mixture

pS(σ) is just a sum of the σ-profiles of the components Xi weighted with their mole

fraction in the mixture xi.

: = ; 3:<=> (7)

Using e(σ,σ’)=(EvdW(σ,σ’) + EHB(σ,σ’) + Emisfit(σ,σ’))/aeff , the chemical potential of a

surface segment with the SCD σ in an ensemble described by normalized distribution

function pS(σ) is given by

? = − @A5BCC D) EF :

#G3: H5BCC@A ?# − # − $% #I J#K (8)

where µS(σ) is a measure for the affinity of the system S to a surface of polarity σ. It is a

characteristic function of each system and is called “σ-potential”. The µS(σ’) is

13

integrated over the complete σ-range, which includes σ of the equation's left hand side.

Eq. (8) is an implicit equation and must be solved iteratively. This is done in

milliseconds on any PC with 2 GHz processor.

The COSMO-RS representations of molecular interactions namely the σ-profiles and σ-

potentials of compounds and mixtures, respectively, contain valuable information -

qualitatively as well as quantitatively. The chemical potential (the partial Gibbs free

energy) of compound Xi in system S is readily available from integration of the σ-

potential over the surface of Xi:

?<= =/?<= + F:<=?J (9)

where µXi

C,S is a combinatorial contribution to the chemical potential. Starting with

version C1.2, the COSMOtherm program includes a new generic expression for the

combinatorial contribution to the chemical potential. The new combinatorial

contribution µXi

C,S results from the derivation of the combinatorial free energy

expression GC,S:

LMN = OPQR; 3 ln T −/RU ln; 3T − R ln; 3V W (10)

The combinatorial contribution µXiC,S to the chemical potential of compound i is:

μMN<= = YZ[\Y= = OP ER ln T +/RU 1 − 2=

2 − ln T + R 1 − _=_ − ln VK (11)

In eq. (11), ri is the molecular volume and qi is the molecular area of compound i. The

total volume and area of all compounds in the mixture are assumed additive:

T = ; 3T (12)

V = ; 3V (13)

The combinatorial contribution µXiC,S eq. (11) contains three adjustable parameters λ0, λ1

and λ2. The µXi

C,S can be replaced with zero, which is useful if compounds are in

question do not have a well-defined surface area and volume such as polymers or

amorphous phases. The chemical potential of eq. (9) is a pseudo-chemical potential

[75], which is the standard chemical potential minus RT ln(xi). The chemical potential

µXi

S of eq. (9) allows for the prediction of almost all thermodynamic properties of

compounds or mixtures, such as activity coefficients, excess properties or partition

coefficients and solubility. In addition to the prediction of thermodynamics of liquids

COSMO-RS is also able to provide a reasonable estimate of a pure compound’s

chemical potential in the gas phase

μZ5<= = Z5<= − MaNba<= − c@1d)@1d<= +/eZ5 (14)

14

where EXiGas and EXi

COSMO are the quantum chemical total energies of the molecule in the

gas phase and in the COSMO conductor, respectively. The remaining contributions

consist of a correction term for ring shaped molecules with nXiRing being the number of

ring atoms in the molecule and ωRing an adjustable parameter as well as parameter ηGas

providing the link between the reference states of the system’s free energy in the gas

phase and in the liquid. Using eqs. (9) and (13) it is possible to a priori predict vapour

pressures of pure compounds. COSMO-RS based on an extremely small number of

adjustable parameters (the seven basic parameters of eq. (4)-(7), (11) and (13) plus nine

τvdW values) some of which are physically predetermined. COSMO-RS parameters are

not specific of functional groups or molecule types. The parameters have to be adjusted

for the QM-COSMO method that is used as a basis for the COSMO-RS calculations

only. Thus the resulting parameterization is completely general and can be used to

predict the properties of almost any imaginable compound mixture or system.

COSMO-RS theory provides also an alternative way to connect molecular and

thermodynamic levels. The moments of the screening charge density distribution

function, presented on Figure 3, the σ-moments, are stated [73] as excellent linear

descriptors derived from theory for regression function relating important material

characteristics (P) to molecular properties:

f = g + gU ∙ i< + g ∙ iU< + gj ∙ i< + gk ∙ ij< + g ∙ ik< + gl ∙ i<

+gm ∙ il< + gn ∙ i$o5U< + gp ∙ i$o5< + gU ∙ i$o5j< + gUU ∙ i$o5k<

+gU ∙ i$o01U< + gUj ∙ i$o01< + gUk ∙ i$o01j< + gU ∙ i$o01k< (15)

where MXi is the ith σ-moment.

15

σ(e/A)

-0,03 -0,02 -0,01 0,00 0,01 0,02 0,03

p( σ

)

0

5

10

15

20

25

30

Bmim cation

BF4 anion

BmimBF4

Figure 3 Screening charge distributions functions of an ionic liquid component and ion pair of 1-butyl-3-

methylimidazolium tetrafluoroborate ([bmim]BF4).

The coefficients (C0-C15) can be derived by multiple regression of the σ-moments with a

sufficient number of reliable experimental data. Some of the 15 σ-moments have a well-

defined physical meaning (e.g. surface area of the molecule: MX0 = M

Xarea, total charge:

MX

1 = MX

charge, electrostatic interaction energy: MX2 = M

Xel, the kind of skewness of the

σ-profile: MX

3 = MX

skew, and acceptor and donor functions: MX

Hbacc1-4, MX

Hbdon1-4, but

some of them (MX4, M

X5, M

X6) do not have simple physical interpretations [73].

The σ-moment approach has been successfully applied to such diverse problems as

olive oil-gas partitioning, blood-brain partitioning, intestinal absorption and soil-

sorption [76].

16

3. Calculation of model elements

3.1. Quantum chemical and COSMO calculations

The course of my COSMO-RS calculations, which are carried out for modelling

of both evaporation and HSPs is illustrated in Figure 4. The starting point is always a

QM-COSMO calculation. However, the time-consuming QM-COSMO calculations

have to be done only once for each compound. The results of the QM-COSMO

calculations (i.e. the charge distribution on the molecular surface) can be stored in a

database. Databases of COSMO files are available at commercial vendors or can be

created according to individual claims. COSMO-RS then can be run from a database of

stored QM-COSMO calculations. For molecules which are not in the database,

geometry optimization and COSMO calculation have to be done. The 3D structures of

molecules or ion-pairs for amine/acid associates and ionic liquids were built by using

GaussView 3.09. The raw 3D structures were exported in Sybyl Mol2 file format to

OpenBabel version 2.2.3 and were converted to Cartesian XYZ format. Molecular

geometries were optimized by TURBOMOLE 6.0 quantum chemical software package

[77]. The amine/acid associates and ionic liquid were considered as neutral ion-pairs,

since charged species cannot be observed without the presence of counter ions, and

measured HSP parameters were defined and reported for bulk phases and not for

individual ions.

17

Molecular

structure

Quantumchemical COSMO

calculation

Energy and screening

charge distribution on

molecular COSMO-surface

Database of

COSMO-files

σ-profile of compounds (COSMO-RS)

Fast statistical

Thermodynamicsσ-potential of the mixture

Activity coefficient of the compounds of the system

Figure 4 Flowchart of a COSMOtherm calculation.

Because the quality, accuracy, and systematic errors of the electrostatics resulting from

the underlying COSMO calculations depend on the quantum chemical method as well

as on the basis set, COSMOtherm needs a special parameterization for each method and

basis set combination. All of these parameterizations are based on molecular structures

18

quantum chemically optimized at the given method and basis set level. The application

of COSMOtherm in chemical engineering and for thermodynamic calculations -

calculation of activity coefficient belongs to both - typically requires high quality

property predictions for mixtures. For such a problem the necessary quantum chemical

level is BP-RI-DFT-COSMO optimization of the molecular structure using the large

TZVP basis set [60]. The molecular electronic energy is computed based on the

accurate prediction of the electron probability density using Density Functional Theory

(DFT) [78]. DFT offers theoretical solution for electron density in a molecular system

but it does not define its geometry or the electronic boundary. Electronic boundaries are

defined with the so called basis sets. A basis set is a collection of vectors that is used to

specify the space where electron density is computed. The mathematical function in the

basis set is a linear combination of one electron basis function centered on the atomic

nuclei. During my quantum chemical computation, the triple zeta polarized valence

(TZVP) basis set was used. The advantage of such a basis set is the three basis functions

for each atomic orbital. If atoms of different sizes are getting close to each other, the

TZVP basis set will allow the orbital to get bigger or smaller. Another advantage of

TZVP is its polarized function that adds orbitals with angular momentum beyond the

atomic limitations [79]. The RI i.e. RI-J approximation is an expansion of the density in

the basis of the Coulomb energy orbital [80]. BP stands for B-P86 DFT functional,

which is a combination of the gradient-corrected exchange-energy functional proposed

by Becke and of the gradient-corrected correlation-energy functional proposed by

Perdew in 1986.

However, because positive and negative charges even in organic salts and ionic liquids

compensate each other, the quantum chemical calculations are restricted to

electronically neutral chemical entities with a total net charge of zero, therefore the first

σ-moment vanishes in Eq. (15) in both models.

19

3.2. The vapour-liquid equilibrium model

At normal conditions, during the evaporation of pools or large drops with initial

diameter of 1.5 mm, the assumption that the concentration is over the liquid phase is

determined by the vapour-liquid equilibrium, is justified. Consequently, the gas phase

concentrations of the components over the liquid phase are determined by vapour-liquid

equilibrium eq. (16), assuming ideal vapour and real liquid phase, neglecting the

Poynting factor correction.

q = =/r=/s=∗su (16)

where yi is the mole fraction of component i in the gas phase, xi is the mole fraction of

component i in the liquid phase, Pi* is the vapour pressure of component i at system

temperature, Pt is the total (equilibrium) pressure and γi designates the activity

coefficient of component i, which is calculated by COSMO-RS theory.

