THE TRAIL OF FRAGRANCES
JOANA DA SILVA MATIAS PEREIRA DISSERTAÇÃO DE MESTRADO APRESENTADA
À FACULDADE DE ENGENHARIA DA UNIVERSIDADE DO PORTO EM ENGENHARIA QUÍMICA
Mestrado Integrado em Engenharia Química
The Trail of Fragrances
Dissertação de Mestrado
de
Joana da Silva Matias Pereira
Desenvolvida no âmbito da unidade curricular de Dissertação
realizado em
LSRE – LCM
Orientador na FEUP: Professor Alírio Rodrigues Co – orientadores na FEUP: Doutora Patrícia Costa
Professor José Miguel Loureiro
Departamento de Engenharia Química
“Superar o fácil, não é mérito,
É obrigação;
Vencer o difícil é glorificante;
Ultrapassar o que até
então era impossível,
é esplendoroso!”
Agradecimentos
Gostaria de agradecer ao Professor Alírio Rodrigues por me ter recebido no laboratório, pela oportunidade que me deu em trabalhar nesta área que desde o início despertou a minha curiosidade e por toda a ajuda prestada, essencial para o desenvolvimento desta tese.
À Doutora Patrícia Costa estou bastante grata, não sópor todo o conhecimento transmitido, mas também pela forma como me recebeu e me acompanhou diariamente. Pelas palavras de encorajamento e incentivo nos momentos em que as coisas não correram bem e pelas palavras de alegria quando correram, um muito obrigada.
Ao Professor José Miguel Loureiro, quero agradecer a oportunidade e o apoio prestado, não só no desenvolvimento desta tese, mas ao longo dos anos que nos acompanhou.
À Professora Maria do Carmo Coimbra quero também manifestar a minha gratidão pela ajuda, que foi muito importante para o desenvolvimento de um dos tópicos abordados nesta tese.
Gostaria ainda de agradecer a todos os que estiveram presentes durante todo o meu percurso académico. Às minhas colegas de laboratório, Ana e Sofia, que me acompanharam desde o primeiro dia, todos os dias; ajudaram-me quando necessário, apoiando-me incansavelmente. Um grande obrigada, pois sem vocês teria sido sem dúvida uma caminhada bem mais difícil de ultrapassar. À Catarina e ao Diogo por todo o apoio que me deram não só nestes meses, mas ao longo dos 5 anos, e também pelos momentos que passamos todos juntos, que vou para sempre recordar.
À minha família, em especial aos meus pais, pelos sábios conselhos, por me ensinarem sempre a lutar pelos meus sonhos e por todo o esforço que fizeram para que eu pudesse terminar este ciclo da minha vida, e iniciar tantos outros que se avizinham.
Por último, gostaria de agradecer ao Rui, sempre presente nesta longa caminhada, pelo apoio constante e sobretudo pela paciência demonstrada, que foi muita, sem dúvida.
Muito obrigada a todos, Joana Este trabalho foi financiado por: Projeto POCI-01-0145-FEDER-006984 - Laboratório Associado LSRELCM - financiado pelo Fundo Europeu de Desenvolvimento Regional (FEDER), através do COMPETE2020 – Programa Operacional Competitividade e Internacionalização (POCI) e por fundos nacionais através da Fundação para a Ciência e a Tecnologia I.P.
Abstract
This thesis aims at studying the trail of perfumes involving the prediction of fragrance diffusion under different scenarios. The first case intended to model the radial diffusion of fragrances, in an infinite medium, based on Fick’s second law. Regarding this topic, two different situations were considered: continuous and non-continuous sources. The Fick’s second law for a continuous source was solved analytically, using the method of Laplace Transforms for a binary and a ternary fragrance mixture, composed of limonene and linalool, and limonene, linalool and vanillin, respectively. By studying these two mixtures, it is possible to see the influence of the liquid mixture in the gas concentration profiles. Regarding the non-continuous source, a numerical solution was performed using the general PROcess Modelling Systems (gPROMS) software. The experimental validation was carried out using three fragrance mixtures: two single mixtures composed of limonene and α-pinene, and a quaternary fragrance mixture with limonene, α-pinene, linalool and geranyl acetate as fragrance components. The experimental gas concentrations of these components were measured in a diffusion tube and quantified using gas chromatography with flame ionization detector (GC-FID). The odor intensity and character of the studied systems were assessed using the Stevens’ power law and the Strongest Component model, respectively. The obtained results show a good agreement between the numerical simulation and the experimental gas concentration data, suggesting the proposed methodology as an efficient tool to assess the performance of fragrance systems over time and distance. The second case simulated a three-dimensional diffusion, but in a finite medium, i.e., a limited space, for example a room. This was performed using Computational Fluid Dynamics (CFD). A multicomponent fragrance system composed of ethanol, limonene, geraniol, linalool and vanillin was studied. The results suggest this methodology as an efficient tool to simulate the performance of a fragrance in a room, which is particularly important when we consider the olfactory marketing.
Finally, the third case focused a preliminary study about a moving source where the fragrance is continuously released over a period of time. Two different situations were considered: one-dimensional and three-dimensional diffusions. Concerning the one-dimensional diffusion, it is presented a theoretical model for a moving source that was validated for a pure component (α-pinene) using a diffusion tube. The experimental results fit well with the predicted data. Regarding the three-dimensional diffusion, it is also presented a theoretical model for the prediction of the gas concentration profiles.
Resumo
Esta tese visa o estudo do rasto de perfumes envolvendo a previsão da difusão de fragrâncias em diferentes cenários. O primeiro caso pretendeu modelar a difusão radial das fragrâncias, num meio infinito, tendo por base a 2ª lei de Fick. Neste caso foram consideradas duas situações distintas: fontes contínuas e não contínuas. A 2ª lei de Fick para uma fonte contínua foi resolvida analiticamente usando as transformadas de Laplace, para uma mistura binária e uma ternária, compostas por limoneno e linalol, e limoneno, linalol e vanilina, respetivamente. Analisando estas misturas, é possível verificar a influência da composição da fase líquida nos perfis de concentração gasosa. Quanto à fonte não contínua, a solução numérica foi obtida usando o software general PROcess Modelling Systems (gPROMS). A validação experimental foi realizada utilizando três misturas de fragrâncias: duas soluções compostas por componentes puros (limoneno e α-pineno) e uma mistura quaternária com limoneno, α-pineno, linalol e acetato de geranilo. As concentrações gasosas experimentais destes componentes foram medidas num tubo de difusão e quantificadas utilizando cromatografia gasosa com um detetor de ionização de chama (GC-FID). A intensidade e caráter do odor dos sistemas perfumados estudados foram avaliados utilizando os modelos Stevens’ power law e Strongest Component, respetivamente. Os resultados obtidos pela simulação numérica e os dados experimentais são concordantes, sugerindo a metodologia proposta como uma ferramenta eficiente para avaliar o desempenho das fragrâncias ao longo do tempo e da distância.
Relativamente ao segundo caso de estudo, foi estudada a difusão de fragrâncias a três dimensões num meio finito, isto é, um espaço limitado, como por exemplo, uma sala. Esta simulação foi realizada utilizando Computational Fluid Dynamics (CFD). Os resultados sugerem esta metodologia como uma ferramenta importante para simular o desempenho de fragrâncias numa sala, o que é particularmente importante para o marketing olfativo.
