[insert Figure 4 here]
LD50
\(LD_{50}\) of the perfume solution is calculated by Eq. 24. It depends
on the toxicity of ingredients (\(LD_{50,i}\)) and the mass fraction
(\(m_{i}\)) converted from the volume fraction \(V_{i}\).
\(LD_{50}=\frac{1}{\sum_{i=1}^{48}\frac{m_{i}}{LD_{50,i}}}\) (24)
\(m_{i}=\frac{V_{i}\bullet\rho_{i}}{\sum_{j=1}^{48}{V_{j}\bullet\rho_{j}}}\)(25)
where \(LD_{50,i}\) and the density (\(\rho_{i}\)) for the 48 ingredient
candidates are given in Table S2.
Flash point
The flash point (\(T_{\text{fp}}\)) of a flammable liquid mixture can be
theoretically determined based on the Le Chatelier’s mixing
rule.53
\(\sum_{i=1}^{48}\frac{\text{FP}P_{i}}{\text{FPLF}L_{i}}=1\) (26)
where \(\text{FP}P_{i}\) and \(\text{FPL}FL_{i}\) are the partial
pressure and lower flammability limit of the i -th ingredient
candidate at the flash point, respectively. \(\text{FPL}FL_{i}\) is
calculated by
\(\text{FPL}FL_{i}=LFL_{i}^{*}-\frac{0.182\times(T_{\text{fp}}-298)}{Hc_{i}}\)(27)
where \(Hc_{i}\) and \(\text{LF}L_{i}^{*}\) are the heat of combustion
and lower flammability limit at 298 K (see Table S2), respectively.\(\text{FP}P_{i}\) is calculated via the vapor-liquid equilibrium in Eq.
28. The UNIFAC model is used to calculate the activity coefficient\(\text{FPγ}_{i}\) at the flash point. The mole fraction \(x_{i}\) is
converted from mass fraction \(m_{i}\). \(\text{FPPsa}t_{i}\) is the
saturated vapor pressure at flash point, which is calculated using the
Antoine equation in Eq. 31.
\(\text{FP}P_{i}\ =\text{FPγ}_{i}\bullet x_{i}\bullet FPP\text{sat}_{i}\)(28)
\(\text{FPγ}_{i}=f_{\text{unifac}}\left(x_{i},T_{\text{fp}}\right)\)(29)
\(x_{i}=\frac{m_{i}}{MW_{i}\bullet\sum_{j}\frac{m_{j}}{MW_{j}}}\)(30)
\(\operatorname{}{\text{FPPsa}t_{i}}=A_{i}-\frac{B_{i}}{C_{i}+T_{\text{fp}}}\)(31)
The molecular weight \(MW_{i}\), UNIFAC parameters, and Antoine
coefficients \(A_{i}\), \(B_{i}\), and \(C_{i}\) for the 48 ingredient
candidates are given in Table S2.
Homogeneous solution
To ensure a homogeneous solution, the volume of selected organic
fragrances must be less than their volume solubility (\(SV_{i,ew}\)) in
the ethanol-water solvent system.
\(\frac{V_{i}}{(V_{47}+V_{48})}\leq SV_{i,ew},\ \ \ i=1,\ldots,46\)(32)
It is found that it is quite hard to calculate \(SV_{i,ew}\) using
rigorous thermodynamic models due to the many missing parameters. In the
literature, several short-cut models have been developed to predict\(SV_{i,ew}\). The log-linear mixture rule below is widely
used.54
\(\log{\text{SV}_{i,ew}=}\log{SV_{i,w}}+\beta\bullet\log\frac{SV_{i,e}}{SV_{i,w}},\ \ \ \ i=1,\ldots,46\)(33)
\(\beta=\frac{V_{47}}{V_{47}+V_{48}}\) (34)
\(\log\frac{SV_{i,e}}{SV_{i,w}}=M\bullet\log K_{ow,i}+N,\ \ \ \ i=1,\ldots,46\)(35)
where \(SV_{i,e}\) and \(SV_{i,w}\) are the volume solubility in ethanol
and water, respectively.\(\ K_{ow,i}\) is the n-octanol/water partition
coefficient of the i -th candidate. \(M\) and \(N\) are the
cosolvent constants. Based on experimental data, their values have been
regressed as 0.81 and 0.85, respectively.