3.3. Calculation of the vapour pressure of the components

The vapour pressure of pure compounds plays important role in the evaporation

modelling. The temperature function of vapour pressure of pure liquid i, the Pi* is

usually given by an Antoine-type equation determined on the basis of experimental

data:

logf∗/fy = z − %|AM| (17)

where AA, BA and CA stand for Antoine-constants, Pi* is the vapour pressure of the liquid

i at temperature T and Pu is the unit of pressure. If the Antoine-constants are not known

they can be calculated using at least five measured vapour pressure data points of

compound i.However, if the experimentally determined five vapour pressure data points

are also not available, it is possible to estimate the necessary vapour pressure data at

various temperature points of compounds by the method of Thek-Stiel from the normal

boiling point [81]:

D)f2∗ = zAN H1.14893 − 1P2 − 0.11719P2 − 0.03174P2 − 0.375D)P2I +

+1.042 − 0.46284zAN ∙ A..|\.\U

.lpU.mj\j.Umjn\ + 0.04 UA − 1 (18)

where P*r = P*/Pc is the reduced vapour pressure, Tr = T/Tc is the reduced temperature,

Tbr = Tb/Tc is the reduced normal boiling temperature, pressure is in mmHg, temperature

is in °C and ATS and hTS are parameters defined as follows:

20

zAN = ∆$@AUA. (19)

ℎAN = Po2 1sUA (20)

The molar heat of evaporation at the normal boiling point, expressed in cal/(g·mol),

may be estimated using Chen’s method [82]:

∆7o = OPo j.pmnAj.pjnU.1sU.mA (21)

where Tb is the normal boiling point and Tbr is the reduced normal boiling point of the

component. The only one adjustable parameter in eq. (18) is αc, which can be

determined by a fitting procedure using the normal boiling point - normal vapour

pressure (101 325 Pa) data pair for compound i.

With the obtained αc Thek-Stiel equation can be used for calculation the necessary

vapour pressure and temperature pairs for getting the Antoine constants.

The variation of molar enthalpy of vaporization with temperature is estimated by

the Watson equation [82]:

∆7 = ∆7U UAUA

.jm (22)

where the enthalpy of vaporization at the normal boiling point is taken as reference

value. COSMOtherm is able to handle the Antoine equation constants for evaluating the

pure component vapour pressure at various temperatures via the *.vap approach,

therefore the above described method was used in my models. COSMOtherm has a

different option for estimating the pure components vapour pressure based on ab initio

calculations. In the lack of any experimental vapour pressure data this option can be

used as an alternative way. I have tested the vapour pressure estimation power of

COSMOtherm against experimentally measured vapour phase concentration values for

aroma compounds having different chemical characters. Figure 5 shows the comparison

of measured vs. calculated vapour phase concentration values. In a wide range of

vapour pressure values the COSMOtherm estimation results in R2 = 0.67 and S.D. =

2.19 ln (µg/l) unit statistical performance on 102 substances with diverse chemical

identity, which is not enough for an accurate prediction for one of the key parameters of

evaporation, but could be an alternative possibility if other methods do not work.

Shortly: better than nothing.

21

Measured ln P* [µg/l]

-4 -2 0 2 4 6 8 10 12

Estim

ate

d ln P

* [µ

g/l]

-8

-6

-4

-2

0

2

4

6

8

10

12

Figure 5 Comparison of experimentally determined vapour phase concentration values vs. COSMOtherm

prediction.

It is clearly visible on Figure 5 that COSMOtherm is not capable to estimate the vapour

pressure of pure compounds accuracy. The ordinate has logarithmic scale, which means

that the vapour pressure of outlier molecules is far from the measured ones. Outliers

have different chemical entity, so domain of application can not be defined exactly. It is

likely that in case of outlier molecules some intermolecular interactions pay important

role, which are not taken into account in the parameterization of COSMO-RS theory.

3.4. Calculation of evaporation flux and transport in gas phase

The flat-surface evaporation model is based on the Maxwell-Stefan diffusion

and convection theory for departure of the particles from the evaporation surface (Fig.

6). The simplified mass transport model describes a process in which the vapours

evaporated from the surface of the liquid phase, are transported by coupled diffusion

and convection to the top of the modelled domain (vessel). The demonstration case for

Maxwell-Stefan diffusion phenomenon is the so called Stefan tube, depicted in Figure 6

(a), is a simple device generally used for measuring diffusion coefficients in binary

vapours.

22

Figure 6 2D and 1D sketches of the Stefan tube.

At the bottom of the tube there is a pool of mixture to evaporate. The vapour that

evaporates from this pool diffuses to the top of the tube, where a stream of air, flowing

across the top of the tube, keeps the mole fraction of diffusing vapour there to be zero.

The mole fraction of vapour above the liquid interface assumed to be in equilibrium.

Because not assumed horizontal flux inside the tube, it is possible to analyse the

problem using a 1D model [83] (see Fig. 6b).

To account for such important phenomena, i.e. the cross-effects, a

multicomponent diffusion approach is required. The standard approach in the theory of

Irreversible Thermodynamics replaces Fickian fluxes by linear combinations of the

gradients of all involved concentrations, respectively chemical potentials. This requires

the knowledge of a full matrix of binary diffusion coefficients and this diffusivity

matrix has to fulfill certain requirements like positive semi-definiteness in order to be

consistent with the fundamental laws from thermodynamics [44]. The Maxwell-Stefan

equations are successfully used in engineering applications, however, the calculation of

the diffusivity matrix is quite complex as well as their experimental determination. At

regular pressure, multicomponent diffusion coefficients can be replaced with Fick-

analogous binary diffusion coefficients, which latter can be estimated by the method of

Wilke and Lee [84]. According to the investigation of Jarvis and Lugg [85] this method

has 4.3% absolute average error tested for about 150 compounds. The binary diffusion

coefficient, Di,j is calculated as:

= Ej.j.pn/b= .K∙UA.b= .¡= ¢£

(23)

23

where Di,j is expressed in cm2/s. The diffusion collision integral, ΩD, is a function of the

characteristic energy (εi,j)

¤¥ = U.ljlA∗. +

.Upj .kmljA∗+

U.jmn U.pplA∗+

U.mlkmk j.npkUUA∗ (24)

where T*=kT/εj. The mean molar mass (Mij), Lennard-Jones characteristic energy (εI,j)

and length values (σij) for i-j mixture are given by the expressions:

i = 2H Ub=

+ Ub IU

(25)

¦ = 4¦¦6. (26)

= ¡=§¨¡ §¨ (27)

The Lennard-Jones scale parameter can be estimated as follows

©ª = 1.18«U/j (28)

where Vm is the liquid molar volume of i at the boiling point, and

¦/¬ = 1.15Po (29)

where εi/k is the Lennard-Jones energy parameter. Because eq. (23) contains empirical

constants, values should express as: Mi in g/mol, Vm in cm3/mol, εi/k in K and σa in Å.

For the estimation procedures listed above the knowledge of the critical data (Tc, Pc) is

necessary. In the lack of experimental data, the critical properties of the compounds can

be predicted by the method of Joback [82] from molecular structure. Because of the

temperature dependency of diffusion coefficient, the recalculation of Maxwell-Stefan

diffusional matrix is necessary for non-isothermal modelling.

24

4. Flat surface evaporation model

4.1. Model description

A model for flat surface evaporation of multi-component real liquid mixture has

been developed. The model is based on activity coefficient calculation from theoretical

chemical structures alone, and it is completely independent of any experimental VLE

data and of any group interaction parameters of the regarded compounds.

The model applies the so-called fractional evaporation method and assumes that:

• chemical reactions do not occur between the species,

• the liquid phase is perfectly mixed,

• side effects can be neglected,

• the solubility of air in the liquid is negligible,

• the whole process takes place under ambient pressure and isothermal conditions,

• the gas phase is ideal, and

• the components are additive.

Due to the mass lost caused by evaporation, the composition of the liquid phase, and

therefore the activity coefficients of the components will permanently change during the

evaporation process. This continuous changing is approached by fractional,

discontinuous evaporation steps during discrete (quanted) execution time intervals (∆t)

using iterative calculation methods. The amount of substance lost of component i, ∆ni,t,

is a function of the evaporation flux, jt, the size of the evaporating surface, A, and the

processing time quantum, ∆t:

∆) = z∆® (30)

The current amount of component i in the liquid phase can be given by:

) = )∆ − ∆) (31)

The momentary liquid phase composition of component i can be calculated by:

3 = 1=u;1 u (32)

where xi,t is the mole fraction of the component i in the liquid phase at time moment t.

The knowledge of the evaporation fluxes of the components at the evaporating surface

allows to apply them in eq. (30) and to calculate the amount of substance lost during the

time steps of the fractional evaporation process. Computational Fluid Dynamics (CFD)

approach was used to obtain particle fluxes by COMSOL Multiphysics commercial

25

software package [83]. Vapour-liquid equilibrium is assumed on the evaporating surface

and eq. (16) is used to calculate the mole fraction yi of component i in the gas phase, see

chapter 3.2.

The flowchart of the calculation procedure is illustrated in Figure 7. Each time

step has a different composition so at each time step have to call COSMOtherm for the

calculation of activity coefficients of compounds and COMSOL Multiphysics for

estimation of particle flux. The repetitions of calculation steps were continued until the

total evaporation with acceptable computational demand.

Figure 7 Flowchart of the flat surface evaporation model.

Because the preliminary CFD calculations have shown that the effects caused by edges

on the evaporation flow are negligible and the liquid phase was assumed to be well

26

stirred, a time sparing and effective strategy has been developed for evaporation

simulation using a 1D model, see Figure 5 (b). The simplifying of the schematic 2D

evaporation domain into 1D simulation model results in the following boundary and

initial condition for solving the governing equation of Maxwell-Stefan diffusion

phenomena by COMSOL Multiphysics. Boundary 1 refers to the liquid-vapour

interface, where the concentration of evaporating components is given by eq. (16).

Boundary 2 symbolizes the vapour-air flow interface, where the concentrations of the

evaporated components (expressed in mole fraction) are fixed to zero. For the liquid

phase, a real liquid mixture approach is applied in eq. (16) based on activity coefficient

calculation by COSMOtherm. The initial concentrations - which are required for a

solution of a partial differential equation (PDE) problem - were the same for all

components, as the initial equilibrium concentrations in the full simulation domain.

Subroutine of the model are written in MATLAB [86], which beyond the modelling

calculations, controls and harmonizes the external software such as COSMOtherm and

COMSOL Multiphysics. A steady-state simulation can be carried out on a common

laptop with 2 GHz processor, and takes approximately one hour time, depending on the

time resolution (duration of one evaporation step, ∆t) and the number of compounds.