Finalmente, o terceiro caso focou um estudo preliminar sobre uma fonte móvel, onde a fragrância é libertada continuamente ao longo do tempo. Foram consideradas duas situações: difusão unidimensional e tridimensional. No que diz respeito à difusão unidimensional, é apresentado um modelo teórico posteriormente validado para um componente puro (α-pineno), usando um tubo de difusão. Os resultados experimentais são bem ajustados pelo modelo teórico. Em relação à difusão tridimensional, é também apresentado um modelo teórico para a previsão dos perfis de concentrações gasosas.
Declaração
Declaro, sob o compromisso de honra, que este trabalho é original e que todas as contribuições não originais foram devidamente referenciadas com identificação da fonte.
Joana da Silva Matias Pereira Setembro 2017
Contents
1. Introduction ... 1
1.1. Motivation and Relevance ... 1
1.2. Thesis Objectives and Layout ... 2
2. Context and State of the Art ... 3
2.1. Flavor and Fragrance Industry ... 3
2.2. Importance of Fragrances ... 3
2.3. Olfactory Marketing ... 4
2.4. Perfume Structure ... 4
2.4.1. Volatility and Chemical Structure ... 5
2.4.2. Solubility ... 6
2.4.3. Substantivity ... 6
2.4.4. Thresholds ... 6
2.5. Odor Intensity and Odor Character Models ... 6
2.5.1. Odor Value (OV) ... 7
2.5.2. Stevens’ Power Law ... 7
2.5.3. Strongest Component Model ... 7
2.6. Fragrance Performance ... 7
2.7. Diffusion Process ... 8
2.8. Trail of a Fragrance ... 9
3. Radial Diffusion in an Infinite Medium ... 10
3.1. Radial Diffusion considering a Continuous Source ... 10
3.1.1. Methods ... 10
3.1.2. Materials ... 11
3.1.3. Results and Discussion ... 12
3.1.3.1. Fragrance System FS1: Limonene and Linalool ... 12
3.1.3.1. Fragrance System FS2: Limonene, Linalool and Vanillin ... 13
3.2. Radial Diffusion considering a Non-Continuous Source ... 14
3.2.1. Materials ... 14
3.2.1.1. Chemicals ... 14
ii
3.2.2.1. Radial Diffusion Model ... 15
3.2.3. Equivalence Relation between the Axial and Radial Diffusion Models ... 17
3.2.4. Experimental Gas Concentrations (Axial Diffusion) ... 20
3.2.5. Results and Discussion ... 21
3.2.5.1. Gas Concentrations ... 21
3.2.5.2. Odor Intensities ... 24
4. Three-Dimensional Diffusion in a Finite Medium, using Computational Fluid Dynamics ... 26
4.1. Materials and Methods ... 26
4.2. Results and Discussion ... 32
5. Fragrance Diffusion from a Moving Source ... 35
5.1. One-Dimensional Diffusion ... 35
5.1.1. Materials ... 35
5.1.2. Methods ... 35
5.1.1.1. Numerical Simulation ... 37
5.1.1.2. Validation ... 37
5.1.2. Results and Discussion ... 39
5.2. Three-Dimensional Diffusion ... 40
5.2.1. Methods ... 40
5.2.1.1. Numerical Simulation ... 40
5.2.2. Results and Discussion ... 41
5.2.2.1. Gas Concentration Profiles as a Function of Distance, at 1.60 m and 1.00 m ... 41
5.2.2.2. Gas Concentration Profiles as a Function of Time, at 1.60 m ... 44
6. Conclusions ... 46
7. Work Evaluation... 47
7.1. Limitations and Future work ... 47
References ... 48 Appendix A ... 51 Appendix B ... 55 Appendix C ... 58 Appendix D ... 59 Appendix E ... 62 Appendix F ... 63 Appendix G ... 67
List of Figures
Figure 1. Triangular structure of a perfume: top, middle and base notes. ... 5
Figure 2. Performance parameters as a function of time and distance. ... 8
Figure 3. Scheme of the simulated gas and liquid phases. ... 10
Figure 4. Gas concentration profiles of fragrance system FS1 over distance, at fixed times: 1 min (left) and 360 min (right) after fragrance release. ... 13
Figure 5. Odor intensity profiles of fragrance system FS1 over distance, at fixed times: 1 min (left) and 360 min (right) after fragrance release. ... 13
Figure 6. Gas concentration profiles of fragrance system FS2 over distance, at fixed times: 1 min (left) and 360 min (right) after fragrance release. ... 14
Figure 7. Odor intensity profiles of fragrance system FS2 over distance, at fixed times: 1 min (left) and 360 min (right) after fragrance release. ... 14
Figure 8. Equivalence relation between the gas numerical 1D axial and 1D radial models (gPROMS). ... 18
Figure 9. Scheme of the procedure used to compare the numerical solution with the gas concentrations obtained through the equivalence relation. ... 19
Figure 10. Theoretical equivalence relation between analytical solutions of 1D axial and 1D radial models. ... 20
Figure 11. Photograph of the diffusion tube, with the respective sampling ports; a- Zoom of the glass container. .. 21
Figure 12. Gas concentration profiles for pure component limonene in air at different distances from the source (SP2, SP3 and SP4) and over time; Δ Radial concentration equivalent to axial concentration in the diffusion tube; - - - Numerical solution of 1D radial model. ... 22
Figure 13. Gas concentration profiles for pure component α-pinene in air at different distances from the source (SP2, SP3 and SP4) and over time. Δ Radial concentration equivalent to axial concentration in the diffusion tube; - - - Numerical solution of 1D radial model. ... 23
Figure 14. Gas concentration profiles of quaternary mixture in air at different distances from the source (SP2, SP3 and SP4) and over time. Δ Radial concentration equivalent to axial concentration in the diffusion tube; - - - Numerical solution of 1D radial model. ... 24
Figure 15. Predicted odor profiles of limonene and α-pinene over time and distance (SP2, SP3, SP4). ... 25
Figure 16. Predicted odor profiles of the quaternary fragrance mixture at different distances (SP2, SP3, SP4) over time; a-c – Zoom of the first hours after release. ... 25
Figure 17. Geometry and the location of the perfume source. ... 26
Figure 18. Options selected during the creation of the mesh. ... 27
Figure 19. a) Mesh of the geometry. b) Inflation near the source. ... 27
Figure 20. Mass fraction of ethanol after one hour (left) and seven and a half hours (right) of release. ... 33
Figure 21. Mass fraction of ethanol specified in the plane z=2.5 m, after one hour (top) and seven and a half hours (bottom) of release. ... 33
Figure 22. Schematic representation of the line, where odor intensity was evaluated for each component. ... 34
Figure 23. Odor intensity of the odorant components of the quinary system, evaluated in the line created in the center of the room, after 1 hour of release; a – Zoom near the source. ... 34
Figure 24. Odor intensity of the odorant components of the quinary system, evaluated in the line created in the center of the room, after 7.5 hours of release; b – Zoom near the source. ... 34
iv
Figure 26. System developed in the laboratory; F1 – Zoom of the piece of fabric used as the source, and the respective
dimensions... 38
Figure 27. Theoretical and experimental gas concentration profiles of α-pinene over distance, at a fixed time of 100 s, of a source moving at 1.34 × 10-2 m/s, and Dα-pin=6.04 × 10-6 m2/s. ... 39
Figure 28. Moving source path, in three dimensions. ... 40
Figure 29. Gas concentration profiles of a source moving along x-axis (at a velocity of 1.34 m/s), assessed at a height of 1.60 m, for times of 10, 50, 100, 200, 300 and 500 s; a-b – Zoom of the Dα-pin=6.04 × 10-6 m2/s. ... 42
Figure 30. Gas concentration profiles of a source moving along x-axis (at a velocity of 1.34 m/s), assessed at a height of 1.00 m, for times of 10, 50, 100, 200, 300 and 500 s; a-c – Zoom of Dα-pin=6.04 × 10-5 m2/s; d-g – Zoom of D α-pin=6.04 × 10-6 m2/s. ... 44
Figure 31. Gas concentration profiles as a function of time of a source moving at 1.34 m/s, evaluated at a height of 1.60 m, and three fixed distances, using three values of the diffusion coefficient: Dα-pin=6.04 × 10-4 m2/s, Dα-pin=6.04 × 10-5 m2/s and D α-pin=6.04 × 10-6 m2/s. ... 45
Figure D 1. Calibration curve of limonene. ... 61
