# Fatty Acid Content in Biomasses: State-of-the-Art and Novel Physical Property Estimation Methods

Authors: Ana M. Sousa; Thalles A. Andrade; Massimiliano Errico; José P. Coelho; Rui M. Filipe; Henrique A. Matos
Journal: International Journal of Chemical Engineering
 (2019)

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## Abstract

In line with the growing environmental awareness developed along the last decades, modern societies are urged to evolve into sustainable economics where the reuse of organic wastes represents the key feedstock for a green transaction. The oil phase obtained from different biomasses has the potential to be a source of food supplements, medicines, cosmetics, or feedstock for biofuel production. In the present work, the composition of 104 different biomasses including seeds, peels, flowers, plants, and leaves has been reviewed for the lipid content. Based on the most frequent fatty acids screened, experimental data for normal boiling point temperature, normal melting point, critical properties, and acentric factor were collected and compared with the most common estimation methods, which are functions of the molecular structure and interaction between different functional groups. New predictive equations have been proposed to reduce the estimation deviation and to provide simple correlations to be used in simulation software when dealing with biomass processes. For all the properties, the estimations proposed have an absolute average deviation equal to or lower than 4.6%.


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## Body

## 1. Introduction

The development of sustainable processes can be pursued through multiple paths, among which decreasing wastes, increasing efficiency in industrial processes, mitigating environmental and human risks, reducing pollutant agents, and fostering energy production from renewable sources play unquestionable roles.Within this context, biomass appears as a valuable feedstock to recover several compounds and/or produce energy to answer the global consumption growth without creating a negative legacy to the next generations [1]. In fact, the widespread employment of biomass not only bears the potential of recovering industrial wastes as a valuable resource but also leverages the energy and industrial productions through efficient and less polluting engineering processes.This work is deemed to study the potential of fatty acids in addressing significant multidisciplinary issues associated with the chemical industry and petroleum exploration [2], incorporated in the broader research initiative aiming to develop integrated processes and product design for sustainable biorefineries [3].Fatty acids are long hydrocarbon chains with a terminal carboxyl group. They can be obtained from rather abundant organic compounds, including organic matter wastes [4, 5] or nonedible oils [6, 7]. Fatty acids are characterized by an outstanding range of applications within the energy domain, such as biofuels [8], and flow improvers for crude oils [9–12], or within other high end industries such as food supplements [13], cosmetics [14], medicines, and drugs [15, 16].Fatty acids, often referred to as fats, are used in the human body to store excess energy, and they constitute the building blocks for membrane cells [17, 18]. Moreover, as recommended by the Food and Agriculture Organization of the United Nations [19], there is convincing evidence that some fatty acids like the linoleic acid and the alpha-linoleic acid are indispensable for human health even if they cannot be synthesized.Considering the broad range of fatty acid applications, an extensive review of their amount and composition in different biomasses has been drawn.Such remarkable potential opens the definition of new industrial processes for fatty acid recovery from different biomasses contributing to valorize poor feedstocks and to reduce their environmental impact when treated as wastes. Such approach paved the way in applying strategies deemed to foster sustainability on an industrial scale.Within a biorefinery approach, the target compounds are recovered or obtained from the biomass through a series of processes classified as thermochemical, biochemical, mechanical/physical, and chemical [20]. It means that, in order to move from the definition of the biomass potential in recovering or obtaining some compounds to the process setup, a reliable model must be available to design the single unit operation and to define the process global performance and profitability. Product and process design rely on the knowledge of the properties of chemical compounds and their mixtures. Although large databases of experimental data are available, covering the complete set of compounds, mixtures, and operating conditions is a huge task that may not be economically feasible. Property estimation models for physical properties are widely used when experimental data are not available.Unlike what can be found in published research concerning thermophysical properties of pure compounds, this work has been focused on the most important temperature-independent physical properties of the main fatty acids found in biomass feedstocks. These properties are also essential in improving the processes modeling, in the formulation of product portfolios and for the composition assessment of the generated pollutants.Published prediction methods were critically analyzed given the scarcity of experimental data for determining the relevant physical properties. Considering current methods shortcomings for the intended applications and using the National Institute of Standards and Technology (NIST) database as a reference, innovative estimation methods were developed. Such correlation’s novelty is not only limited to a higher accuracy but also finds a significant advantage in its simplicity, given the use of molecular weight (MW) and the maximum number of double bonds (DBs) in the molecule as input. To sum up, a reliable method for the estimation of fatty acid (FA) physical properties was developed. Therefore, a set of six new correlations have been proposed to reduce the estimation deviation of boiling point, melting point, critical properties, and acentric factor.Such a tool will be of paramount importance in proposing strategies that congregate high-value addition in industrial processes with the use of abundant biomass and the reuse of potentially polluting waste from food or other related processes, such as the waste from olive oil or wine manufacture.