4.2. Test calculations of flat surface evaporation model

Model validation is possibly one of the most important steps in the model

development. Validation examines the agreement between simulated and experimental

(which also can be loaded with errors) results. Depending on the aims and opportunities

the developed model can be accepted or submit for further improvement. Therefore, the

developed flat surface evaporation model has been tested against experimental data

from literature [11, 87].

4.2.1. Testing of 1D evaporation model for a binary liquid mixture

The binary evaporating liquid system of acetone and methanol in air has been

extensively investigated, measuring both diffusion coefficients and composition at

various positions within Stefan tubes. This makes it an ideal model to valid the CFD

code and the solution algorithm with independent measured data from literature [87].

For such multicomponent system, Maxwell-Stefan equation stands for the concentration

gradient of compound i at isothermal conditions:

27

Y=Y = ; 4=¯ =6

¥= 1°U (33)

where Dji is the Maxwell-Stefan diffusion coefficients, ci is the concentration of

compound i, s is the space dimension, n is the number of components, x is the mole

fraction and j is the molar flux. Equation (33) can be solved with numerical calculation

procedure based on the Finite Element Method (FEM). The simulation results for the

model system are shown in Figure 8, where steady-state mole fractions of acetone (-),

methanol (…) and air (---) in the gas phase are plotted as a function of the distance from

the liquid surface.

Distance [m]

0,00 0,05 0,10 0,15 0,20 0,25

Mole

fra

ctio

n [

1]

0,0

0,2

0,4

0,6

0,8

1,0

Calculated - Acetone

Calculated - Methanol

Calculated - Air

Experimental - Acetone

Experimental - Methanol

Experimental - Air

Figure 8 Simulation results for 1D evaporation of acetone + methanol system in a Stefan tube filled with air.

It is clearly visible on Figure 8 that the simulated values from the Maxwell-Stefan

diffusion model agree well with the measured ones in the domain where experimental

data are available. It means that the Maxwell-Stefan equation can describe the mass

transport over the surface of the evaporating liquid system.

28

4.2.2. Testing of 2D evaporation model for multicomponent liquid mixtures

The good predictive ability of the simulation model was tested on experimental

evaporation data of 2-5 components mixtures. The measured evaporation data are taken

from the paper by K. Okamoto et al. [11] for equimolar mixtures of five aliphatic and

aromatic hydrocarbons: n-pentane, n-hexane, n-heptane, toluene and p-xylene. The test

sets of mixed solvents consist of 6 liquid mixtures of 2-5 components systems: n-

pentane and n-hexane; n-pentane and n-heptane; n-pentane and toluene; n-pentane, n-

hexane and n-heptane; n-pentane, n-hexane and toluene; n-pentane, n-hexane and

toluene; n-pentane, n-hexane, n-heptane toluene and p-xylene). Evaporation rate was

measured as mass loss by using an electronic balance (Sartorius - CP4202S) with an

accuracy of 0.01 g. A tarred square pan (base area: 0.1 m2) was loaded on the balance,

and a liquid was poured into the tray, and then the weight loss was measured. The data

were recorded on a PC every 10 seconds until the mass loss fraction reached 0.7. The

measurements were conducted under a fume hood. The fume hood fan was not

operated, and a liquid sample was evaporated under no wind condition. The evaporation

rates were measured at temperature of 293 K.

For the simulations the vapour pressures of pure components at 293 K have been taken

from the database of Thermodynamics Research Center [88] and are given in Table 1.

These data are used for the calculation of Antoine parameters, for making the model

more flexible and applicable at diverse temperatures.

Table 1 Experimental vapour pressures of test compounds used in evaporation simulation at 293 K

Name p* [kPa]

n-Pentane 56.1

n-Hexane 16.1

n-Heptane 4.7

Toluene 2.9

p-Xylene 0.9

The estimated Maxwell-Stefan diffusivity matrix, which is required for the modelling of

gas phase transport, is reported in Table 2 for the most challenging five component

mixture. The matrix is symmetric; therefore only the elements above the diagonal are

reported.

29

Table 2 Estimated Maxwell - Stefan diffusion coefficients for five-component mixture contains n-pentane, n-

hexane, n-heptane, toluene, p-xylene at 293 K and atmospheric pressure (1 bar)

Component air n-pentane n-hexane n-heptane toluene p-xylene

air - 7.22E-06 6.36E-06 5.72E-06 5.72E-06 5.92E-06

n-pentane - - 4.79E-06 4.24E-06 4.24E-06 4.37E-06

n-hexane - - - 4.04E-06 4.73E-06 4.16E-06

n-heptane - - - - 4.56E-06 4E-06

toluene - - - - - 4.09E-06

p-xylol - - - - - -

Diffusion coefficients DAB [m s-2

]

Using the estimated activity coefficients values and the vapour pressure data of the pure

components, the evaporation process of the selected liquid mixtures were simulated (see

Fig. 7) and the evaporation flux as a function of the weight loss during the evaporation

has been calculated. The evaporation flux Φ is defined as the rate of the evaporating

mass flow across a unit area, i.e. the mass lost from the crucible per unit surface in a

time unit (kg·m-2·s-1) due to the evaporation. The mass loss fraction α is given as the

ratio of the evaporated mass to the initial mass of liquid sample w0.

= ±±±

(34)

The calculated evaporation flux values were compared with the measured data of the

mixtures of 2-5 components systems. Figures 9-11 show the comparison of the

calculated and experimental [11] evaporation fluxes as a function of the mass loss

fraction of three two-component mixtures: n-pentane and n-hexane; n-pentane and n-

heptane; n-pentane and toluene. It can be concluded that the estimated evaporation

fluxes agree well with the experimental ones except for the starting phase (α < 0.1) of

the vaporization of n-pentane and n-heptane mixture.

30

α [1]

0,0 0,2 0,4 0,6 0,8 1,0

Φ [

(kgm

-2s

-1]

0,0

0,1

0,2

0,3

0,4

0,5

Calculated

Measured

Figure 9 Comparison of the calculated (―) and experimental () evaporation flux values as a function of the

weight loss fraction of a two-component mixture of n-pentane and n-hexane.

On Figure 9 systematic deviation can be observed between the trends of calculated and

measured curves, which could be probably due to the error of the estimated activity

coefficients of compounds. Additionally, binary diffusion coefficients in the Maxwell-

Stefan diffusivity matrix are estimated values, which could be loaded also with errors.

The errors of the two estimation methods could have significant effect on the calculated

evaporation profiles. Thus, the exact reason of the model deviation could not be clearly

discussed.

31

α [1]

0,0 0,2 0,4 0,6 0,8 1,0

Φ [

(kgm

-2s

-1]

0,0

0,1

0,2

0,3

0,4

0,5

0,6

Calculated

Measured


weight loss fraction of a two-component mixture of n-pentane and n-heptane.

α [1]

0,0 0,2 0,4 0,6 0,8 1,0

Φ [

(kgm

-2s

-1]

0,0

0,1

0,2

0,3

0,4

Calculated

Measured


weight loss fraction of a two-component mixture of n-pentane and toluene.

32

Figures 12 and 13 compare the predicted and measured evaporation flux values during

the evaporation of three-component mixtures containing n-pentane, n-hexane and n-

heptane; n-pentane, n-hexane and toluene components. It is clearly seen that the

equilibrium evaporation model cannot describe properly the initial period (α < 0.1) of

the evaporation of the mixtures, where probably non-equilibrium conditions dominate.

However, the agreement between the calculated and obtained evaporation flux data is

appropriate in the significant part of the evaporation of mixtures, which makes probably

that after a short onset interval the vaporization is governed by quasi-equilibrium

parameters.

α [1]

0,0 0,2 0,4 0,6 0,8 1,0

Φ [

(kgm

-2s

-1]

0,0

0,1

0,2

0,3

0,4

0,5

Calculated

Measured


weight loss fraction of a three-component mixture of n-pentane, n-hexane and n-heptane.

33

α [1]

0,0 0,2 0,4 0,6 0,8 1,0

Φ [

(kgm

-2s

-1]

0,0

0,1

0,2

0,3

0,4

0,5

Calculated

Measured


weight loss fraction of a three-component mixture of n-pentane, n-hexane and toluene.

The simulation of the evaporation process of the five-component mixture (n-pentane, n-

hexane, n-heptane, toluene and p-xylene) can be considered as a challenging test for the

model because of the high differences between the volatilities of the components and

the corresponding continual changes of the gas phase and liquid phase compositions as

evaporation proceeds, as shown in Figure 14. The mole fraction of volatile components

(n-pentane and n-hexane) quickly decreases in the liquid phase. The calculations

indirectly confirm the presumption that the components evaporate from the mixture in

the order of their vapour pressures, as expected.

34

α [1]

0,0 0,2 0,4 0,6 0,8 1,0

xi [

1]

0,0

0,2

0,4

0,6

0,8

1,0

Pentane

Hexane

Heptane

Toulene

Xylene

Figure 14 Calculated changes in mole fractions of the components in the liquid mixture containing n-pentane,

n-hexane, n-heptane, toluene and p-xylene.

The continuous altering of the molecular environments around the molecules and their

molecular interactions makes absolutely necessary the recalculations of the activity

coefficients during the vaporization. Figure 15 presents the plots of the estimated

activity coefficients as a function of the mass loss fraction of a five-component mixture

contains n-pentane, n-hexane, n-heptane, toluene and p-xylene.

35

α [1]

0,0 0,2 0,4 0,6 0,8 1,0

γ i [1

]

1,0

1,1

1,2

1,3

1,4

1,5

1,6

Pentane

Hexane

Heptane

Toulene

Xylene

Figure 15 Calculated activity coefficients as a function of the weight loss fraction during the evaporation of the

five-component mixture containing n-pentane, n-hexane, n-heptane, toluene and p-xylene.

The estimated activity coefficients of the less volatile aromatics, toluene and p-

xylene converge nearly linearly to the unit as their concentrations increase in the

mixture. However, the activity coefficients and therefore the partial pressures of the

alkanes rise as their mole fractions decrease in the liquid phase due to their evaporation

from the liquid. The last points on the activity coefficients plots of the alkanes represent

the values of the limiting activity coefficients at infinite dilution and the endpoints of

their evaporation, where the molecules of the evaporated components disappear from

the mixture.