Figure D 2. Calibration curve of α-pinene. ... 61
Figure E 1. Equivalence relation for limonene and ethanol, for two different times (2 h and 5 h). ... 62
Figure F 1. Mass fraction of limonene after one hour (left) and seven and a half hours (right) of release. ... 63
Figure F 2. Mass fraction of limonene specified in the plane z=2.5 m, after one hour (top) and seven and a half hours (bottom) of release. ... 63
Figure F 3. Mass fraction of geraniol after one hour (left) and seven and a half hours (right) of release. ... 64
Figure F 4. Mass fraction of geraniol specified in the plane z=2.5 m, after one hour (top) and seven and a half hours (bottom) of release. ... 64
Figure F 5. Mass fraction of linalool after one hour (left) and seven and a half hours (right) of release. ... 65
Figure F 6. Mass fraction of linalool specified in the plane z=2.5 m, after one hour (top) and seven and a half hours (bottom) of release. ... 65
Figure F 7. Mass fraction of vanillin after one hour (left) and seven and a half hours (right) of release. ... 66
Figure F 8. Mass fraction of vanillin specified in the plane z=2.5 m, after one hour (top) and seven and a half hours (bottom) of release. ... 66
List of Tables
Table 1. Properties of the studied fragrance components: Molecular Formula, Molecular Weight (Mi), Vapor Pressure
(𝑃𝑖𝑠𝑎𝑡), Odor Threshold (OT), Olfactory Power Law Exponent (ni), Density (ρl,i) and Diffusion Coefficients in Air
(Di,air)... 12
Table 2. Mole fractions (xi) of the fragrance components present in the liquid phase, of systems FS1 and FS2. ... 12
Table 3. Properties of the studied fragrance components used in this work: Molecular Formula, Molecular Weight (Mi), Vapor Pressure (𝑃𝑖𝑠𝑎𝑡), Odor Threshold (OT), Olfactory Power Law Exponent (ni), Density (ρl,i) and Diffusion Coefficients in Air (Di,air). ... 15
Table 4. Predicted gas concentration values, considering axial and radial diffusion, assessed at three different times (2 h and 5 h), for limonene and ethanol. ... 18
Table 5. Average (φ) of the ratio of 𝐶𝑎𝑥𝑖𝑎𝑙𝑔 and 𝐶𝑟𝑎𝑑𝑖𝑎𝑙𝑔 as a function of distance for both limonene and ethanol. ... 19
Table 6. Mole fractions (xi) of each fragrance component in the liquid phase of the studied fragrance systems. ... 21
Table 7. Number of groups of the ith type and group contribution of each component. ... 29
Table 8. Molecular weight (Mi), density (ρg,i), and viscosity (μi) of each component in the gas phase. ... 30
Table 9. Diffusivity coefficient of each component in air. ... 31
Table 10. Mass fractions in the gas phase of each odorant component, evaluated using the respective liquid volume (Vi). ... 32
Table 11. Properties of the studied fragrance component used in this work: Molecular Formula, Molecular Weight (Mi), Vapor Pressure (𝑃𝑖𝑠𝑎𝑡), Odor Threshold (OT), Olfactory Power Law Exponent (ni), Density (ρl,i) and Diffusion Coefficients in Air (Di,air). ... 35
Table 12. Electrical components and respective manufacturers. ... 35
Table 13. Typical values for molecular diffusivity (Di,air) and film thickness (δ) for gases and liquids. ... 36
Table 14. Parameters used in the 1D numerical simulation: diffusion coefficient (Dα-pinene), time (t), evaporation rate (μα-pinene) and source velocity (vs). ... 37
Table 15. Parameters used in the 3D numerical simulation: evaporation rate (μα-pinene) and source velocity (vs). .... 41
Table B 1. UNIFAC groups, subgroups, number of groups of each type j in molecule i (𝜈𝑗(𝑖)) and the group parameters of molecular van der Waals volumes (𝑅𝑗(𝑖)) and surface area (𝑄𝑗(𝑖)). ... 56
Table C 1. Atomic contributions for the determination of the molecular diffusion volumes [50]. ... 58
vi
Notation
A
glArea of gas-liquid interface
m
2a
mnUNIFAC binary interaction parameter
K
C’
Group contribution
C
equilibriumGas concentration in equilibrium with liquid
C
gasGas concentration
g/m
3C
TTotal gas concentration
g/m
3D
Diffusion coefficient
m
2/s
J
Mass diffusion flux
kg/(m
2s)
k
Mass transfer coefficient
m/s
K
Equilibrium ratio
M
Molecular weight
g/mol
n
Power law exponent
n*
Number of groups of the ith type
n’
Number of moles in the liquid phase
mol
ODT
Odor Detection Threshold
g/m
3ORT
Odor Recognition Threshold
g/m
3OT
Odor Threshold
g/m
3OV
Odor Value
P
Pressure
Pa
p
Static pressure
Pa
P
cCritical pressure
bar
Peak Area
Area of peaks in the CG chromatograms
counts
P
opOperating pressure
Pa
P
satVapor pressure
Pa
q
Molecular surface area parameter
Q
jGroup area parameter
r
Radial distance from the source
m
r
Three-dimensional position vector
r’
Molecular van der Waals volume parameter
r
0Moving source path vector
R
jGroup volume parameter
R
maxMaximum radial distance
m
R
sRadius of the liquid source
m
t
Time
s
T
Temperature
K
T
cCritical temperature
K
T
rReduced temperature
K
v
Velocity vector
V
Volume of liquid
mL
V
columnVolume that flows to the column
mL
V
injVolume injected in the CG-FID
mL
V
mMolecular diffusion volumes
v
sVelocity of the moving source
m/s
V
ventVolume vented through the split vent
mL
x
Molar fraction of the liquid phase
X
mMole fraction of group m in the mixture
y
Molar fraction of the gas phase
Y
Mass fraction of the gas phase
Greek Letters
γ
Activity coefficient
Γ
Group residual activity coefficient
δ
Film thickness
m
θ
Molecular surface area fraction
μ
Gas viscosity
kg/(m s)
μ
mDipole moment
debyes
viii
μ
molarReleasing rate (molar)
mol/(m
2s)
μ
rReduced dipole moment
ν
jNumber of groups of type j
ρ
gGas density
kg/m
3ρ
lLiquid density
g/mL
φ
Parameter of ratio of 𝐶
𝑎𝑥𝑖𝑎𝑙𝑔and 𝐶
𝑟𝑎𝑑𝑖𝑎𝑙𝑔ϕ
Molecular volume fraction
ψ
Odor intensity
ψ
mnUNIFAC parameter
Indexes
axial
Model 1D axial
C
Combinatorial part
g
Gas
i
Component
i,0
Component, initial value
l
Liquid
R
Residual part
radial
Model 1D radial
List of Acronyms
CFD
Computational Fluid Dynamics
1. Introduction
1.1. Motivation and Relevance
Fragrances are commonly incorporated into a wide variety of consumer products, like perfumes, soaps, household cleaners, or detergents, due to their capacity to mask unpleasant odors, thus converting them into more attractive consumer products [1]. Fragrances are also employed to create a favorable impression on other people: it is expected that a person who smells good will induce a positive response from others in different occasions [2]. Moreover, fragrances help people to build their self-image and self-esteem, as well as to increase their confidence [3]. Thus, human behavior is significantly influenced by the sense of smell which is closely related to emotions and memories [4]. Today, many sectors are focused on olfactory marketing, i.e., the use of scents to influence consumers’ choices, for example, in terms of money spent in a store, clients’ attraction and creation of a memorable brand [5]. However, the formulation of a successful fragrance is a complex task, involving dozens of odorant chemicals, solvents, surfactants, preservatives and other ingredients that can compromise the release and diffusion of fragrances and, at a late stage, influence its perception by the human nose. This is the reason why it is of great interest to study the behavior of each odorant present in a formulation in order to predict and control its performance, thus enhancing the probability of a perfume’s success. However, nowadays the formulation of fragrances is mainly a costly trial-and-error process that involves testing hundreds of mixtures until the desired scent is reached. In this context, Product Engineering has been used to introduce some scientific knowledge into an empiric and experimental area. Through the use of scientific principles, it is possible to predict the odor of complex fragrance mixtures through the liquid composition and using psychophysical parameters [6].