## 2. Review on Fatty Acid Content in Biomasses

A biorefining process is based on the usage of selected biomass(es) treated in a smaller or larger unit, locally or centrally, for the production of valuable final products or intermediates with applications in pharmaceutical, food and beverages, or even nutraceutical fields [21]. Subsequently, waste biomass effluents could be treated to produce lower added-value products, e.g., insecticides or pesticides, or used as feedstock for biofuel production.A total of 104 different types of oils obtained by extraction of different biomasses were collected from the scientific literature to identify the most common and frequent fatty acids. The list of oil sources and main fatty acid composition is reviewed in Table1. The information was collected and summarized from different published works, and when available, the minimum and maximum concentration values were reported. In some cases, the percentage of oil content in the biomass matrix was also included. This value could be useful to determine a possible upper bound oil extraction yield from the biomass matrix. In Table 1, 16 fatty acids were listed since, according to the sources reviewed, they appear in the oil in a significant amount.Table 1
Fatty acid content in the selected biomasses expressed in weight.
Biomass
Oil content (%)
Caprylic acid, C8 : 0 (%)
Capric acid, C10 : 0 (%)
Lauric acid, C12 : 0 (%)
Myristic acid, C14 : 0 (%)
Palmitic acid, C16 : 0 (%)
Palmitoleic acid, C16 : 1 (%)
Stearic acid, C18 : 0 (%)
Oleic acid, C18 : 1 (%)
Linoleic acid, C18 : 2 (%)
α-Linolenic acid, C18 : 3 (%)
Arachidic acid, C20 : 0 (%)
Eicosenoic acid, C20 : 1 (%)
Eicosadienoic acid, C20 : 2 (%)
Behenic acid, C22 : 0 (%)
Erucic acid, C22 : 1 (%)
Lignoceric acid, C24 : 0 (%)
Others (%)
References
Alexandrian laurel (Calophyllum inophyllum)
NA
13.9
0.2
15.1
40.3
25.6
0.2
0.3
[22]
Almond
NA
5.07–6.78
0–0.5
0–1.4
57.54–73.94
19.32–35.18
0.04–0.10
0–0.9
[23, 24]
Andiroba (Carapa guianensis)
NA
27
1
7
49
16
[25]
Argemone mexicana
30–40
0–0.1
14.1–15.3
0–1.3
3.7–9.8
34.3–45.8
29.0–44.2
0–0.3
0–0.3
0–0.2
[26]
Avocado seeds
1.1–1.6
0–0.28
0.20–0.60
0.54–1.30
17.7–20.85
1.79–4.60
0.90–1.50
17.41–24.1
35.30–38.89
4.10–6.58
0.04–0.80
0.45–2.00
0.40–1.11
0–0.12
1.68–3.30
[27, 28]
Babassu kernel (Orbignya sp.)
60–70
0–5
0–6
35–48
13–16
9–10
2–4
14–17
2–5
0–5
[25, 29, 30]
Bay laurel fruit
NA
18.3
3.5
21.8
0.5
3.8
30.9
20.5
0.4
0.3
[31]
Bay laurel leaves
NA
26.2
4.5
25.5
0.3
3.1
10.6
11.3
17.1
1.4
[24, 31]
Bitter almond
NA
0.1
10.3
3.9
33.9
46.0
4.8
0.2
[32]
Black currant (Ribes nigrum L.)
2.2
6.4
1.6
16.1
57.8
13.2
0.2
4.7
[33]
Burdock seeds
15–20
7
20
69
4
[34]
Camelina
38–47
0.05–0.2
5.1–6.8
0–0.3
2.7–3.4
14.3–19.7
14.3–19.6
28.6–38.4
0.25–1.83
11.9–16.8
1.3–2.2
0.2–1.4
2.3–4.2
0–0.14
0–0.8
[35–38]
Carapa seeds
51
0.5
25.3
10.5
57.8
5.9
[39, 40]
Carob seeds
1.73–1.84
10.3–12.0
0.6
3.5–4.6
28.1–31.8
49.1–51.0
1.4–1.7
0.6
1.7–2.5
[41]
Castor oil
42–54
0.84–1.30
0.78–3.60
2.92–4.90
1.27–5.40
0–0.78
0–0.66
0–0.63
84.7–89.6a
[42–47]
Chia (Salvia hispanica L.)
25–38
5.5–8.9
2.7–4.8
5.3–8.4
15.7–26
50.0–69.3
[48–50]
Chokeberry (Aronia melanocarpa L.)
1.93
5.1
1.1
21.4
71.1
0.5
0.8
[33]
Chlorophyceae (Botryococcus sp.)