Using the proposed evaporation model one can predict the evaporation fluxes of

the individual components of the mixtures, which are illustrated in Figure 16.

36

α [1]

0,0 0,2 0,4 0,6 0,8 1,0

Φ [

(kgm

-2s

-1]

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

Pentane

Hexane

Heptane

Toulene

Xylene

Figure 16 Calculated evaporation fluxes of the individual components of the five-component mixture as a

function of the mass loss fraction during the evaporation.

The estimated order of the evaporation fluxes is in agreement with the expectation that

the components of higher volatility possess grater evaporation rate. Actually, the high

predicted evaporation fluxes of n-pentane and n-hexane dominate during the first period

of the vaporization of the multi-component mixture, while the estimated evaporation

fluxes for the medium or non-volatile components are nearly constant over a wide range

of the evaporation process. The non-ideal behaviour of the mixture (γ > 1) plays an

important role in the evaporation characters of the components: the estimated positive

deviations of the partial pressures from the Raoult’s law accelerate the volatilization of

the components as their concentrations converge to zero.

Figure 17 allows the comparison between the predicted and measured

cumulative evaporation flux values of five-component mixture of n-pentane, n-hexane,

n-heptane, toluene and p-xylene. The evaporation model presented in this work

describes properly the whole evaporation range of the mixture, except for a very short

non-equilibrium region at the beginning of the volatilization. Furthermore, the

simulation predicts correctly the small brake in the plot of the evaporation flux caused

by the end of the volatilization of n-pentane at weight loss fraction of about α ≈ 0.35.

37

α [1]

0,0 0,2 0,4 0,6 0,8 1,0

Φ [

(kgm

-2s

-1]

0,00

0,05

0,10

0,15

0,20

0,25

0,30

Calculated

Measured


weight loss fraction of the five-component mixture of n-pentane, n-hexane, n-heptane, toluene and p-xylene.

4.3. Summary of flat surface evaporation model

The developed flat surface evaporation model based on vapour-liquid

equilibrium theory of non-ideal solutions and the Maxwell-Stefan diffusion and

convection theory is appropriate tool to make quickly computational simulation for

investigation of evaporation of multi-component mixtures. The method is flexible

because just the so called .cosmo files and vapour pressure of compounds are required

for simulation. Despite the deviation of calculated and measured evaporation profiles,

the model can be characterized as realistic, since measuring techniques have significant

shakiness in case of more complex mixtures.

The model possesses acceptable predictive ability for quasi-equilibrium

evaporation characteristics of real liquids. The good simulation results are demonstrated

by comparing the estimated evaporation fluxes with the measured ones of several 2-5

components mixtures. However, it has to be noted, that the model cannot describe

properly the initial period (α < 0.1) of the evaporation of the mixtures, where probably

non-equilibrium conditions dominate.

The proposed evaporation model needs small number of input parameters.

However, it is also concluded that the model is sensitive to the reliable vapour pressure

38

data of pure compounds. The use of the flat surface evaporation model presented in this

work is an important tool by providing evaporation parameters (evaporation flux, mass

lost, liquid and gas phase composition, etc.) for real solvents in process design, in safety

engineering, in chemical, fuel, flavour and fragrance industries.

39

5. Droplet evaporation model

5.1. Model description

Droplet evaporation has more interest in the science than the flat surface

evaporation due to its importance in Otto- and Diesel engine design. True enough that it

is more complex than flat surface evaporation; therefore, numerous additional effects

should be taken into account.

In my doctoral work I developed a model for droplet evaporation which is

similar to the flat surface evaporation model applying the so called fractional

evaporation method. The model assumes that:

• chemical reactions do not occur between the species,

• the droplet (liquid phase) is perfectly mixed,

• the whole process takes place under ambient atmosphere,

• the droplet is perfectly spherical during the evaporation,

• the gas phase is ideal,

• the components are additive,

• the solubility of air in the liquid is negligible, and

• heat transfer by radiation is also negligible.

Beyond the so called fractional evaporation method, this model also applied the real

mixture approach and the Maxwell-Stefan diffusion theory. However, because of the

relatively small amount of the liquid to evaporate, droplets are subjected to cooling due

to the enthalpy change of vaporization. A second additional effect also origins from the

droplet shape, especially from the small droplet curvature; this is the so called Kelvin

effect.

During droplet evaporation, Maxwell-Stefan diffusion and convection theory are

used for describe the departure of the particles from the evaporation surface as it is

shown in Figure 18. The mass transport model describes a process in which the vapours

evaporated from the surface of the droplet, are transported by coupled diffusion and

convection to the ambient air.

40

Figure 18 The 3D and 1D sketches of the droplet evaporation model.

Boundary layer Thickness of the boundary layer is essential for application of Maxwell-

Stefan diffusion theory; however, it is very hard to find appropriate data in the literature

for the thickness of boundary layer. Biance et al. [89] reported a study where they

deduce that the film thickness is about 10 µm. In Ref. [90] the authors describe that the

film on the gas side of a gas-liquid interface is usually very thin ~ 100 µm. It is shown

in the paper of Bogdanic et al. [91] that the experimental data for the thickness of

boundary layer is 8 ± 2 µm. Averaged the available data, a boundary layer thickness of

28 µm is used in my evaporation model.

The flowchart of the calculation procedure of droplet evaporation model is

illustrated in Figure 19. Each time step has a different composition so at each time step -

means fractional evaporation step, outer iteration cycle in Figure 19 - have to call

COSMOtherm for the calculation of activity coefficients of compounds and COMSOL

Multiphysics for particle flux estimation similarly to the flat surface evaporation model.

Additionally, the cooling effect must be taken into account for calculation of rate of

evaporation, because the temperature at the droplet surface is less than the ambient

temperature and can be changed during the evaporation. Therefore, due to the iteration

procedure for surface temperature prediction, the vapour pressure, activity coefficient,

diffusion coefficient and particle flux calculations are also repeated in the inner iteration

cycle. The iteration procedure for surface temperature prediction is quite fast, it reaches

the termination criteria of εT = 0.1 °C about in ten cycles. The repetitions of calculation

steps were continued until the total evaporation with acceptable computational demand.

41

Figure 19 Flowchart of the droplet evaporation model.

Basically, there are three different approaches to calculate the change of droplet

temperature such as Infinite Conductivity Models (ICMs) [16-19, 27], Finite

Conductivity Models (FCMs) [92] and Effective Conductivity Models (ECMs) [93]. In

ICMs, due to the perfect mixed droplet, the temperature is changing in time, but not in

space, a global energy balance is used for temperature estimation of evaporating

42

surface. FCMs assume that temperature of the droplet is changing in time and as also in

radial direction (shell by shell) and the internal liquid circulation is ignored. ECMs are

the extended versions of FCMs, where internal liquid circulation is assumed.

During my doctoral work I developed a new method to estimate the droplet

temperature during the evaporation process, which belongs to ICMs. Considering

relatively large droplets, the assumption of perfectly mixed liquid phase inside of

droplets can be valid, because of the formed vortices due to the levitation technique

[18]. Assuming quasi-equilibrium conditions around the droplets, the temperature of the

droplet surface - and also the temperature of the whole droplet - can be established by

balancing the heat required for evaporation, the heat content of the droplet and the heat

gained by conduction from the warmer surrounding air to the droplet. For global energy

balance calculation in evaporating systems the evaporation enthalpy and specific heat

capacity data for components, furthermore thermal conductivity data for air are

necessary.

Experimentally determined latent heat of evaporation ∆Hv and specific heat

capacities cp data are reported in [94]. The latent heat of evaporation decrease steadily

with temperature, therefore, for other temperatures the reported values should be

corrected. The widely used correlation between ∆Hv and T is the Watson relation, see

eq. (22). Specific heat capacities for liquids do not vary significantly with temperature

[82] except for temperatures over Tr = 0.7 to 0.8, so cp can be assumed as constant in

temperature range of evaporation at atmospheric pressure. The heat required for

evaporation during a quanted time unit ∆t can be calculated with eq. (35)

²75 = ; ²)U ²7 (35)

where ∆ni,t is the amount of substance lost of component i. Gross heat content of the

droplet can be calculated as Ht = Σ (cp,i ni,t )Tt, where Ht is the current heat content of

the droplet at the given temperature Tt. The heat gained by conduction from the warmer

surrounding air to the droplet is calculated applying Fourier’s first law:

³01 = −¬´P (36)

where Qcond is the conducted heat and k is the thermal conductivity of air. This equation

can be solved also with COMSOL Multiphysics, and the resulted heat flux can be used

to estimate the heat balance of the evaporating droplet. For quantifying the thermal

behaviour of the droplet, the current heat content, the evaporation heat and the

conduction heat are balanced and the following equation is obtained:

43

P = PAuµu − ¶·¯¸¹$º»;»=41=uµu¹1=u6 (37)

Because of the heat of evaporation and partial vapour pressures of mixture components

depend on surface temperature, eq. (37) has to be solved by iteration.

The partial pressure of the vapour (Pd) at the surface of a small droplet with a diameter

of dp, is greater than the saturated vapour pressure (P*) over the flat surface of the

liquid, because of the surface tension at the liquid/gas surface (σLG) and can be

estimated by the Kelvin equation:

f = f∗exp/k¡§¿bÀ@A (38)

The Kelvin effect is significant only for particles with a diameter less than 0.1 µm.

Nevertheless, it is implemented into the droplet evaporation model, because it has effect

on one of the most important input parameters (vapour pressure of pure compounds) of

the model.

5.2. Tests of the droplet evaporation model

Prediction ability of the droplet evaporation model was tested against

experimental data. Measured evaporation data are taken from the paper by Brenn et al.

[27] for four challenging mixtures with diverse composition such as methanol, ethanol,

1-butanol, n-heptane, n-decane and water. Experiments were carried out using an

acoustic levitator to investigate the evaporation behaviour of single (individual) droplets

of multi-component liquids. The experimental setup of the levitator is depicted in Figure

20.

Figure 20 Experimental setup to measure droplet evaporation behaviour.