Regarding the prediction of fragrance diffusion in the present study a radial diffusion model for fragrance mixtures, in an infinite medium, is proposed based on Fick’s second law and, considering two cases: a continuous source and a non-continuous one. For the continuous source, the Fick’s second law was solved analytically for a binary mixture composed of limonene and linalool, and a ternary one composed of limonene, linalool and vanillin. Regarding the non-continuous source, the numerical solutions were obtained using the general PROcess Modeling Systems (gPROMS) software. To validate this topic, fragrance mixtures containing odorants with different physicochemical properties (limonene, α-pinene, linalool and geranyl acetate) were prepared. Their gas concentrations in the air were experimentally measured using the diffusion tube and analyzed by means of gas chromatography with flame ionization detector (GC-FID). The gas concentrations were converted into perceived odor using the psychophysical model Stevens’ power law [7] and the odor character assessed by the Strongest Component (SC) model [1]. Finally, the experimental results were compared with those predicted by the numerical simulation.
The fragrance diffusion in a finite medium was also assessed, using Computational Fluids Dynamics (CFD) by means of the software Fluent included in the commercial package ANSYS® 17.0. CFD provides a
qualitative and quantitative prediction of fluid flows by means of mathematical modeling (partial differential equations), numerical methods (discretization and solution techniques) and software tools
Introduction 2 (solvers, pre- and post-processing utilities) [8]. The reason why it is of great interest to use CFD in Perfume Engineering is the fact that it makes possible to assess the fragrance propagation in any known environment. In this work, a fragrance mixture composed of ethanol, limonene, geraniol, linalool and vanillin was used in the simulation, and its performance was studied inside a room with dimensions of 10 m × 5.0 m × 2.5 m. As far as we know, all the methodologies available in the literature to predict the diffusion of fragrances in the air are based on static fragrance sources. Therefore, this study presents, for the first time, a preliminary study on fragrance release and diffusion from a moving source, with the objective of predicting the trail left by a person after applying a perfume. It was developed a one-dimensional model, which was numerically simulated using the software Matlab®, considering a fragrance system composed of α-pinene continuously
released from a moving source. The experimental validation was carried out in a diffusion tube, already existent in the laboratory. Finally, a three-dimensional model was developed, considering a source moving through the space at a constant velocity. This model was also numerically simulated for a fragrance system of α-pinene using Matlab®.
1.2. Thesis Objectives and Layout
The main goal of this thesis consists in studying the trail of a perfume. To reach that objective radial fragrance diffusion either considering infinite or finite media, with a fixed source is studied. Then, the prediction of fragrance diffusion is extended to a moving source (simulating the trail left by a person after applying perfume), either in one or three dimensions.
This thesis is divided in seven chapters. The first chapter ‘Introduction’ presents the motivation and relevance of this work, as well as the main objectives and layout. A general review of the main subjects is presented in chapter 2: ‘Context and State of the Art’. In chapter 3, ‘Radial Diffusion in an Infinite Medium’ are presented two different cases: continuous and non-continuous sources. A theoretical model for the continuous source, and a theoretical model and respective experimental validation for the non-continuous source were developed to predict the gas concentrations of fragrance components, considering an infinite medium in both cases. Chapter 4: ‘Three-Dimensional Diffusion in a Finite Medium, using Computational Fluid Dynamics’ describes the numerical simulation performed, as well as the obtained results for a multicomponent fragrance mixture, with the objective of predicting the behavior of fragrance components within a finite medium (i.e., a room). Chapter 5: ‘Fragrance Diffusion from a Moving Source’ presents two approaches: one-dimensional and three-dimensional diffusion of a fragrance component from a moving source, with the purpose of predicting the gas concentrations left in the trail. The main conclusions of this work are exhibited in chapter 6: ‘Conclusions’ and chapter 7: ‘Limitations and Future Work’ presents the work evaluation, as well as suggestions for a future work. It is important to mention that this division of the chapters aimed to simplify and facilitate reader’s understanding once there are different topics addressed in this work.
2. Context and State of the Art
2.1. Flavor and Fragrance Industry
The intense demand for products containing flavors and fragrances made the Flavor & Fragrance (F&F) industry as one of the most profitable industries over the last decades, allowing the production of high-value products for application in consumer products [9]. During the past years, F&F has been growing steadily, expecting a 4% annual growth for 2017, becoming a nearly 25 $ billion sector [10]. This market, regardless of its magnitude, is mainly controlled by a group of 10 world companies, where the top 5 account for more than 60% of the market share (Givaudan – 20.6%, Firmenich – 15.1%, International Flavors & Fragrances – 11.9%, Symrise – 9.6% and Takasago – 6.4%). Thus, it can be said that, despite there are several small companies operating within it, this market is closed and strong, ruled by a small group of companies, concentrated in Europe, USA and Japan [1].
Three main subdivisions can be distinguished in the F&F industry [11]:
essential oils and natural extracts (mainly complex mixtures, obtained from natural resources by different processes);
aroma chemicals (uniform compounds that can be either natural or of synthetic origin);
formulated flavors and fragrances (complex blends of aromatic materials – essential oils, natural extracts or aroma chemicals), that constitutes the core of the F&F business, with 76% of the sales volume.