6.19
20.56
8.20
4.56
44.74
6.36
15.55
[51]
Chlorophyceae (Chlorella sp.)
14.05
1.60
0.86
35.72
0.85
8.53
21.50
15.30
0.36
15.27
[51]
Chlorophyceae (Chlorococcum sp.)
8.21
0.86
21.65
0.90
0.50
20.60
21.30
2.81
0.42
16.53
12.66
[51]
Chlorophyceae (Dunaliella salina)
6.25
36.40
5.30
50.81
7.46
[51]
Coconut (copra)
NA
8.0–8.9
6.0–6.2
47.0–48.8
17.0–19.9
7.8–9.7
0.1
3.0
4.4–7.0
0.8–2.2
[24, 42, 52]
Corn
NA
11.67–11.80
1.85–2.00
24.80–25.16
60.60–61.30
0–0.48
0–0.24
0.3
[24, 42, 43]
Cottonseed
NA
0.67–3.45
8.83–28.70
0.42–4.05
0.89–14.72
10.3–30.7
20.6–62
[42, 43, 53–55]
Crambe abyssinica
35
2.07
0.70
18.86
9.00
6.85
2.09
0.80
58.51
1.12
[43, 56, 57]
Crataegus monogyna
NA
0.26
0.20
13.21
2.23
33.78
44.60
1.37
1.01
[58]
Cumaru (Dipteryx odorata)
NA
23
7
37
29
4
[25]
Cuphea viscosissima
NA
0.3
64.7
3.0
4.5
7.0
0.9
12.2
6.7
[59]
Cyanophyceae (Calothrix sp.)
3.42
26.95
6.55
3.99
55.52
6.94
[51]
Cyanophyceae (Leptolyngbya sp.)
3.23
13.74
1.53
4.73
69.52
5.36
1.56
3.57
[51]
Cyanophyceae (Lyngbya sp.)
2.52
1.23
26.49
5.37
3.24
54.53
4.00
5.15
[51]
Cyanophyceae (Nostoc muscorum)
3.22
26.89
54.01
7.85
11.24
[51]
Cyanophyceae (Oscillatoria acuta)
4.47
13.61
8.53
68.68
9.17
[51]
Cyanophyceae (Oscillatoria marina)
6.61
0.21
18.74
2.01
4.74
63.15
8.90
0.65
1.60
[51]
Cyanophyceae (Spirulina platensis)
7.75
22.53
2.34
5.21
53.54
7.74
5.52
[51]
Cyanophyceae (Synechococcus sp.)
4.2
8.74
82.12
1.73
5.45
1.94
[51]
Ethiopian mustard (Brassica carinata)
NA
2.9–3.1
1.0
9.7–33.9
8.2–16.8
1.1–16.6
0.7
10.9
42–42.5
9.6
[60–62]
Euphorbia lathyris L.
46
5.53–6.80
0–0.50
0–1.98
78.20–81.46
3.71–4.62
2.78–7.85
0–0.50
0–0.20
[63, 64]
Forsythia suspensa
30
5.65
1.99
18.68
72.89
0.79
[65]
Goji berry (Lycium barbarum L.)
NA
15.1–21.8
1.0–3.1
2.61–4.78
16.7–22.4
37.0–44.0
1.0–3.0
0.48–6.78
[66]
Grape seeds
NA
0–0.09
7.2–10.35
3.8–6.67
12.80–29.1
52.26–65.33
0–0.6
0–0.38
0–0.10
[58, 67]
Hazelnut kernel (Corylus avellana)
51–75
4.9–5.1
0.2–0.4
2.1–2.6
76.9–83.6
8.5–13.1
0.2
0–0.2
0–0.3
0–0.4
[24, 42, 68]
Jatropha curcas
50–60
0–0.31
13.23–14.9
0.85–0.88
5.40–9.5
40.5–45.79
32.27–36.99
0–0.3
0–1.93
[60, 64, 69, 70]
Jojoba (Simmondsia Chinensis)
45–55
0.30–3.50
3.10–9.00
0.60–2.20
28.20–46.30
0.20–3.10
0.30–3.80
20.80–51.50
0–14.5
1.40–4.40
0.50–19.70
[71–73]
Karanja (Pongamia pinnata)
30–40
7.4–11.6
3.8–7.5
51.6–65.6
15.4–16.6
0–4.4
3.4–12.7
[60, 70, 74]
Kusum (Schleichera trijuga)
NA
0.31
15.54
10.35
11.11
27.08
6.14
15.79
6.17
0.08
0.01
7.42
[75]
Lemon seeds
33.4–41.9
23.5–29.4
0.4–0.9
4.1–6.5
24.8–29.3
33.2–36.3
3.2–7.8
0.5–1.5
[76]
Lesquerella fendleri
0.3
1.0
0.4
1.8
13.6
6.7
10.2
0.8
0.9
64.3b
[47]
Linseed
35–45
4.92–5.1
0.3
2.41–2.5
18.9–19.7
18.03–18.1
54.94–55.1
[42, 43, 60]
Mahua (Madhuca indica)
35–40
0–0.14
16.0–28.2
18.97–25.8
37.2–51.0
8.9–19.5
0–0.16
[60, 70, 77]
Meliaceae (Aphanamixis polystachya)
NA
18.4
0.3
11.8
18.3
26.7
23.2
0.5
0.2
0.6
[78]
Mesua ferrea seeds
55–57
0.05–2.13
10.85–13.64
0–0.09
13.65–14.19
50.71–55.93
13.68–19.47
0–0.32
0–1.3
0.18–2.92
0–0.35
0–0.19
[79–81]
Michela champaca
45.0–45.5
0–0.57
15.8–32.52
6.9–13.39
2.5–8.88
6.03–22.3
30.72–42.5
0–0.71
[82, 83]
Milk thistle
NA
7.3–8.4
4.6–6.8
22.8–28.9
49.7–53.6
0–0.3
2.9–4.3
0.8–0.9
2.3–2.9
0.6–0.8
[84]
Milkweed seed (Calotropis gigantea)