44

The transducer constantly emits sound waves at 56 kHz frequency, which produces a

quasi-steady pressure distribution in the resonator, with pressure nodes and antinodes.

The quantity of liquid mixture to be tested was taken into a microliter syringe and

introduced into the standing wave, thereby levitating the droplet. The levitated droplets

were back lighted by a white light source. Sharp images of the shadows of the droplets

were obtained through a CCD camera. The whole levitator was placed in an acrylic

glass box, where a controlled temperature of about 302 K ± 2 K and a relative humidity

of 2% or 3% were maintained throughout the experiments.

For my simulations the vapour pressures of pure components have been taken

from the database of Thermodynamics Research Center [88]. These vapour pressure

data are used for the calculation of Antoine parameters. The TRC vapour pressure data

of mixture components at 298 K are given in Table 3.

Table 3 Experimental vapour pressures of test compounds used in droplet evaporation simulation at 298 K

Name p* [kPa]

Methanol 16.809

Ethanol 7.8082

n-heptane 6.0523

1-butanol 0.84843

n-decane 0.18201

The Maxwell-Stefan diffusivity matrix is given in Table 4 for the most challenging five

component mixture. The matrix is symmetric; therefore only the elements above the

diagonal are presented.

Table 4 Estimated Maxwell - Stefan diffusion coefficients of five-component mixture containing methanol,

ethanol, 1-butanol, n-heptane and n-decane at 298 K and atmospheric pressure (1 bar)

Diffusion coefficients DAB [m s-2

]

Component air methanol ethanol 1-butanol n-heptane n-decane

air - 1.85E-05 1.43E-05 1.03E-05 8.27E-06 6.59E-06

methanol

- 8.36E-06 6.08E-06 4.95E-06 3.93E-06

ethanol

- 4.69E-06 3.81E-06 3.02E-06

1-butanol

- 2.72E-06 2.14E-06

n-heptane

- 1.73E-06

n-decane -

45

Using the estimated activity coefficients values and vapour pressures calculated by

Antoine equation, the evaporation process of droplets of the selected liquid mixtures

were simulated and the normalised droplet diameter as a function of time has been

calculated.

Figure 21 allows the comparison of the calculated (―) and experimental ()

evaporation profile of four-component droplet containing initially 20% methanol, 30%

ethanol, 30% 1-butanol and 20% n-heptane. After a short period at the beginning of the

evaporation (until the first 20 seconds) where the measured and calculated profiles run

together, the model slightly underpredicts the evaporation rate and therefore the

decrease of normalised diameter of droplet. It is probably due to the error of the

estimated activity coefficients. In the second half of the evaporation the model slightly

overpredicts the experimental evaporation rates.

Time [s]

0 20 40 60 80 100 120 140 160

(d/d

0)2

[-]

0,0

0,2

0,4

0,6

0,8

1,0

Measured

Calculated

Figure 21 Comparison of the calculated (―) and experimental () normalized droplet diameter changes as a

function of the time during the evaporation of four-component droplet containing initially 20% methanol,

30% ethanol, 30% 1-butanol and 20% n-heptane (Ts = 302 K, p= 1 bar).

Figure 22 compares the predicted and measured normalized droplet diameter

changes as a function of the time during the evaporation of the five-component mixture

containing initially 20 %(V/V) of methanol, ethanol, 1-butanol, n-heptane and n-decane.

It can be concluded that the estimated evaporation profile agrees well with the

46

experimental one. The model can properly describe the evaporation behaviour during

the droplet evaporation, even in the initial period. Due to the small volume to evaporate

and the optimal surface/volume ratio (the droplet is perfectly spherical during the

evaporation), the conditions can reach the equilibrium quickly. The total evaporation

time is also estimated well, which means that the suggested approach for evaporation of

droplets of multicomponent mixtures is able to describe this phenomenon. The curve

clearly exhibits the presence of various slopes in the evolution of the normalised

surface, which represent the influence of various components with different volatilities.

In Figure 22 three distinct slopes can be identified, which marks the evaporation of

various components. It can be concluded that during the evaporation of five components

mixture containing methanol, ethanol, 1-butanol, n-heptane and n-decane with relatively

high vapour pressures, vaporization is governed by quasi-equilibrium parameters.

Vapour pressures of compounds cover a wide range - two orders of magnitude -

therefore this mixture can be considered as a very challenging test for the model, which

is able for estimations with appropriate precision.

Time [s]

0 100 200 300 400

(d/d

0)2

[-]

0,0

0,2

0,4

0,6

0,8

1,0

Measured

Calculated


function of the time during the evaporation of five-component droplet containing initially 20-20 %(V/V) of

methanol, ethanol, 1-butanol, n-heptane and n-decane (Ts = 302 K, p= 1 bar).

47

Comparison of the calculated (―) and experimental () evaporation profile of

five-component droplets containing initially 30 %(V/V) methanol, 20 %(V/V) ethanol,

20 %(V/V) 1-butanol, 15 %(V/V) n-heptane, and 15 %(V/V) n-decane is shown in

Figure 23. Three distinct slopes can also be identified. The total evaporation time is

estimated perfectly, however, the model slightly overpredicts the evaporation rate in the

whole evaporation process.

Time [s]

0 100 200 300 400

(d/d

0)2

[-]

0,0

0,2

0,4

0,6

0,8

1,0

Measured

Calculated


function of the time during the evaporation of five-component droplets containing initially 30% methanol,

20% ethanol, 20% 1-butanol, 15% n-heptane, and 15% n-decane by volume (Ts = 302 K, p= 1 bar).

Figure 24 shows the comparison of the calculated (―) and experimental ()

evaporation behaviour of five-component droplets initially containing 20 %(V/V)

methanol, 10 %(V/V) ethanol, 10 %(V/V) 1-butanol, 40 %(V/V) n-heptane, and 20

%(V/V) n-decane. In the evaporation process of this mixture two different slopes can

be recognized. On the first one, until 70 seconds the measured and predicted

evaporation profiles run together, which means that the model predicts well the

evaporation of volatiles compounds. However, the evaporation rates of less volatile

components are slightly underestimated.

48

Time [s]

0 100 200 300 400

(d/d

0)2

[-]

0,0

0,2

0,4

0,6

0,8

1,0

Measured

Calculated


function of the time during the evaporation of five-component droplets containing initially 20% methanol,

10% ethanol, 10% 1-butanol, 40% n-heptane, and 20% n-decane by volume (Ts = 302 K, p= 1 bar).

Using the results of model calculations for the five-component mixture containing

initially 20-20 %(V/V) methanol, ethanol, 1-butanol, n-heptane and n-decane (Figure

22) it has been demonstrated that beyond the simulation of the total evaporation time

and normalised diameter, which are easily measurable, the model can also compute

such intermediate results, like the evolution of liquid phase mole fraction, activity

coefficients and droplet temperature, which are difficult to determinate. However,

calculation of these properties may help to understand the basics of the evaporation of

multicomponent mixtures.

Figure 25 shows the calculated change of mole fractions of compounds in the

liquid phase during the evaporation. According to the mole fractions three diverse

ranges can be identified. The first one keeps until 50 seconds, while the amounts of

most volatile compounds (methanol, ethanol and n-heptane) decrease quickly. In the

second phase - starts from 50 seconds and goes to 100 seconds - the mole fraction of 1-

butanol decreases in parallel with the increasing of n-decane content. In the last section,

after 100 seconds, the evaporation of n-decane dominates.

49

Time [s]

0 100 200 300 400

xL [

1]

0,0

0,2

0,4

0,6

0,8

1,0

methanol

ethanol

1-butanol

n-heptane

n-decane

Figure 25 Calculated changes in the mole fractions of the components in the liquid mixture containing initially

20-20 % (V/V) methanol, ethanol, 1-butanol, n-heptane and n-decane (Ts = 302 K, p= 1 bar).

The calculations confirm again the presumption that the components evaporate from the

mixture in the order of their vapour pressures. These three regions can also be observed

in Figure 26, which shows the temperature profile of the droplet during the evaporation.

50

Time [s]

0 100 200 300 400

Te

mp

era

ture

[K

]

260

270

280

290

300

310

Figure 26 Change of the evaporation temperature of a droplet, which contains initially 20-20 %(V/V)

methanol, ethanol, 1-butanol, n-heptane and n-decane (Ts = 302 K, p= 1 bar).

The evolution of the droplet temperature with time is rather complex, however, it is not

surprising. At the beginning of the evaporation process, the temperature of the mixture

decreases almost until the wet-bulb temperature of the equimolar mixture of methanol,

ethanol and n-heptane due to the quick evaporation of volatile components. The first

evaporation steep is followed by a slightly increasing temperature profile due to the

evaporation of 1-butanol. The last temperature section is almost constant and close to

the surrounding temperature, because of the smaller volatility of n-decane, which

evaporates in this range, see Figure 25.

The continuously altering of the molecular environments around the molecules and their

molecular interactions makes absolutely necessary the recalculations of the activity

coefficients during the vaporization. Figure 27 shows the plots of the estimated activity

coefficients of the compounds as the function of evaporation time. Based on the

predicted activity values, it can be clearly concluded that real mixture approach is really

necessary for modelling the evaporation of multicomponent systems containing

molecules of diverse chemical characters. Because of the application of the assumption

of vapour-liquid equilibrium in the model, activity coefficient values have as much

large effect on evaporation rates as pure component vapour pressures, considering Eq.

(16).

51

Time [s]

0 100 200 300 400

γ i [1

]

0

10

20

30

methanol

ethanol

1-butanol

n-heptane

n-decane

Figure 27 Calculated activity coefficients as a function of time during the evaporation of the five-component

mixture containing initially 20-20 %(V/V) methanol, ethanol, 1-butanol, n-heptane and n-decane.

5.3. Summary of the droplet evaporation model

The developed droplet evaporation model is based on vapour-liquid equilibrium

theory of non-ideal solutions and the Maxwell-Stefan diffusion and convection theory.

Test calculations are carried out with four 4-5 components mixtures and the results are

compared against measured data taken from the literature. It can be concluded

according to the test results, that the model is an appropriate tool to make flexible and

realistic estimation for the evaporation behaviour of not ventilated droplets of multi-

component mixtures.