Odor type in top note
The fragrance molecules in a perfume solution first evaporate into the
air through the liquid-gas interface. Then, the molecules diffuse in the
air (assumed to be stagnant) and are detected at certain distance away.
The processes of evaporation, diffusion, and detection have been
modelled using chemical engineering principles and
psychophysics.38,52,55 Perfume evaporation is
simulated using Eq. 36 with an initial condition. The liquid molar
changes are equal to the moles of ingredients transported through the
interface (i.e., \(z=0\)).
\(\frac{dn_{i,t}}{\text{dt}}=C_{T}\bullet D_{i}\bullet A_{\lg}\ \bullet\left.\ \frac{\partial y_{i,t,z}}{\partial z}\right|_{z=0}\)(36)
Initial condition: \(n_{i,t=0}=n_{p}\bullet x_{i}\)
After discretization, Eq. 37 is obtained.
\(\frac{n_{i,t+t}-n_{i,t}}{t}=C_{T}\bullet D_{i}\bullet A_{\lg}\ \bullet\frac{{y_{i,t,z=z_{1}}-y}_{i,t,z=0}}{z_{1}}\)(37)
where \(n_{p}\) is the initial number of moles of perfume solution.\(C_{T}=P/RT\) is a constant.\(\ D_{i}\) and \(A_{\lg}\) are the
diffusivity of i -th candidate and interfacial area, respectively.\(t\) and \(z_{1}\)are the time interval and the first distance
interval, respectively. These parameters are given in Table S2.\(n_{i,t}\) is the number of moles of the i -th candidate in the
liquid at time \(t\). \(y_{i,t,z}\) is the molar fraction of i -th
ingredient candidate in the air at time \(t\) at distance z . It
is calculated via vapor-liquid equilibrium.
\(y_{i,t,z=0}=\gamma_{i,t}\bullet x_{i,t}\bullet\frac{\text{Psa}t_{i}}{P}\)(38)
\(\gamma_{i,t}=f_{\text{unifac}}(x_{i,t},T_{r})\) (39)
\(x_{i,t}=\frac{n_{i,t}}{\sum_{i=1}^{48}n_{i,t}}\) (40)
where \(\gamma_{i,t}\) and \(x_{i,t}\) are the activity coefficient and
mole fraction of i -th ingredient candidate at time t ,
respectively. \(\text{Psa}t_{i}\) is the saturated vapor pressure at
room temperature \(T_{r}=298\ K\).
After evaporation, fragrance diffusion is modelled based on Fick’s 2nd
law of diffusion with one initial condition and two boundary conditions
(Eq. 41).
\(\frac{\partial y_{i,t,z}}{\partial t}=D_{i}\bullet\frac{\partial^{2}y_{i,t,z}}{\partial z^{2}}\)(41)
Initial condition: \(y_{i,t=0,z}=0\)
Boundary conditions: Eq. 38, \(y_{i,t,z=z_{\max}}=0\)
The initial condition assumes that no fragrances exist in the air before
diffusion begins (i.e., \(t=0\)). The boundary conditions indicate
that vapor-liquid equilibrium is maintained at the interface at any time
(i.e., Eq. 38) and no fragrances exist beyond the maximum distance
(\(z_{\max}=2m\)). This model is discretized using a non-uniform
distance grid (Table S2) for reducing the computational difficulty.
After discretization, we get
\(\frac{y_{i,t+t,z}-y_{i,t,z}}{t}=D_{i}\bullet\frac{\frac{y_{i,t,z+z_{j+1}}-y_{i,t,z}}{z_{j+1}}-\frac{y_{i,t,z}-y_{i,t,z-z_{j}}}{z_{j}}}{0.5\times(z_{j+1}+z_{j})}\),\(z\in[0,z_{\max}]\) (42)
where \(z_{j}\) and \(z_{j+1}\) are the distance intervals,
respectively.
Any fragrance with a different concentration leads to a different
intensity. Many theoretical models (e.g., Weber-Fenchner law, power law,
and linear law) have been proposed for quantifying odor intensity. The
power law is chosen here because it fits experimental data well. The
intensity of the i -th odorant is defined as the ratio of its
concentration in the air (\(c_{i}\) in g/m3) to its
odor recognition threshold value (\(\text{OR}T_{i}\)), raised to a power\(oe_{i}\).52 With this, the odor intensity in the top
note is determined based on the mole fraction of fragrances in the air
at 5 minutes (\(t_{\text{tn}}\)) after application at a distance of 0.2
m (\(z_{\text{tn}}\)).