NA
15.5–16.7
0–0.3
9.0–10.50
30.3–47.2
27.1–36.3
0–0.8
0–0.6
0–0.1
0–0.4
[85]
Moringa seed
38–40
0–0.1
6.5–13.8
1.1–2.0
4.7–6.0
72.1–72.2
1.0–2.5
0–0.9
3.8–4.0
0.9–2.0
[86, 87]
Mustard seed
NA
0.05
5.54
0.21
1.51
8.83
10.79
20.98
1.21
5.27
0.70
1.09
37.71
1.68
4.43
[36]
Neem
20–30
0–0.26
13.6–18.1
0–0.2
14.2–24.1
44.5–61.9
2.3–18.3
0.2–3
0.8–3.4
[60, 88–91]
Niger (Guizotia abyssinica)
30
9.2
10.1
9.0
71.7
[92]
Olive kernel
NA
5.0
0.3
1.6
74.7
17.6
0.8
[24, 42]
Olive pomace
8.5–14.9
0–0.04
9.37–16.20
0.68–0.80
2.68–3.00
69.3–79.0
7.20–12.70
0.30–0.87
0–0.58
0–0.40
0–0.22
[93–95]
Opium poppy (Papaver somniferum L.)
NA
9.79
0.13
1.93
11.94
74.47
1.7
[96]
Orange peel
NA
38.6
4.1
57.2
0.1
[97]
Palm (Elaeis sp.)
NA
0–0.3
0–1.1
35–43
0–0.3
4–7
40–44
10–14
0–0.2
0–0.3
0–1.1
[25, 35, 42, 98]
Corozo palm/Macaúba (Acrocomia aculeata)
NA
5.0
50.9
13.1
7.6
3.0
17.9
2.5
[99]
Peanut (Arachis hypogaea)
40–50
6.7–11.4
1.73–3.99
47.06–78.2
4.4–32.0
0.19–0.93
0–3.0
0–1.9
2.52–3.93
0–0.73
1.23–1.83
0–0.1
[42, 43, 68, 100, 101]
Peppers (Capsicum)
NA
10.6–14.4
2.7–3.9
5.4–7.6
73.9–77.9
[102]
Pequi (Caryocar sp.)
NA
40
2
47
4
7
[25]
Polanga (Calophyllum inophyllum)
NA
13.9
0.2
15.1
40.3
25.6
0.2
0.3
4.4
[22]
Poon (Sterculia foetida)
NA
0.1
0.2
18.4–23.0
0.3
7.3–8.2
1.0–16.4
11.0–45.9
0.7–2.3
2.3–6.5
[103, 104]
Poppy seed
49.9–52.4
8.2–12.6
0–0.1
0–4.0
13.3–23.4
60.2–74.3
0–0.5
[24, 105]
Pot marigold seeds (Calendula officinalis L.)
NA
3.86–4.55
0–2
4.44–6.25
28.50–31.86
51.47–57.63
[106]
Pumpkin seeds
45
0–0.01
0–0.11
10.2–12.51
0–0.15
5.43–6.0
36.1–37.07
43.72–45.2
0.18–2.6
0–0.39
0–0.11
0–0.12
0–0.02
0–0.06
[107, 108]
Putranjiva roxburghii
NA
0.03
10.23
0.07
10.63
48.65
27.50
0.87
1.05
0.30
0.24
0.03
0.31
0.09
[109]
Quinoa (Chenopodium quinoa)
NA
0.14–0.16
8.3–8.9
0.64–0.68
20.8–24.9
50.7–54.3
7.7–8.4
0.55–0.59
1.53–1.75
0.15–0.17
0.96–1.01
1.87–1.93
0.44–0.46
[110]
Rapeseed (Brassica napus) or canola oil
40–44
0–0.24
3.49–4.82
0–0.36
0.85–2.10
58.9–64.4
18.8–22.3
7.6–11.1
0–0.70
0–1.35
0–1.45
0–1.09
0–0.12
0–2.48
[35, 42, 43, 54, 63, 68, 69, 111, 112]
Rice bran (Oryza Sativa)
10–26
0.5
16.2
2.1
42.8
37.2
1.2
[113]
Rosehip seed (Rosa canina L.)
0.82
2.5
0.4
17.8
2.6
8.8
52.6
2.1
1.6
3.5
8.0
[33]
Rubber (Hevea brasiliensis) seed
35–60
0–2.2
0.23–10.6
0.23–0.25
5.69–12
12.7–42.08
39.6–52.84
2.38–26
0.66–0.97
[54, 114, 115]
Safflower seed
NA
7.3–8.6
1.9–1.93
11.58–13.6
77.2–77.89
[24, 42, 43]
Sapium sebiferum
NA
5.45
3.71
2.13
13.78
30.71
38.87
0.59
4.76
[64]
Sal (Shorea robusta) seed
10–12
0.58
3.69–8.6
34.2–47.0
34.2–44.8
1.2–2.7
3.5–12.2
[60, 116, 117]
Sea buckthorn leaves
NA
1.6
19.6
1.5
3.2
7.1
9.5
51.1
3.2
3.2
[118]
Sea buckthorn peels
NA
0.4
32.9
28.6
1.8
16.1
8.8
4.4
7.0
[118]
Sesame seeds
35–58
7.0–13.1
3.5–7.0
35.0–52.8
30.2–50.0
[24, 42, 43, 119–121]
Soapnut
NA
4.67
0.37
1.45
52.64
4.73
1.94
7.02
23.85
1.45
1.09
0.47
0.32
[69]
Soybean (Glycine max)
18–22
10–14
0–0.3
2–4.1
23–24.1
52–56.2
0–8
0–6
[25, 35, 36, 42, 43, 54, 68, 70, 98, 122]
Spruce bark
NA
0.25
0.87
11.91
3.00
1.25
8.65
34.22
35.60
4.3
[58]
Stillingia oil (Sapium sebiferum Roxb.)
13–32
3.55
0.31
5.93–7.72
2.05–2.46
15.24–15.93
29.05–29.77
39.54–41.20
1.75
[123, 124]
Sunflower seed
NA
6.08–6.8
0.1
2.9–3.26
16.93–17.7
72.9–73.73