2.2. Importance of Fragrances
Fragrances are commonly incorporated into a wide variety of products, like perfumes, soaps, household cleaners, or detergents, and for different purposes [1]. Aromatherapy is an example of fragrance application, in which some specific odorant components are recommended to treat specific mental problems. Fragrances can be used to manipulate interpersonal relations, cure psychiatric problems, or even stimulate intellectual abilities [2]. They are also employed to create a favorable impression on other people: it is expected that a person who smells good will induce a positive response from others in different occasions, like selling a product or applying for a job [2]. Rovesti and Colombo [12] reported that some fragrances like water-mint, verbena and lavender can alter worker’s mental state, or increase their productivity by 10-15%. Additionally, fragrances stimulate an interesting environment and counteracts boredom, once odors play an important role in keeping our minds alert and engaged [2]. Moreover, they help people to build their self-image and self-esteem, as well as increase their confidence [2]. Thus, human behavior is significantly influenced by the sense of smell, which is closely related to emotions and memories. Odor memory is about odor memories that people have experienced, or memories that are associated or evoked by that odors. Herz and Engen [4] concluded that evoked memories have special characteristics that are connected to emotional quality, once memories induced by odors appear to be more powerful that memories elicited by other
Context and State of the Art 4 stimuli. Also, a research corroborates a very close relationship between the sense of smell and memory, discovering that a memory of a smell is 100 times more possible than a memory of something saw, heard or touched [13]. Therefore, a scent memory is a powerful tool for service providers, which can be used to develop loyalty to a brand or service [14].
2.3. Olfactory Marketing
The close relationship between the sense of smell and memory can bring benefits for olfactory marketing, i.e., the use of scents in a business context to influence employees and consumers’ behavior on an emotional level.
Since 2007, olfactory marketing has been considered as a trend, and many sectors decided to extensively investigate the possible effects that this type of marketing have in consumers, either in terms of money spent in their stores, clients’ attraction or creation of a memorable brand [13]. In the specific case of stores, the presence of a fragrance as a part of the retail environment has the intention of affecting the attitudes and behaviors of consumers for the benefit of the retailer and it is called Ambient Scent. Basically, it is a specific scent that consumers can feel in a store, connected to the store environment itself [13]. A research study concluded that in a scented shoes store, 84% of people were more likely to appreciate shoes, or even to buy them. In the same study, it was reported that consumers would pay 10-15% more for a product, in a pleasant ambient [15]. In fact, these scents have the ability of influence costumers’ intentions to visit and revisit a store, as well as contribute for a favorable perception of a store and, indirectly, of product quality [16] [17]. However, when a scent becomes too strong, consumers start to react more negatively, meaning that the intensity of a scent is an imperative requirement to be taken into account [18].
2.4. Perfume Structure
Calkin and Jellinek [19] defined a perfume as a ‘blend of odoriferous materials, which are perceived as having its own unique and aesthetically appropriate identity’. Basically, perfumes are complex homogeneous liquid mixtures of fragrant ingredients and solvents with their own olfactory identity, that can be sprayed by their users into their skin and clothes [9].
Fragrance components belong to the group of organic compounds, with a low molecular weight (M < 300 g/mol). At atmospheric pressure and room temperature, these molecules have a vapor pressure high enough to be partially in the vapor phase, so that they can be perceived by human’s olfaction [20]. These components are classified into three main fragrant notes: top, middle and base notes. Top notes are the most volatile, which means that they are the ones smelled in the first place, and they last no longer than a few minutes [21]. As the evaporation takes place and top notes fade away, middle notes start to be noticed, and this occurs only a couple hours after the application. These notes are the body of the perfume. Finally, when the odor changes, after several hours of its application (more than 8 hours, sometimes even days) base notes appear. These are the notes that have the lowest volatility, which means that they can last in the
air for hours or days. Base notes play a very important role, once they can be used as fixatives, i.e., substances that influence molecular interactions, with the purpose of changing evaporation rates of the top and middle notes [21]. The structure of a perfume can be represented in a pyramid (Figure 1) with the following distribution: 15-25% of top notes, 20-40% of middle notes, and at the bottom, 45-55% of base notes [22].
Figure 1. Triangular structure of a perfume: top, middle and base notes.
There are some technical requirements that a perfume must necessarily have in order to perform reasonably, such as [19]:
sufficiently strong to be noticed; diffusive;
persistent to last;
retain its character to be noticed many hours after of its application; chemically stable in the end product.
The formulation of a successful fragrance is a complex task, involving dozens of odorant chemicals – approximately 50 to 100, with different physicochemical properties (volatility, chemical structure, solubility, substantivity), and different psychophysics properties like, thresholds, that are described below [19] –, solvents, surfactants, preservatives and other ingredients that can compromise the release and diffusion of fragrances and, at a last stage, influence its perception by the human nose.
2.4.1. Volatility and Chemical Structure
When odorant molecules enter the nasal cavity and touch the olfactory receptors located there, odor perception takes place, which means that fragrant components have to be in the vapor phase to be perceived. Therefore, evaporation is a fundamental process to perfumery, and volatility is one of the most important properties in this area, being defined as the odorant components’ readiness to pass into the vapor phase. In general, the larger the molecular size (the number of carbon atoms), the lower volatility of the compound.
Context and State of the Art 6 However, when a functional group containing oxygen is present, the volatility is greatly reduced. It can be said that, the smaller the molecule, the greater is the effect of the functional group upon volatility. The effect of lower volatility in molecules with functional groups containing oxygen can be explained by the polarization of the electrical charge within the molecule. Polar molecules are mutually attracted, reducing the readiness to separate, lowering the volatility [19].
2.4.2. Solubility
The relation between the attraction forces among molecules of a substance and the attraction forces that occurs between the molecules of the solvent allows the determination of the solubility of the substance in different solvents. If these two forces are similar in strength and in kind, solubility is high, if they are different, solubility is low. So, it can be said that polar solvents are good solvents for polar substances, and poor for nonpolar substances [19].
2.4.3. Substantivity
This property plays an important role in the persistence of the perfume on the skin, hair and textiles fibers. For instance, the persistence of a perfume on wool fibers is very different from the persistence on other fibers, like nylon or cotton, due to the difference of the molecular structure of fibers and the opportunity or not of hydrogen bond formation [19].
2.4.4. Thresholds
The term threshold refers to a certain value that must be exceeded for a result to occur or to be manifested. Regarding odor thresholds (OT), two parameters are often defined: odor detection threshold (ODT) and odor recognition threshold (ORT). The difference between the two lies in the fact that the first represent the lowest concentration at which a fragrant component can be noticed by human nose, or at which significant detection takes place, while the second one denotes the lowest concentration at which an odorant can be recognized for what it is [19].
2.5. Odor Intensity and Odor Character Models
An odor intensity model is required to predict or calculate the perceived magnitude of odors from a mixture using the vapor concentrations. There are several models based on psychophysics, like Odor Value (OV), Stevens’ power law (ψ) and Strongest Component model, as described below.
2.5.1. Odor Value (OV)
Odor value (OVi) concept can be defined as the ratio of the concentration of an odorant species in the vapor
phase (𝐶𝑖𝑔𝑎𝑠) and the odor threshold in the air (OTi) (Equation 2.1), and measures the odor intensity of a
fragrant species in the headspace (what we smell). The odor threshold was already studied over the past years by different authors, so it can be found in data series from the literature, for a large number of fragrant species [1].
2.5.2. Stevens’ Power Law
Stevens’ power law is also an odor intensity model commonly used (Equation 2.2). This method assumes that the sensation is proportional to the stimulus raised to a power n [7]:
where ni is the power law exponent for each component.