The ability of calculation of some hardly measurable properties, such as the

changes of liquid phase mole fractions, vapour phase compositions, activity coefficients

or droplet temperature, makes the model a useful tool at many fields of engineering.

52

6. Estimation of Hansen Solubility Parameters

For special mixtures, which containing polymers or ionic liquids, the direct

COSMO-RS calculation of activity coefficient could result in unrealistic values. This is

due to the built in parameterization of COSMO-RS theory, which aspires to expand the

abilities of the model as general as possible. In this case, the so called σ-moment

approach can be an alternative possibility. Klamt [73] also proposed to apply σ-

moments as input independent variables in prediction models. Among others, an

estimation model for octanol-water partition coefficient, which uses σ-moment

approach, has been developed side by side of the direct thermodynamic calculation.

Application of the σ-moments in QSPR models as molecular descriptors provide the

possibility of the second-order parameterization of COSMO-RS for special classes of

compounds to get more realistic estimation results. Note, that using σ-moment as

descriptors has the disadvantage that no temperature dependence is available.

The Hansen solubility parameters are related to the molecular interactions of

compounds, which among others, depends on polarity and shape of molecules.

Consequently, the HSPs could be correlated by QSPRs using independent variables

connected to the intermolecular forces. Considering the physics behind the COSMO-RS

sigma function, it is tempting to use the σ-moments as QSPR descriptors to model

solubility as such, and to predict the Hansen solubility parameters in a more

deterministic/phenomenological way.

Multiple linear regression (MLR) were employed for generating my first

predictive models assuming that HSPs are directly related to a linear combination of the

σ-moments. However, the statistical results of MLRs, the low values of squared cross-

correlation coefficients and the high values of mean absolute errors of the fits for the

dispersion, polar and hydrogen bonding components of Hansen’s solubility parameters

indicate that the multilinear σ-moment approaches are not suitable for correlation of the

components of HSP. My observations agree well with those of Katritzky et al. [95], who

pointed out that the real world is rarely “linear” and most QSAR/QSPR relationships are

nonlinear in nature. These hidden nonlinearities between the property and the

descriptors can be detected and described by artificial or computational neural networks

(ANN, CNN) included in nonlinear approaches [96-101]. Therefore, in my doctoral

work I developed a novel method which can be applied for the prediction of Hansen

Solubility Parameters using COSMO-RS sigma-moments as molecular descriptors and a

53

non-linear modelling strategy. Thanks to the COSMO-RS theory, the models can be

used for prediction even if just the molecular structure is available and not the

synthetized compound.

6.1. Data and σ-moment sets for modelling

Development of a method for the prediction of HSPs was the aim of this

research activity, which generally applicable on various chemicals. The experimental

HSPs component values were taken from the official HSPs chemical database [49, 60]

and from selected references [55, 102-105]. A training/validation set of 128 molecules

with chemically diverse characters, including a wide range of molecular size,

complexity, polarity and hydrogen bond building ability (alkanes, alkenes, aromatics,

haloalkanes, nitroalkanes, amines, amides, alcohols, ketones, ethers, esters, acids,

organic salts, ionic liquids) was selected to cover a wide numerical range of HSPs

component values, i.e. δd values ranging from 14.3 to 24.7 MPa1/2, δp values ranging

from 0 to 29.2 MPa1/2, and δh values ranging from 0 to 35.1 MPa1/2. This represents a

challenging set because of the structural diversity, the several multifunctional groups

present in large molecules, organic salts and the ionic liquids. A test set consists of 17

compounds with various functional groups and polarity.

Two different groups have been determined according to the physical meaning

of the σ-moments.. First is the set of the five basic moments, the so called Klamt’s set,

and another one, which consists of all 14 σ-moments. In some extents, the σ-moment

approach has some similarities with the Abraham empirical solvation model [73, 106].

Table 5 shows the calculated five basic σ-moments for selected molecules, ion-pairs and

organic salts.

54

Table 5 Basic σ-moments for selected chemical entities calculated by COSMOtherm

Chemical entities MX0/nm2 MX

2 MX3 MX

Hbacc3 MXHbdon3

4-Amino-benzoic acid 1.667 113.3 -27.36 2.65 5.719 Benzene 1.214 27.81 -0.436 0 0 Benzoic acid 1.529 75.42 -15.91 1.34 3.938 [bmim]PF6 2.407 209.7 192.2 25.04 1.561 γ-Butyrolactone 1.169 64.98 40.34 2.647 0 Diethylethanolamine+acetic acid 2.25 145.2 111.2 13.95 2.361 Hexane 1.569 7.92 0.434 0 0 Ibuprofen 2.575 85.3 -9.78 1.34 3.942 Lactose 3.091 297.9 29.51 13.61 12.82 Na-benzoate 1.696 230.3 -170.9 12.08 0 Na-diclofenac 3.063 269.2 -156.7 14.2 0.527 Salicylic acid 1.585 79.77 -27.46 0.863 4.64 Tetrahydro-furfurylalcohol 1.395 64.9 48.03 4.995 0.618 Urea 0.911 122.7 16.32 8.096 5.416

6.2. Nonlinear QSPR model

The nonlinear QSPR models were developed with artificial neural networks.

Neural networks are composed of simple elements operating in parallel. These elements

are inspired by biological nervous systems. As in nature, the connections between

elements largely determine the network function. It is possible to train a neural network

to perform a particular function by adjusting the values of the connections (weights)

between elements.

Three-layered feed-forward networks with back-propagation training function

were chosen as nonlinear regression model using the Neural Network Toolbox 7 of

MATLAB 7.11.0.584 (R2010b) version [107] and an in-house developed MATLAB

routine for process automation. Two sets of σ-moments were also included in these

models. The number of neurons in the input and output layers was automatically

determined by the number of input and output variables (5 and 14 σ-moments and one

HSPs component, respectively). To define the ANN’s topologies and to determine the

numbers of neurons in the hidden layer, several ANN’s with different architectures were

developed by simultaneous building of the ANN models and their validation, for which

the correlation coefficients (R) between input and output variables was compared. A

central symmetric sigmoid transfer function was employed in the hidden layer and a

55

linear transfer function in the output layer. The network architectures (using 5 and 14 σ-

moments) are illustrated in Figure 28 and 29.

i1

o1

i2

i3

i4

i5

h1

h2

h3

h4

h5

h6

h7

h8

h9

h10

h11

h12

Input (5) Hidden (12) Output (1)

Figure 28 Visualization of architecture of the optimized ANN’s with 5 σ-moments using 5-12-1 network

topology.

56

Figure 29 Visualization of architecture of the optimized ANN’s with 14 σ-moments using 14-13-1 network

topology.

Each network calculation was started many times with random initial values to avoid

convergence to local minima. The architectures which showed the highest R values for

the training and validation sets were chosen for the final models. Models were

constructed using the training set of compounds and a validation subset was used to

provide an indication of the model performance using Levenberg-Marquardt back

propagation training algorithms and mean squared error performance function. Since the

models are nonlinear, the determination of the regression coefficients required iterative

processes. To avoid “overtraining” phenomena, the ANN models obtained were firstly

internally validated once by the leave-many-out cross-validation technique and finally

externally validated. 113 data points were chosen for training, 15 compounds were

selected for post-training analysis (internal validation) and the 17 molecules of the test

set were used for testing (external validation).

57

The MLR and ANN models were statistically evaluated by the squared

correlation coefficient of the experimental versus both fitted and predicted values (R2)

and mean absolute error which calculated as:

iz = ; ÁÂ= »B¸.Â= BÃ».ÁÄ (39)

where i stands for the number of component and j is d, p or h, respectively.

6.3. Test of HSPs estimation methods

The multivariate nonlinear QSPR models developed in this work were based on

the optimized ANN topology and parameters. The final ANN architectures contained 12

and 13 neurons in the hidden layer, according to the two sets of σ-moments. Standard

visualisations of ANN’s topology are plotted in Figure 28 and 29. After optimization of

the ANN’s architecture, the networks were trained by using the training set for the

adjustment of weights and bias values. The external validation set was used to monitor

the quality of generalisation ability of the neural networks at each learning cycle. After

the training of the ANNs was completed, the optimized weights and biases were set in

the networks and the best-trained neural networks were saved. The total MAE and R2

values obtained by the trained ANNs on training set are summarized in Table 6.

Table 6 Statistical data of multiple nonlinear regressions for QSPRs models based on ANN with 5 and 14 σ-

moments as independent variables. R2 is the squared correlation coefficient and MAE is mean absolute error.

Statistics Set ANN5σ ANN14σ

δd δp δh δd δp δh

R2 Training 0.86 0.9 0.93 0.91 0.92 0.97 Test 0.85 0.91 0.92 0.87 0.91 0.94

MAE (MPa1/2)

Training 0.48 1.66 2.21 0.37 1.45 0.98 Test 1.37 1.85 2.58 1.09 1.7 1.96

As apparent from the statistical results of both ANN models depicted in Table 6, the

multivariate ANN based nonlinear QSPR models for the correlation of HSPs

components and the σ-moments are acceptable, even if only the five basic σ-moments

with well-defined physical meaning (ANN5σ) are used. The MAE values for HSPs data

of the compounds in the training set are comparable to the experimental errors of

different methods [105]. However, as expected, the nonlinear QSPR model with 14 σ-

moments (ANN14σ) produced slightly better results for all the three HSPs components.

58

In order to evaluate the prediction power of nonlinear QSPR models, the trained and

validated ANNs were used to calculate the HSPs of test set molecules, which were not

involved in the regression process. The computed correlation coefficient (0.85 ≤ R2 ≤

0.94) and mean absolute error (1.09 MPa1/2 ≤ MAE ≤ 2.58 MPa1/2) values obtained for

δd, δp and δh (Table 6) of the test-set compounds confirm that both ANN5σ and

ANN14σ models satisfactorily predict all three HSPs components, when applied to an

external dataset. However, despite of the less number of independent variables, the

ANN5σ model possesses about the same prediction power as the ANN14σ method.