[24, 43, 54]
Syringa (Melia azedarach L.)
NA
9.7–10.1
3.31–3.67
21.69–22.31
63.27–64.89
0.34–0.43
0.19–0.21
0–0.36
[125]
Tallow (animal fat)
NA
23.3
0.1
19.3
42.4
2.9
0.9
2.9
[24]
Tallow (Triadica sebifera L.)
NA
55.78
0.54
5.92
9.13
28.55
0.1
[126]
Terminalia catappa
NA
0–0.1
28.3–35.0
0–0.9
4.9–5.0
30.0–32.0
28.0–32.8
0–1.7
[39, 127]
Terminalia (Terminalia belerica Robx.) seed oil
31–43
11.6–32.8
0–0.5
3.9–6.4
31.3–61.5
18.5–28.8
0.3–0.8
[128–130]
Tiger nut (Cyperus esculentus)
78.69
1.7
15.4
5.3
65.8
5.5
0.2
6.1
[131]
Tobacco seed (Nicotiana tabacum L.)
38
0.09–0.17
3.0–10.96
0–0.20
3.34–5.0
12.14–16.0
67.75–72.98
0.69–4.20
0.20–0.25
0.12–0.13
0.07–0.12
0–0.04
0.69–3.12
[132–135]
Tung (Vernicia montana)
NA
2.3
2.4
5.6
6.3
0.1
0.2
0.9
0.1
82.2c
[136–140]
Walnut (Juglans regia)
52–70
7.2
0–0.2
1.9–2.6
15.1–18.5
56.0–60.7
12.8–16.2
0–0.2
0–0.5
[24, 42, 68]
Wheat germ
NA
0–0.13
17.4–18
0.17–0.23
0–0.7
16.1–16.5
55.1–57.7
7.9
0–0.28
[141, 142]
Wheat grain
NA
11.4
0.4
20.6
1.0
1.1
16.6
56.0
2.9
1.8
[24]
Waste coffee ground
10–15
33.99
10.91
7.16
23.27
4.04
20.6
[143]
Yellow oleander seed (Thevetia peruviana)
60–65
15.03–23.28
0–0.39
6.16–10.71
43.72–44.35
19.85–31.99
0–0.79
0.79–2.41
0–0.49
[144–146]
Zanthoxylum bungeanum
24–28
10.3
10.8
1.1
35.9
24.8
15.9
1.2
[147]
aCastor oil contains 89.6 ricinoleic acid. bLesquerella oil contains 57.6 lesquerolic acid. cTung oil contains 82.2 elaeostearate acid.Ricinoleic acid, lesquerolic acid, and elaeostearate acid, the main constituents of castor oil, lesquerella oil, and tung oil, respectively, were not reported since they do not appear in the other biomasses listed. Information about their quantity is provided in Table1 notes.Figure1 shows the average of each fatty acid yield in the biomasses and their frequency of appearance.Figure 1
Percentage and frequency of each fatty oils in the biomasses reported in Table1.The fatty acid yield average was obtained by considering the values reported in Table1. For those cases where a yield interval is indicated, the average was calculated. The frequency of appearance expresses the occurrence of fatty acid in the set of biomasses.Regarding the saturated fatty acids, palmitic acid (C16 : 0) and stearic acid (C18 : 0) are the ones with the highest content and frequency in all the bio-oils examined. Considering the unsaturated fatty acids, oleic acid (C18 : 1), linoleic acid (C18 : 2), and linolenic acid (C18 : 3) are the most plentiful compounds identified in the reported bio-oils. However, since the average yield reported in Figure1 was evaluated considering the distribution of the single fatty acids, a more revealing indicator is the average minimum and average maximum, reported in Figure 2 as a grey rectangle. The minimum and maximum averages were published together with the absolute minimum and absolute maximum. On the one hand, taking caprylic acid as an example, its average yield range is from 0.2% to 5.2%, which is a very narrow range, while the absolute maximum is 11.4%. On the other hand, when palmitoleic acid, oleic acid, and linoleic acid are considered, their amount can extensively vary.Figure 2
Average minimum and maximum (grey rectangle) and absolute maximum and minimum (interval lines).