2.5.3. Strongest Component Model
The combination of different smells that have different rates of evaporation, combined with different perceived odors by the receptors cells located in the human nose, results in different odor character of a perfume mixture. The Strongest Component (SC) model is used to evaluate the odor character of a fragrance mixture (Equation 2.3), and defines that within a mixture of N odorants, the one that have the highest odor intensity will dominate the odor of the mixture [23].
2.6. Fragrance Performance
The performance of a fragrance can be characterized using parameters commonly employed by perfumers like impact, tenacity, diffusion and volume (Figure 2). Impact is evaluated during the first moments near the application; diffusion represents the efficacy of a perfume at some distance from the source; tenacity
𝑂𝑉𝑖 = 𝐶𝑖 𝑔𝑎𝑠 𝑂𝑇𝑖 (2.1) 𝜓𝑖 = (𝐶𝑖 𝑔𝑎𝑠 𝑂𝑇𝑖) 𝑛𝑖 (2.2) 𝜓𝑖 = 𝑚𝑎𝑥{𝜓𝑖}, ∀ 𝑖 = 1, … , 𝑁 (2.3)
Context and State of the Art 8 measures the persistency of a perfume near the source, but for long times after its application; and volume is the effectiveness of a perfume over distance, some time after its application [24].
Figure 2. Performance parameters as a function of time and distance.
2.7. Diffusion Process
Diffusion is the process by which random molecular motions result in the transport of matter from one part of a system to another, from higher to lower concentrations. Isotropic diffusion happens when molecules have no preferred direction of motion towards one or another [25].
The magnitude of the diffusion coefficient is indicative of the atoms diffusion rate and this rate depends on several factors, such as [26]:
Temperature – particles have more energy and touch each other more frequently with increasing temperature, resulting in the increase of diffusion rates;
Diffusing species and host material – molecules present in the host material act as a barrier to diffusion. Collisions between host material molecules and diffusing particles result in the reduction of the diffusion rate.
Regarding fragrances, the diffusion process occurs after the odorant evaporates from the liquid, thus it is also important to address this process. Evaporation results from a phase change (liquid to vapor). It is affected by several factors such as [26]:
Temperature – when temperature increases, liquid molecules begin to move faster and collide more, until some escape to the headspace;
Exposed area – the greater the area of the interface liquid-vapor, the greater the evaporation; Vapor pressure – it is the pressure that a vapor exerts in equilibrium with liquid. Thus, when the
vapor pressure increases, the probability of evaporation also increases;
Strength of intermolecular forces – when molecules in the liquid create strong bonds with each other, the evaporation rate decreases.
Concerning the prediction of fragrance diffusion, a theoretical model considering that the diffusion of fragrant molecules in the gas phase occurs in the axial direction (one-dimensional diffusion) was developed and validated by our research group in a recent past [27]. This model is mathematically described by the Fick’s second law and the diffusion simulated considering an ideal gas phase and a nonideal liquid mixture, in which the concentration and volume of the liquid change as a function of time. In this context, Product Engineering has been used to introduce some scientific knowledge into an empiric and experimental area, based not only on basic concepts from Chemical Engineering and transport phenomena, but also on psychophysics (a fusion of psychology and physics, where the physical stimuli and their properties relate to a sensory process; it can also refer to methods of analyzing an organism perception) [9]. Through the use of scientific principles, it is possible to predict the odor of complex fragrance mixtures through the liquid composition and using psychophysical parameters [6].
2.8. Trail of a Fragrance
Fragrances, even invisible, have the ability to say so much about the tastes and personality of its wearer, though remaining completely silent. They have the potential to leave a lasting impression on anyone who catches a passing scent, being the reason why we are so interested on choosing a perfume capable of leaving a scented trail, when buying one.
The word sillage, in French, is the technical word to describe the motion of a fragrance, like the trail left by a boat as it moves through water. It is not about how long a fragrance lasts on the skin, but how far it travels away and diffuses around from the wearer. There are different levels of sillage: a strong one means that a perfume performs well, while minimal sillage fragrances are the ones that stay close to the skin, creating a more intimate scented aura. The preference by one or another depends largely on the personality of its wearer [28].
Radial Diffusion in an Infinite Medium 10
3. Radial Diffusion in an Infinite Medium
3.1. Radial Diffusion considering a Continuous Source
3.1.1. Methods
The first approach to evaluate the diffusion of fragrances consists in studying the radial diffusion of odorant components, continuously released from a liquid source, in an infinite medium. Considering an isotropic diffusion in the gas phase, which means that the diffusion rate is the same regardless of which direction the component is diffusing in [25], a mass balance was applied over the gas control volume presented in Figure 3. The mathematical model used to describe this problem was based on Fick’s second law for radial diffusion, where the variable 𝑟 represents the distance from the source, and can be related to the cartesian coordinates 𝑥, 𝑦 and 𝑧 by the following relation: 𝑟2= 𝑥2+ 𝑦2+ 𝑧2. The unsteady-state mass balance is
described by the following partial differential equation (PDE) (Equation 3.1):
where 𝑡 is the time (s) and 𝐷𝑖,𝑎𝑖𝑟 is the diffusion coefficient of component 𝑖 in air (m2/s). The derivation of
this PDE is presented in Appendix A.
Figure 3. Scheme of the simulated gas and liquid phases.
As initial conditions (IC), it was defined that for 𝑡 = 0: Gas phase: 𝐶𝑖𝑔𝑎𝑠 = 0
The two boundary conditions (BC) (𝑡 > 0) for the resolution of the PDE are: 𝑟 = 𝑅𝑠 ∶ 𝐶𝑖𝑔𝑎𝑠 = 𝐶𝑖𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑟 → ∞: 𝐶𝑖𝑔𝑎𝑠= 𝑙𝑖𝑚𝑖𝑡𝑒𝑑 𝜕𝐶𝑖𝑔𝑎𝑠 𝜕𝑡 = 𝐷𝑖,𝑎𝑖𝑟 𝑟2 𝜕 𝜕𝑟(𝑟2 𝜕𝐶𝑖𝑔𝑎𝑠 𝜕𝑟 ) (3.1) s
where 𝐶𝑖𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 represents the gas concentration in equilibrium at the interface with liquid (g/m3),
calculated by Equation 3.2 and 𝑅𝑠 is the radius of the liquid source.
where 𝑥𝑖 represents the mole fraction of component 𝑖 in the liquid phase, 𝛾𝑖 is the activity coefficient of component 𝑖, 𝑃𝑖𝑠𝑎𝑡 represents the saturated vapor pressure for component 𝑖 (Pa), 𝑀
𝑖 is the molecular mass
of component 𝑖 (g/mol), 𝑅 is the universal gas constant (m3 Pa/(mol K)) and 𝑇 is the temperature (K).
The activity coefficients were predicted using the thermodynamic UNIversal Functional Activity Coefficient (UNIFAC) method, and the respective equations are presented in Appendix B.
The analytical solution of Equation 3.1 was obtained using the method of Laplace transforms. This method is very useful to determine the solution of various problems in mathematical physics. Its application to the diffusion equation removes the time variable, leaving an ordinary differential equation, which solution yields the transform of the concentration as a function of the space variable 𝑟. This is then interpreted, according to certain rules, to give an expression for the concentration in terms of space variable and time, satisfying the initial and boundary conditions [25]. Appendix A also presents the application of this method to obtain the analytical solution of Equation 3.1 (Equation 3.3).