\(\psi_{i}=\left(\frac{c_{i}}{\text{OR}T_{i}}\right)^{oe_{i}}\) (43)
\(c_{i}=y_{i,t_{\text{tn}},z_{\text{tn}}}\bullet MW_{i}\bullet C_{T}\)(44)
Given multiple odorants, the one with the highest intensity is more
strongly sensed and can be regarded as the major odor type. Thus, the
dominant odor type in top note is expressed as
\(OTTN=i,\ \ if\ \psi_{i}=\psi_{\max}\operatorname{=}\left\{\psi_{i}\right\}\)(45)
Heuristics
Following Table 3, constraints for the Eau de parfum formulation are
derived from dozens of modern Eau de parfum recipes.51It is found that the suggested number of ingredients for each fragrance
note can be represented by Eq. 46-48. Eq. 49 shows that Eau de parfum
usually contains 10-20% organic fragrances. The suggested volumetric
proportions for top note and middle note are 15-25% and 30-40%,
respectively (Eq. 50-51). The suggested volume fraction of water is
9-13%.49,52
\(3\leq\sum_{i=1}^{17}S_{i}\leq 6\) (46)
\(3\leq\sum_{i=18}^{33}S_{i}\leq 6\) (47)
\(2\leq\sum_{i=34}^{46}S_{i}\leq 5\) (48)
\(0.1\leq\sum_{i=1}^{46}V_{i}\leq 0.2\) (49)
\(0.15\bullet\sum_{i=1}^{46}V_{i}\leq\sum_{i=1}^{17}V_{i}\leq 0.25\bullet\sum_{i=1}^{46}V_{i}\)(50)
\(0.3\bullet\sum_{i=1}^{46}V_{i}\leq\sum_{i=18}^{33}V_{i}\leq 0.4\bullet\sum_{i=1}^{46}V_{i}\)(51)
\(0.09\leq V_{48}\leq 0.13\) (52)
Iterative Model Adoption and Optimization Solution
Strategy
The identified rigorous mechanistic models for \(LD_{50}\), flash point,
and odor type, the short-cut model for transparency, the surrogate model
for sensorial rating as well as the heuristics in Eq. 46-52 are
integrated to form the perfume formulation problem below.
\(\operatorname{}q_{s}\) (53)
s.t. Eq. 21-23 ANN-based surrogate model for \(q_{s}\)
Eq. 16-18 Design targets
Eq. 24-45 Mechanistic models
Eq. 46-52 Heuristics
Eq. 19-20 Design variables
This problem is implemented in GAMS 24.7 on a laptop with Intel 3.30 GHz
CPU. The global solver BARON is used first and then the local solver SBB
is employed if no optimal solutions are obtained from BARON.
GDP reformulation
Because of the complexity of the identified models and the number of
intermediate variables, the problem is directly programmed using GDP.
The disjunction is explicitly expressed as
\(\par
\begin{bmatrix}Y_{i}\\
\par
\begin{matrix}VL_{i}\leq V_{i}\leq VU_{i}\\
\par
\begin{matrix}Eq.25\\
\par
\begin{matrix}Eq.27-31\\
Eq.37-44\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{bmatrix}\bigvee\par
\begin{bmatrix}\neg Y_{i}\\
\par
\begin{matrix}V_{i}=0\\
\par
\begin{matrix}m_{i}=0\\
\par
\begin{matrix}\text{FPLF}L_{i},FPP_{i},FP\gamma_{i},x_{i},FPPsat_{i}=0\\
n_{i,t},y_{i,t,z},\gamma_{i,t},x_{i,t},c_{i},\psi_{i}=0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{bmatrix}\) (54)
The GDP problem is further reformulated using the big-M approach with
the solver JAMS and then solved by SBB. Different initial guesses are
utilized. The second column of Table 5 lists the computational
statistics. It contains 46 discrete variables, 9783 single variables,
and 18230 equations. It takes 3459 seconds to obtain a local optimal
solution.