## 3. Fatty Acid Classification

Oil extracts from biomasses comprise saturated and unsaturated fatty acids with long carbon chain, which can be employed in integrated processes for sustainable biorefineries. To highlight the differences between the molecules, the different fatty acids were identified based on the number of carbon atoms (CN) and the number of carbon-carbon double bonds (DB).While saturated acids are solid at room temperature, the unsaturated are in the liquid state. Furthermore, saturated and unsaturated fatty acids also differ regarding their structure. While saturated fatty acids have a linear molecule, the introduction of one or more double bonds promotes the molecule twisting and the possibility of geometrical isomerism. Figure3 reports the difference in the structure for the fatty acid with 18 carbon atoms and 0, 1, and 2 double bonds.Figure 3
C18 : 0, stearic acidC18H36O2 (a); C18 : 1, oleic acid C18H34O2 (b); C18 : 2, linoleic acid C18H32O2 (c).
(a)
(b)
(c)Following the analysis summarized in Figures1 and 2, the most common saturated and unsaturated fatty acids identified are presented in Table 2 together with the corresponding CAS number and their molecular weight.Table 2
Fatty acids selected for the property evaluation classified by name, CAS number, and molecular weight.
CN : DB
Molecular formula
Common names and IUPAC name
CAS no.
MW (g/mol)
C8 : 0
C
8
H
16
O
2
Caprylic acid,n-octanoic acid
124-07-2
144.21
C10 : 0
C
10
H
20
O
2
Capric acid,n-decanoic acid
334-48-5
172.27
C12 : 0
C
12
H
24
O
2
Lauric acid,n-dodecanoic acid
143-07-7
200.32
C14 : 0
C
14
H
28
O
2
Myristic acid, tetradecanoic acid
544-63-8
228.38
C16 : 0
C
16
H
32
O
2
Palmitic acid, hexadecanoic acid
57-10-3
256.43
C18 : 0
C
18
H
36
O
2
Stearic acid,n-octadecanoic acid
57-11-4
284.48
C20 : 0
C
20
H
40
O
2
Arachidic acid, eicosanoic acid
506-30-9
312.54
C22 : 0
C
22
H
44
O
2
Behenic acid, docosanoic acid
112-85-6
340.59
C24 : 0
C
24
H
48
O
2
Lignoceric acid, tetracosanoic acid
557-59-5
368.65
C16 : 1
C
16
H
30
O
2
Palmitoleic acid, (9Z)-hexadec-9-enoic acid,cis-9-hexadecenoic acid
373-49-9
254.41
C18 : 1
C
18
H
34
O
2
Oleic acid, (9Z)-octadec-9-enoic acid,n-octadecenoic acid, [omega-9 fatty acid (18 : 1 ω9)]
112-80-1
282.47
C20 : 1
C
20
H
38
O
2
Paullinic acid, 13-eicosenoic acid, [omega-7 fatty acid (20 : 1ω7)]
29204-02-2
310.52
Gondoic acid,cis-gondoic acid, cis-11-eicosenoic acid, 11-eicosenoic acid, 11Z-eicosenoic acid, cis-11-icosenoic acid, (11Z)-icos-11-enoic acid, [omega-9 fatty acid (20 : 1 ω9)]
5561-99-9
Gadoleic acid, 9-eicosenoic acid, [omega-11 fatty acid (20 : 1ω11)], (E)-icos-2-enoic acid
17735-94-326764-41-0
C22 : 1
C
22
H
42
O
2
Erucic acid, (Z)-docos-13-enoic acid,cis-13-docosenoic acid
112-86-7
338.58
C24 : 1
C
24
H
46
O
2
Nervonic acid, (Z)-tetracos-15-enoic acid, [omega-9 fatty acid (24 : 1ω9)]
506-37-6
366.62
C18 : 2
C
18
H
32
O
2
Linoleic acid, (9Z,12Z)-9,12-octadecadienoic acid, [omega-6 fatty acid (18 : 2ω6)]
60-33-3
280.45
C18 : 3
C
18
H
30
O
2
α-Linolenic acid, cis-9,12,15-octadecatrienoic acid, [omega-3 fatty acid (18 : 2 ω3)]
463-40-1
278.44
Gamma-linolenic acid, gamolenic acid, all-cis-6,9,12-octadecatrienoic acid, [omega-6 fatty acid (18 : 2 ω6)]
506-26-3
α-Calendic acid, (8E,10E,12Z)-octadeca-8,10,12-trienoic acid, [omega-6 fatty acid (18 : 2 ω6)]
5204-87-5