In this equation, erfc is the complementary error function.
3.1.2. Materials
In this study, Equation 3.3 was graphically represented for two fragrance mixtures: a binary mixture composed of a top (limonene) and a middle (linalool) note, and a ternary mixture, with a top (limonene), a middle (linalool) and a base note (vanillin). Table 1 shows some relevant properties of each component used in this study.
Equation 3.3 allowed to represent the concentration profiles over time and distance, released from a liquid source. The radius of the liquid source was defined as 3 cm, corresponding to a liquid volume of approximately 110 mL. 𝐶𝑖𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚= 𝑥𝑖 𝛾𝑖𝑃𝑖 𝑠𝑎𝑡 𝑀 𝑖 𝑅 𝑇 (3.2) 𝐶𝑖𝑔𝑎𝑠(𝑡, 𝑟) =𝐶𝑖 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚𝑅 𝑠 𝑟 erfc ( 𝑟 − 𝑅𝑠 2√𝐷𝑖,𝑎𝑖𝑟𝑡 ) (3.3)
Radial Diffusion in an Infinite Medium 12 Table 1. Properties of the studied fragrance components: Molecular Formula, Molecular Weight (Mi), Vapor Pressure (𝑃𝑖𝑠𝑎𝑡), Odor
Threshold (OT), Olfactory Power Law Exponent (ni), Density (ρl,i) and Diffusion Coefficients in Air (Di,air).
a From Chemspider Database. [29]; b Vapor pressures for pure components were obtained at 296.15 K; c From Teixeira
et al. (2013) [27]; d OTs were geometrically averaged from data available in van Gemert, L.J. [30]; e From Devos et
al. [31]; f Estimated from Fuller et al. [32]; g The median power law exponent value in the compilation of data from
Devos et al. [31]; h From Mackay et al. [33]; i From Teixeira et al. (2011) [34]; j OT was geometrically averaged from
data available in Murnane et al. [35]; k From PubChem, U.S National Library of Medicine. [36].
Table 2 shows the mole fraction of each fragrance component in the studied fragrance systems. Please note that mole fractions of the systems were chosen taking into consideration the recommended proportions for top, middle and base notes [22].
Table 2. Mole fractions (xi) of the fragrance components present in the liquid phase, of systems FS1 and FS2.
3.1.3. Results and Discussion
3.1.3.1. Fragrance System FS1: Limonene and Linalool
For the case of the binary mixture composed of limonene and linalool, the gas concentrations were evaluated at two different fixed times – 1 min and 360 min (6 hours) – and considering that components are continuously released from a liquid source and that the diffusion is radial (Figure 4). It can be seen that the gas concentration in equilibrium with the liquid, determined by Equation 3.2, is higher for limonene than for linalool. Also, it is possible to see that for a fixed time of 1 min, the concentration of limonene and
Fragrance Component Molecular Formula a 𝑴𝒊 (g/mol) a 𝑷𝒊𝒔𝒂𝒕 (Pa) b 𝑶𝑻 (g/m3) 𝒏𝒊e 𝝆𝒍,𝒊 (g/mL) 𝑫𝒊,𝒂𝒊𝒓 (m2/s) f Limonene C10H16 136.2 205.4 c 6.19 × 10-4 d 0.37 0.879 a,c 6.04 × 10-6 Linalool C10H18O 154.3 22.1 c 9.26 × 10-6 d 0.35 g 0.858 a,c 5.84 × 10-6 Vanillin C8H8O3 152.2 2.44 × 10-2 h,i 7.29 × 10-7 j 0.31 1.060 k 6.49 × 10-6 Fragrance Component xi FS1 FS2 Limonene 0.425 0.193 Linalool 0.575 0.262 Vanillin - 0.545
linalool nearly reaches 10 cm away from the source. On the other hand, after 360 min of release, limonene has already spread for a distance of 1 m.
Figure 4. Gas concentration profiles of fragrance system FS1 over distance, at fixed times: 1 min (left) and 360 min (right) after fragrance release.
For a complete evaluation of fragrance performance, the gas concentrations were converted into odor intensities (Figure 5) through psychophysics models, specifically the Stevens’ power law (Equation 2.2) and the SC model (Equation 2.3). It is possible to conclude that linalool is the dominant component over distance, either after 1 or 360 min.
Figure 5. Odor intensity profiles of fragrance system FS1 over distance, at fixed times: 1 min (left) and 360 min (right) after fragrance release.
3.1.3.1. Fragrance System FS2: Limonene, Linalool and Vanillin
A ternary mixture, composed by a top (limonene), a middle (linalool) and a base (vanillin) notes, was also assessed, either in terms of gas concentrations (Figure 6) or odor intensities (Figure 7). The results show the effect of the base note present in the liquid mixture, causing a decrease in the gas concentrations of limonene and linalool. Comparing Figures 5 and 7, it can be seen that the behavior of limonene and linalool is very similar, being linalool the strongest component in both fragrance mixtures. The main difference
t=1min Distance (m) 0.0 0.2 0.4 0.6 0.8 1.0 i 0 10 20 30 40 50 Limonene Linalool t=360 min Distance (m) 0.0 0.2 0.4 0.6 0.8 1.0 i 0 10 20 30 40 50 Limonene Linalool t=1min Distance (m) 0.0 0.2 0.4 0.6 0.8 1.0 G as concentrat ion (g/mL) 0 2x10-6 4x10-6 6x10-6 8x10-6 Limonene Linalool t=360 min Distance (m) 0.0 0.2 0.4 0.6 0.8 1.0 G as concentrat ion (g/mL) 0 2x10-6 4x10-6 6x10-6 8x10-6 Limonene Linalool
Radial Diffusion in an Infinite Medium 14 between both profiles lies on the fact that the odor intensities of limonene and linalool are lower in the presence of the base note, proving that the liquid phase composition influences the release of fragrance components, and consequently, their corresponding odor intensities.
Figure 6. Gas concentration profiles of fragrance system FS2 over distance, at fixed times: 1 min (left) and 360 min (right) after fragrance release.
Figure 7. Odor intensity profiles of fragrance system FS2 over distance, at fixed times: 1 min (left) and 360 min (right) after fragrance release.