Comparison of R2 values of the ANN5σ model does not show significantly better

performance using the training set than those from using the test set, revealing that no

over-fitting did occur. The residual mean square method [108], proposed by Héberger

[109], was used as statistical characterization to confirm that there is no significant

difference between training and test sets of ANN5σ model:

Å = Q; q − qÆÄ°U W/Ç − : (40)

where p is the number of parameters and N is the number of measured points. The F test

[108] was used to compare the two sets; the calculated value of Fc =s2tr/s

2ts was

compared to the tabulated value, Ftab (N-p, N-p, 0.95 ). This variance test confirmed that

there is no significant difference between the training and test sets including dispersion,

polar and hydrogen bonding HSPs.

The differences in MAE values of the test set are also close to those of the training set

for δp and δh and only for δd is slightly higher because of some valuable outlying points,

which have more influence on the correlation than others - see Figure 30. This is

assuming, using multivariate QSPRs with only the basic COSMO σ-moment descriptors

(MX0 = MX

area, MX2 = MX

el, MX3 = MX

skew, MXHbacc3, MX

Hbdon3) over-parameterization

was avoided when training the ANNs.

It can be concluded from the above that in the prediction of Hansen solubility

parameters the five basic theoretical Klamt descriptors encode almost the same

chemical information on molecular interactions, as the total σ-moment set. This

confirms the statements of Abraham and Imbrahim [106] and Klamt [73] that the

solvent space is approximately five-dimensional, therefore a small number of

descriptors, probably no more than five, is enough to describe the most important

intermolecular interactions. A principal drawback of proposed neural networks is that

59

they are too complex to allow a straightforward interpretation of the interrelationships

between HSPs components and the σ-moments.

The good agreement between the observed dispersion, polar and hydrogen bonding

HSPs components of the compounds in training set and those fitted by ANN5σ is

demonstrated in Figures 30-32 and for a series of characteristic molecules of the

training set in Table 7.

Measured δd [MPa0.5]

12 14 16 18 20 22 24 26 28 30

Estim

ate

d δ

d [M

Pa

0.5]

12

14

16

18

20

22

24

26

28

30

Training/validation set

Regression line

Test set

Diagonal

Figure 30 Fitted and predicted (ANN5σ) Hansen dispersion solubility parameters as function of experimental

data for the training and test sets.

60

Measured δp [MPa0.5]

-5 0 5 10 15 20 25 30 35

Estim

ate

d δ

p [M

Pa

0.5

]

-5

0

5

10

15

20

25

30

35


Regression line

Test set

Diagonal

Figure 31 Fitted and predicted (ANN5σ) Hansen polar solubility parameters as function of experimental data

for the training and test sets.

Measured δh [MPa0.5]

-5 0 5 10 15 20 25 30 35 40

Estim

ate

d δ

h [M

Pa

0.5]

-5

0

5

10

15

20

25

30

35

40


Regression line

Test set

Diagonal

Figure 32 Fitted and predicted (ANN5σ) Hansen hydrogen bonding solubility parameters as function of

experimental data for the training and test sets.

The regression lines (- - - -) of the predicted vs. observed data almost coincide

with the diagonal () of the plot (1:1 relationship). This confirms the good

61

prediction quality of the nonlinear ANN5σ models and the absence of significant bias.

The estimated HSPs values obtained by ANN5σ models and the experimental ones are

compared in Table 7. The quantitative predictions for the HSPs components are quite

accurate in a wide range of values for the dispersion, polar and hydrogen bonding HSPs

components. Even chemical entities with high HSPs components are predicted well and

the model is able to quantitatively differentiate between compounds with high and low

HSPs values. This demonstrates the usefulness of the nonlinear multivariate QSPR

models with five σ-moments for the estimation of HSPs of very strongly polar chemical

species, which is particularly interesting from a practical standpoint. The majority of

estimated values were close to or within the experimental error associated with the

determination of solubility parameters [47].

Numerical ranges of HSPs component values of the training sets, i.e. δd values

ranging from 14.3 to 24.7 MPa1/2, δp values ranging from 0 to 29.2 MPa1/2, and δh

values ranging from 0 to 35.1 MPa1/2 determine the applicability domain of the model.

Within this domain, the models possess acceptable predictive power to estimate the

HSPs components of compounds which are not included into the building of the

models.

62

Table 7 Comparison of experimental HSPs components to those obtained by fitting and estimation using

multivariate nonlinear QSPR models with 5 σ-moments (ANN5σ)

Dispersion Polar

Hydrogen bonding

MPa1/2 MPa1/2 MPa1/2

Name Calc. Exp. Calc. Exp. Calc. Exp.

Aceticacid-2-ethylhexylestera 15.4 15.8 3.8 2.9 2.8 5.1

Acrylic acida 17.4 17.7 7.3 6.4 12.3 14.9

4-aminobenzoic acida 17.2 17.3 13.2 14.3 15.6 14.4

Benzoic acida 17.1 17.6 8.6 10.1 10.8 10.7

Benzyl alcohola 18 18.4 6.2 6.3 12.4 13.7

[bmim]PF6a 21.1 21 18.6 17.2 8.6 10.9

Bis(2-chloroethyl)ethera 18.9 18.8 7.3 9 4 5.7

Citric acida 20.9 20.9 9.4 8.2 20.6 21.9

γ-butyrolactonea 17 19 14.6 16.6 6.1 7.4

Dibutylphthalatea 18.5 17.8 8.2 8.6 2.3 4.1

Dipropyleneglycola 16.5 16.5 10.7 10.6 16.7 17.7

Ethylenecyanohydrina 18.6 17.2 20.1 18.8 15.1 17.6

Formic acida 14.2 14.3 12.1 11.9 14.9 16.6

Hexafluoro-1-propanola 17.3 17.2 5.4 4.5 12.5 14.7

Hexamethylphosphoramidea 18.1 18.5 6.3 8.6 9.4 11.3

Na-benzoatea 16.3 16.3 27.2 29.2 9.8 13

Na-diclofenaca 16.4 16.3 17.9 18 10.4 13.5

N,N-dimethylacetamida 17.5 16.8 12.4 11.5 8.8 10.2

Propylenecarbonatea 18.5 20 17.9 18 6.1 4.1

Salicylic acida 17 16.6 11.5 12.4 10.5 14.6

Sorbitola 19.4 19 8.9 10.3 27 33.5

Sucrosea 24.7 24.7 10.9 11.3 33.1 35.1

Tetrahydrofurfurylalcohola 17.3 17.8 8.1 8.2 10 10.2

Trichloromethanea 17.8 17.8 3.1 3.1 6 5.7

Tricresyl phosphatea 18.7 19 10 12.3 2.7 4.5

Triethyl phosphatea 15.8 16.7 10.1 11.4 7.2 9.2

Trimethyl phosphatea 17.3 16.7 12.6 15.9 8.1 10.2

Acetonitrileb 14.5 15.3 15 18 5.5 6.1

[bmim]BF4b 26 23 18.6 19 8.8 10

2-butoxyethanolb 15.1 16 6.7 5.1 8.1 12.3

Butyl acetateb 16 15.8 6.5 3.7 3.8 6.3

Butyl benzyl phthalateb 20.4 19 13.3 11.2 3.1 3.1

63

Table 7 continued Comparison of experimental HSPs components to those obtained by fitting and estimation

using multivariate nonlinear QSPR models with 5 σ-moments (ANN5σ)

Diethylethanolamine/acetic acidb 16.1 16 20.8 20.3 18.4 18.4

Diethyl etherb 15.6 14.5 3.4 2.9 3.6 5.1

Dimethyl-ethanolamineb 16.3 16.1 6.9 9.2 14.5 15.3

Dipropyleneglycolb 15.9 16.5 10.6 10.6 16.5 17.7

Ethanolamineb 18.2 17 18.5 15.5 12 21.2

Ethylbenzeneb 18.8 17.8 3 0.6 0.5 1.4

Hexafluoro-i-propanolb 17.6 17.2 7.1 4.5 12.4 14.7

Ibuprofenb 19.2 16.4 11.3 6.4 10.6 8.9

Lactoseb 28.1 24.2 11.9 11.2 32.4 34.9

Mannitolb 20 19 11.1 10.3 27.2 33.5

Piroxicamb 19.6 16.8 19.1 21.4 5.9 6.6

Ureab 19 20.9 20.2 18.7 18.1 26.4 asome characteristic, randomly selected compounds are taken from the training set

bcompounds and data of the test set

To the best my knowledge, there are no other QSPR studies of HSPs in the literature

dealing with data sets comprising base/acid molecular associates and ionic liquids, and

therefore my nonlinear σ-moment HSPs models can not be compared directly to the

models of other authors. However, the statistical confidence of the prediction by ANN

based QSPR are comparable with other methods which are applied to less challenging

datasets. A numerical comparison of the predictive ability (measured by the mean

absolute estimation error) of the HSP estimation methods is shown in Table 8.

Table 8 Comparison of the estimation errors of representative HSP prediction methods

Estimation method Mean Absolute Errors (MPa1/2)

δd δp δh

ANN5σ/QSPR COSMO σ-moment method 1.37 1.85 2.58

CED MD method [59] 0.98 3.84 5.96

Equation-of-state model [58] 0.77 0.72 0.16

Group contribution method [61] 0.41 0.86 0.8

The equation-of-state model [58] and the group contribution method [61] with specific

fitted constants and molecular fragments perform the best estimation results. The

64

accuracies of the prediction methods using quantum chemical or molecular dynamic

methods, like the CED MD method [59] and the ANN5σ/QSPR method are lower,

probably due to the generalities of these methods to deal also with complex mixtures.

6.4. Summary of the models for HSPs prediction

In this chapter, nonlinear models were presented, which were built up using

artificial neural networks and were able to derive flexible QSPR correlation models

between the COSMO σ-moments and Hansen solubility parameters over a wide range

of HSPs component values. The reliability of these models was confirmed by statistical

analysis of the training and test data sets, which clearly indicates the superiority of the

ANN. A QSPR model set developed via ANN and using only the five basic COSMO σ-

moments (the so-called Klamt descriptors) having well-defined meaning as molecular

descriptors, is proposed as optimal method for the estimation of dispersion, polar and

hydrogen bonding. This nonlinear QSPR set exhibits very good ability to estimate the

HSPs components within the test set as confirmed by the relatively low MAE values (in

the range of 1.37-2.58 MPa1/2) and high correlation coefficients (0.85 ≤ R2 ≤0.92). The

COSMO σ-moments, included in these models as molecular descriptors, can be

calculated purely by quantum chemical methods based on the molecular structure, and

provide useful information related to various molecular structural features that can

participate in solution processes. Furthermore, the results provide new insights in the

sigma function of COSMO-RS and support the view that the solvent space can be fully

characterized by a limited set of parameters. The use of the multivariable nonlinear

QSPR correlation equation models presented in this work is an important tool by

providing Hansen solubility parameters for solvents in process design, for molecules in

early drug discovery or in the CAMD of new chemical entities with high polarity, even

if they should involve unusual chemical functionality or ion-pairs.