## 4. Physical Properties

Temperature nondependent properties, namely, normal boiling point, normal melting point, critical properties, and acentric factor, were evaluated in this section.Using the NIST database as a reference, the values obtained from the different estimation models and the values from the proposed correlations were compared using different deviation indexes:(i)
The absolute deviation (AD) is defined as(1)AD=estimated  value−experimental  value.(ii)
The relative deviation (RD) and the absolute relative deviation (ARD) are as follows, respectively:(2)RD=estimated value−experimental valueexperimental value×100,(3)ARD=RD.(iii)
The average absolute deviation (AAD) is given as follows:(4)AAD=1n∑i=1NAD.(iv)
The root mean square deviation or relative deviation (RMSD) is obtained using the following equation:(5)RMSD=∑i=1NAD2N.(v)
The average absolute relative deviation (AARD) is defined by the following equation:(6)AARD=1n∑i=1NARD.


### 4.1. Normal Boiling Point

The normal boiling point (TB) is defined as the temperature in which the vapor pressure equals the value of 1 atmosphere. The normal boiling point is used to compare different liquids and represents an essential input for the estimation of other properties like the critical parameters.


#### 4.1.1. Estimation Methods

To determine the normal boiling point, several methods are available in the literature. The most relevant ones are summarized in Table3.Table 3
Methods for the normal boiling point estimation.
Method
Parameter(s) required
Joback and Reid [148]
Molecular structure
Constantinou and Gani [149]
Molecular structure
Marrero-Marejón and Pardillo-Fontdevila [150]
Molecular structure andMW
Marrero and Gani [151]
Molecular structure
Nannoolal et al. [152]
Molecular structureEach of these methods has been applied for the set of fatty acids identified in Table2.The Joback and Reid method [148] estimates the normal boiling point considering the group incremental value (TBi) multiplied by the number of times the group appears in the compound (Ni). The estimation formula is reported in the following equation:(7)TB=198.2+∑iNi⋅Tbi.The coefficients used for the estimation of the normal boiling point are reported as Supplementary Material in TableA1.Constantinou and Gani [149] established the two-level evaluation method reported as follows:(8)TBK=204.359⋅ln∑iNi⋅Tb1i+W⋅∑jMj⋅Tb2j.The method is based on the first- and second-order group contribution. In the former, each group has a single contribution independently of the type of compound involved (TB1i). The latter (TB2j) has the function to provide more structural information about the portions of the structure where the description through the first-order groups is insufficient. The model constant W assumes the value zero if only the first-order group estimation is taken into account, or one if the second order is also considered. Ni and Mj are the group occurrences in the compound. The coefficients used to determine the normal boiling point are reported in Tables A2 and A3 of the Supplementary Material.Marrero-Marejón and Pardillo-Fontdevila proposed a family of models to estimate the normal boiling point of pure organic compounds based on the compound chemical structure and molecular weight [150]. The normal boiling point is evaluated according to equation (9), where Ni is the number of times that type i occurs with contributions tbbk:(9)TB=MW−0.404∑iNi⋅tbbki+156.0.In this case, the contributions are based on group interaction, taking into account the interactions between bonding groups in the molecule. TableA4, in the Supplementary Material, presents the coefficients used.Marrero and Gani [151] defined an estimation of the normal boiling point based on three levels, as follows:(10)TBK=222.543⋅ln∑iNi⋅Tb1i+∑jMj⋅Tb2j+∑kOj⋅Tb3k.The first level includes the simple groups used for describing a wide variety of organic compounds. The second level involves groups that permit a better description of polyfunctional compounds and differentiation among isomers. Finally, the third level provides more structural information about molecular fragments of compounds whose description is insufficient through the first- and second-level groups. The coefficients to determine the normal boiling point can be found in TablesA5 and A6 of the Supplementary Material.The estimation of the boiling point proposed by Nannoolal et al. [152] is reported as follows:(11)TBK=∑iNiCin0.6583+1.6868+84.3395,where Ci is the group contribution of group i and n is the number of atoms in the molecule (excluding the hydrogen atoms). The method is valid for nonelectrolyte organic compounds and was developed using experimental data for about 2850 compounds stored in the Dortmund data bank. Table A7 of the Supplementary Material presents the coefficients correspondent to this method.