3.2. Radial Diffusion considering a Non-Continuous Source
3.2.1. Materials
3.2.1.1. Chemicals
R-(+)-Limonene (CAS No. 5989-27-5, purity 97%) was obtained from Sigma-Aldrich, (-) α-pinene (CAS No. 7785-26-4, purity ≥ 98%) was obtained from Fluka, linalool (CAS No. 78-70-6, purity ≥ 97%) was obtained from Sigma-Aldrich and geranyl acetate (CAS No. 105-87-3, purity 98%) was obtained from Acros Organics. All reagents were used as received without further purification. Table 3 presents some relevant physicochemical properties of these components.
t=1min Distance (m) 0.0 0.2 0.4 0.6 0.8 1.0 G as concentrat ion (g/mL) 0 2x10-6 4x10-6 6x10-6 Limonene Linalool Vanillin t=360 min Distance (m) 0.0 0.2 0.4 0.6 0.8 1.0 G as concentrat ion (g/mL) 0 2x10-6 4x10-6 6x10-6 Limonene Linalool Vanillin t=1min Distance (m) 0.0 0.2 0.4 0.6 0.8 1.0 i 0 10 20 30 40 50 Limonene Linalool Vanillin t=360 min Distance (m) 0.0 0.2 0.4 0.6 0.8 1.0 i 0 10 20 30 40 50 Limonene Linalool Vanillin
Table 3. Properties of the studied fragrance components used in this work: Molecular Formula, Molecular Weight (Mi), Vapor
Pressure (𝑃𝑖𝑠𝑎𝑡), Odor Threshold (OT), Olfactory Power Law Exponent (ni), Density (ρl,i) and Diffusion Coefficients in Air (Di,air).
a From Chemspider Database [29]; b Vapor pressures for pure components were obtained at 296.15 K; c From Teixeira
et al. (2013) [27]; d OTs were geometrically averaged from data available in van Gemert, L.J. [30]; e From Devos et
al. [31]; f Estimated from Fuller et al. [32]; g Estimated from Antoine Equation [37]; h The median power law exponent
value in the compilation of data from Devos et al. [31]; i From Elsharif and Buettner (2016) [38]; j From PubChem,
U.S National Library of Medicine [36].
3.2.2. Methods
3.2.2.1. Radial Diffusion Model
As an attempt to create a model for predicting the radial diffusion of a liquid fragrance mixture, a small liquid volume (1 mL) evaporating over time and distance was considered.
Gas phase
The diffusion in the gas phase was considered isotropic [25] and Figure 1 shows the gas control volume in which the mass balance of the gas phase was applied. The Radial Fragrance Diffusion model is based on Fick’s second law for radial diffusion, where the variable 𝑟 also represents the distance from the source. The unsteady-state mass balance is described by the following partial differential equation (PDE) (Equation 3.4):
where yi is the mole fraction of component i in the gas phase and it is calculated from:
Fragrance Component Molecular Formula a 𝑴𝒊 (g/mol) a 𝑷𝒊𝒔𝒂𝒕 (Pa) b 𝑶𝑻 (g/m3) 𝒏𝒊 e 𝝆𝒍,𝒊 (g/mL) a,c 𝑫𝒊,𝒂𝒊𝒓 (m2/s) f Limonene C10H16 136.2 205.4 c 6.19 × 10-4 d 0.37 0.879 6.044 × 10-6 α-Pinene C10H16 136.2 513.4 c 2.40 × 10-4 d 0.49 0.879 6.044 × 10-6 Linalool C10H18O 154.3 22.1 c 9.26 × 10-6 d 0.35 h 0.858 5.838 × 10-6 Geranyl acetate C12H20O2 196.3 5.38 g 6.18 × 10-5 i 0.35 h 0.910 5.263 × 10-6 Ethanol C2H6O 46.04 7050 c 1.21 × 10-1 d 0.58 0.789 j 1.23 × 10-5 𝜕𝑦𝑖 𝜕𝑡 = 𝐷𝑖,𝑎𝑖𝑟( 𝜕2𝑦 𝑖 𝜕𝑟2 + 2 𝑟 𝜕𝑦𝑖 𝜕𝑟) (3.4)
Radial Diffusion in an Infinite Medium 16 where CT is the total gas concentration [𝐶𝑇 =
𝑃
𝑅𝑇, where P is the total pressure in the system (Pa)]. CT is
constant since the gas behavior is considered ideal.
Liquid phase
The liquid phase was considered a nonideal mixture of odorant components. As the liquid composition changes during the evaporation of the fragrance, the mass balance to the liquid phase considers the number of moles that are transferred through the gas-liquid (Agl) interface, as described by the following ordinary
differential equation (ODE):
where n’i is the number of moles of component i in the liquid phase (mol).
It is important to note that once a small volume of liquid is being considered, it was assumed that the radius of the liquid source is constant over time.
Initial and boundary conditions
As initial conditions (IC), it was defined that for t = 0: Gas phase: yi = 0
Liquid phase: n’i = n’i,0
where n’i,0 represents the initial number of moles in the liquid phase.
For the resolution of the PDE two boundary conditions (BC) were defined (t > 0): 𝑟 = 𝑅𝑠: 𝑦𝑖 =𝛾𝑖𝑃𝑖𝑠𝑎𝑡
𝑃 𝑥𝑖
𝑟 = 𝑅𝑚𝑎𝑥: 𝑦𝑖 = 0
where Rmax is the maximum distance in the radial coordinate. The activity coefficients (γi) were numerically
predicted using the thermodynamic UNIFAC method. Table B 1 (Appendix B) shows the UNIFAC groups 𝑦𝑖 =𝐶𝑖 𝑔𝑎𝑠 𝐶𝑇 (3.5) 𝑑𝑛′𝑖 𝑑𝑡 = 𝐷𝑖,𝑎𝑖𝑟 𝐴𝑔𝑙 𝐶𝑇 𝜕𝑦𝑖 𝜕𝑟|𝑟=𝑅𝑠 (3.6)
and subgroups of each molecule i, the number of groups of each type j in molecule i (𝜈𝑗(𝑖)) and the parameters of molecular van der Waals volumes (𝑅𝑗(𝑖)) and surface area (𝑄𝑗(𝑖)).
Numerical Simulation
The unsteady-state concentration profiles over time and distance for each species are obtained from Equations. 3.4 and 3.6 together with the IC and BC. The numerical solutions were obtained using the gPROMS software version 4.2.0. The radial co-ordinate was discretized using the second order Backward Finite Difference Method (BFDM), with 1500 points and a relative and absolute tolerance equal to 10-5.
Once the problem is expected to exhibit large gradients at the start of the domain, a logarithmic transformation was used to place more nodes near the lower bound of the domain.
3.2.3. Equivalence Relation between the Axial and Radial Diffusion Models
The proposed methodology was validated using the diffusion tube due to its simplicity and accuracy to validate axial diffusion [27]. However, as it only describes the axial diffusion of components, it was necessary to find an equivalence relation between the axial and radial diffusion models. This equivalence relation was obtained for two single mixtures composed of limonene and ethanol for different times (2 h and 5 h) and distances (0.13, 0.38, 0.63, 1.13 and 1.63 m), using the predicted gas concentrations obtained through the axial model described by Teixeira et al. (2013) (which were validated by the diffusion tube) and the predicted gas concentrations assessed by gPROMS considering a radial diffusion. The 𝐶𝑎𝑥𝑖𝑎𝑙
𝑔
𝐶𝑟𝑎𝑑𝑖𝑎𝑙𝑔 ratio
results for different times and distances allowed concluding that this ratio is only dependent on the distance (Table 4).
Figure E 1 from Appendix E shows the equivalence relation between 𝐶𝑎𝑥𝑖𝑎𝑙𝑔 and 𝐶𝑟𝑎𝑑𝑖𝑎𝑙𝑔 for each time and component.
The equivalence relation was then obtained by plotting the average of 𝐶𝑎𝑥𝑖𝑎𝑙
𝑔
𝐶𝑟𝑎𝑑𝑖𝑎𝑙𝑔 = 𝜑𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 (Table 5) as a
function of distance (Figure 8). Therefore, this relation is given by:
Radial Diffusion in an Infinite Medium 18 Figure 8. Equivalence relation between the gas numerical 1D axial and 1D radial models (gPROMS).
Table 4. Predicted gas concentration values, considering axial and radial diffusion, assessed at three different times (2 h and 5 h), for limonene and ethanol.
