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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.
Evaporation models for multicomponent mixtures
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
Evaporation models for multicomponent mixtures
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
λ1 adjustable parameter of COSMO-RS theory
λ2 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
Figure 10 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-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
Figure 11 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 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
Figure 12 Comparison of the calculated (―) and experimental () evaporation flux values as a function of the
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
Figure 13 Comparison of the calculated (―) and experimental () evaporation flux values as a function of the
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
Figure 17 Comparison of the calculated (―) and experimental () evaporation flux values as a function of the
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
Figure 22 Comparison of the calculated (―) and experimental () normalized droplet diameter changes as a
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
Figure 23 Comparison of the calculated (―) and experimental () normalized droplet diameter changes as a
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
Figure 24 Comparison of the calculated (―) and experimental () normalized droplet diameter changes as a
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= Bû.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
Training/validation set
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
Training/validation set
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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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].
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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].
77
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.
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