65

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74

8. Tézisek

8.1. Síkfelületű párolgásra vonatkozó modell kifejlesztése

Újszerű eljárást fejlesztettem ki síkfelületű reális folyadékelegyek izoterm

egyensúlyi párolgásának modellezésére. Az eljárás a párolgó folyadékfázis

komponensei aktivitási tényezőinek becslésére alkalmazott COSMO-RS elmélet, a

gázfázisba átkerült molekulák transzportjának modellezésére szolgáló Maxwell-Stefan

egyenlet és a CFD szimuláció általam elsőként alkalmazott kombinálásán alapul. A

kísérleti adatokkal történt összehasonlítás alapján megállapítottam, hogy módszer jól

használható többkomponensű reális folyadékelegyek párolgási anyagmennyiség-

áramsűrűsége időbeli változásának számítására síkfelületű, egyensúly-közeli párolgás

esetén [T1-T3, T6-T10 and T12].

8.2. Csepp-párolgási modell kidolgozása

Kidolgoztam egy eljárást, amely együttesen eddig nem alkalmazott

módszerekkel modellezi a többkomponensű reális elegyek alkotta folyadékcseppek

egyensúlyi, nem-izoterm párolgását. Az eljárás a párolgó cseppek

hőmérsékletprofiljának energia-mérlegen alapuló számításánál újszerű módon, CFD

szimulációval számolja a környezetből a csepp felé irányuló konduktív hő transzportot,

a COSMO-RS elméletet alkalmazza a párolgó folyadékfázis komponensei aktivitási

tényezőinek becslésére, és a Maxwell-Stefan egyenlettel írja le a gázfázisba átkerült

molekulák transzportját. Kísérleti adatokon történt tesztelés alapján megállapítottam,

hogy módszer jól használható többkomponensű reális folyadékelegyek gömbszerű

cseppjei egyensúly-közeli, gázáramlás nélküli párolgása során bekövetkező

méretváltozásának időbeli előrejelzésére, és a cseppek várható élettartamának

becslésére. [T3, T6-T10].

8.3. QSPR modellek kidolgozása a Hansen-féle oldhatósági paraméterek becslésére

A Hansen-féle oldási paraméterek becslésére új nemlineáris QSPR modelleket

dolgoztam ki, amelyekben újszerű módon, független változóként a molekulák COSMO-

RS elmélethez kapcsolódó felületi töltés-sűrűség eloszlásának (σ-profiljának) jellemző

momentumait, az ún. σ-momentumokat alkalmaztam. Neurális hálózatok

alkalmazásával kimutattam a Hansen-féle oldási paraméterek és a σ-momentumok

75

közötti szoros nemlineáris korrelációt. Kísérleti adatokkal történt összehasonlítás során

megállapítottam, hogy az általam javasolt QSPR modellek alkalmasak változatos

funkciós csoportokkal és eltérő kémia sajátságokkal rendelkező molekulák és ionpárok

(alkánok, alkének, aromások, halo- és nitro-alkánok, aminok, amidok, alkoholok,

ketonok, éterek, észterek, savak, amin-sav ion-párok és ionos folyadékok) Hansen-féle

oldási paramétereinek becslésére. [T4-T5, T11].

76

9. Theses

9.1. Development of flat surface evaporation model

A novel method has been developed for modelling the isothermal equilibrium

evaporation of real liquid mixtures having flat surface. The model based on the

innovative combination of COSMO-RS theory for the estimation of activity coefficient,

the Maxwell-Stefan equation and CFD simulation. The method is well applicable for

calculation of cumulative evaporation fluxes as a function of the time during the quasi

equilibrium evaporation of multi-component liquids [T1-T3, T6-T10 and T12].

9.2. Development of droplet evaporation model

A new method has been suggested for modelling the non-isothermal equilibrium

evaporation of droplets of real multi-component liquid mixtures using creative

combination of various methods. The model estimates the heat balance of droplet with a

novel way, where CFD simulation is used to calculate the heat conducted into the

droplet, applies COSMO-RS for the estimation of activity coefficient of components of

evaporating liquid mixtures and describes the transport of evaporated molecules in the

gas phase with Maxwell-Stefan diffusivity equations. The model is suitable for

prediction of evaporation rate and lifetime of droplets of multi-component real mixtures

during quasi equilibrium evaporation without forced convection [T3, T6-T10].

9.3. Model development for estimation of Hansen solubility parameters

New nonlinear models have been proposed for the prediction of Hansen

solubility parameters using the sigma-moments calculated by COSMO-RS theory as

independent variables in nonlinear quantitative structure-property relationships. Strong

nonlinear correlations between sigma-moments and Hansen solubility parameters have

been established by artificial neural networks. It can be concluded from the comparison

of experimental data and simulation results that the proposed QSPR models are suitable

for the prediction of solubility parameters of chemicals having a broad diversity of

chemical characters such as alkanes, alkenes, aromatics, haloalkanes, nitroalkanes,

amines, amides, alcohols, ketones, ethers, esters, acids, ion-pairs: amine/acid associates

and ionic liquids [T4-T5 and T11].

78

10. Kapcsolódó publikációk és közlemények - Related publications

A tézisekben megfogalmazott általánosítható, új tudományos és szakmai

megállapításokat publikáló közlemények:

Publications containing the new scientific results of this thesis:

T1. G. Járvás, C. Quellet, A. Dallos, COSMO-RS based CFD model for flat surface

evaporation of non-ideal liquid mixtures International Journal of Heat and Mass

Transfer 54 (2011) 4630-4635 (IF: 1,898)

doi:10.1016/j.ijheatmasstransfer.2011.06.014

T2. G. Járvás, A. Dallos: Illatanyagok terjedésének vizsgálata levegőben,

számítógépes szimuláció kísérletekkel. XII. Nemzetközi Vegyészkonferencia,

Csíkszereda (Románia), október 3-8. Kiadvány. (2006)

T3. G. Járvás, A. Dallos: Modeling of Evaporation of Droplets of Multicomponent

Liquid Mixtures using COSMO-RS. COSMO-RS Symposium, Maria in der Aue,

Wermelskirchen, Germany, March 30- April 1 (2009)

T4. G. Járvás, C. Quellet, A. Dallos: Estimation of Hansen solubility parameters

using multivariate nonlinear QSPR modeling with COSMO screening charge

density moments. Fluid Phase Equilibria 309 (2011) 8-14 (IF: 2.253)

doi:10.1016/j.fluid.2011.06.030

T5. G. Járvás, A. Dallos: Estimation of Hansen solubility parameters using

multivariate nonlinear QSPR modeling with COSMO screening charge density

moments. Conferentia Chemometrica 2011, Sümeg, Hungary, 2011. September

19-21. Book of Abstracts P18, ISBN 978-963-9970-15-1

T6. G. Járvás, A. Kondor, A. Dallos: Diffusion Evaporation Model of Multi-

component Mixture Droplets. COMSOL Conference, Budapest, Hungary,

November 24, Book of Abstracts. P33 (2008)

T7. G. Járvás, A. Kondor, A. Dallos: Investigation of evaporation of layers and

droplets of bioethanol-blended reformulated gasolines. 35th International

Conference of Slovak Society of Chemical Engineering, Tatranské Matliare,

79

Slovakia, May 26-30, Proc. 104, ISBN 978-80-227-2903-1, Ed.: J. Markos

(2008)

T8. A. Kondor, G. Járvás, A. Dallos: Investigation of Transport of Fragrances in Air.

European COMSOL Conference 2007, Grenoble, Oct. 23-24, Proceedings

(ISBN: 978-0-9766792-5-7) (2007)

T9. G. Járvás, A. Kondor, A. Dallos: A Novel Method to Modeling the Evaporation

of the Multicomponent Mixtures. European COMSOL Conference 2007,

Grenoble, Oct. 23-24 (ISBN: 978-0-9766792-5-7) (2007)

T10. G. Járvás, A. Kondor, A. Dallos: Computer Simulation of Evaporation and

Transport of Multicomponent Mixtures in Air Using Comsol Multiphysics and

COSMOtherm. MATH/CHEM/COMP 2007 Conference on the interfaces among

mathematics, chemistry and computer sciences. Dubrovnik, Croatia, June 11-16,

Book of Abstracts. P34 (2007)

T11. G. Járvás, A. Kondor, A: Dallos: Estimation of Hansen solubility parameters

using QSPR model with COSMO screening charge density moments.

Conferentia Chemometrica 2007, Budapest September 2-5, ISBN 978-963-

7067-17-4, Abstract Book, P10 (2007)

T12. G. Járvás, A. Kondor, A. Dallos: Investigation of evaporation and transport of

perfume ingredients in air with computer simulation using COMSOL

MULTIPHYSICS and COSMOtherm. COMSOL Users Conference, Prague,

Czech Republic, Oct. 27, Proc. 16 (2006)

80

11. Acknowledgement

During my Ph.D. studies it has always been a pleasure to listen to anecdotes of

elderly lecturers especially when it was about their masters, their Professors. I was

impressed by the way how respectfully they mentioned their professional work and

personal properties. At the beginning of my work I found it rather hard to identify this

feeling but by now I understand them. Here I grab the occasion to thank Dr. Dallos for

his help he gave me when making this dissertation. Now I have someone to tell

anecdotes about...

Furthermore, I acknowledge the financial support of this work by the Hungarian

State and the European Union under the TAMOP-4.2.1/B-09/1/KONV-2010-0003 and

TÁMOP-4.2.2/B-10/1-2010-0025 projects, and the grant of Foundation for Engineer

Education of Veszprém.

evaporation models for multicomponent...

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