#### 4.1.2. Normal Boiling Point Estimation Results

The results for the normal boiling point estimated using the methods described are summarized in Table4. The values extracted from the NIST database were accessed through the Aspen Property software, and they were used as a reference for the comparison with the estimated values. The values obtained were checked for consistency, and it was noticed that the normal boiling point for the C24 : 0 (673.7 K) was lower than the value reported for C22 : 0 (685.0 K). For this reason, the value of 704.4 K reported by Yaws [153] was used.Table 4
Comparison of experimental and estimated normal boiling temperature (K).
CN : DB
NIST
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [152]
C8 : 0
512.6
528.2
512.2
518.2
517.8
516.1
C10 : 0
543.2
573.9
540.8
554.3
546.8
549.6
C12 : 0
572.1
619.7
565.9
588.3
572.5
580.0
C14 : 0
599.5
665.4
588.3
620.6
595.6
607.9
C16 : 0
619.8
711.2
608.5
651.4
616.4
633.7
C18 : 0
643.0
757.0
626.8
681.0
635.5
657.9
C20 : 0
672.9
802.7
643.7
709.5
653.1
680.6
C22 : 0
692.9
848.5
659.2
737.1
669.4
702.1
C24 : 0
704.4∗
894.2
673.7
763.8
684.5
722.4
C16 : 1
632.5
715.4
607.0
653.4
617.5
625.8
C18 : 1
635.9
761.1
625.5
682.9
636.5
650.4
C20 : 1
642.4
806.9
642.4
711.4
654.0
673.5
C22 : 1
665.7
852.6
658.1
738.8
670.2
695.3
C24 : 1
672.6
898.4
672.6
765.5
685.3
715.9
C18 : 2
624.1
765.3
624.1
684.8
637.5
643.0
C18 : 3
622.7
769.4
622.7
686.7
638.5
635.5
∗Value reported by Yaws [153].Figure4 indicates the absolute relative deviation between the different methods when compared to the NIST values.Figure 4
Absolute relative deviation comparison for the normal boiling point estimation.From Figure4, it is possible to notice that the Joback and Raid method always overestimates the normal boiling point, and the overestimation increases with the molecular weight. In general, for the same number of atoms of carbon in the molecule, the overestimation increases with the unsaturation. The same trend, but with a reduced absolute relative deviation, is observed for the Marrero-Marejon and Pardillo-Fontdevila method. The opposite tendency is observed for the method developed by Constantinou and Gani, where the normal boiling point is underestimated for the saturated fatty acids with the highest molecular weight. A similar behaviour, but with a reduced margin of deviations, is observed for the Marrero and Gani method. The Nannoolal et al. method has a positive deviation for the unsaturated fatty acids that seem to be independent from the molecular weight. However, for unsaturated fatty acids, the overestimation increases with the molecular weight.Table5 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by the NIST database.Table 5
Deviations of the different methods for the normal boiling point estimation.
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [152]
AAD (K)
119.60
11.55
43.23
10.05
15.43
RMSD (K)
133.05
16.40
49.56
12.33
18.65
AARD (%)
18.5
1.8
6.7
1.6
2.4
MIN RD (%)
3
−5
1
−3
−11
MAX RD (%)
34
0
14
3
6Among all the estimation methods explored, the two-level and the three-level approach proposed by Marrero and Gani provided a more reliable estimation of the normal boiling point for the FAs considered.


#### 4.1.3. Novel Normal Boiling Point Estimation Method

Anand et al. [154] defined the normal boiling point for fatty acids methyl esters (FAMEs), as a function of molecular weight (equation (12)). The same approach was followed to propose an alternative correlation for the fatty acids:(12)TBFAME=AFAMEMW2+BFAMEMW+CFAME.Using a similar equation, it was possible to obtain the coefficients to determine the normal boiling points for the fatty acids considered by applying the generalized reduced gradient method to minimize the deviations. In fact, observing the normal boiling point values, as a function of the molecular weight, it is possible to conclude that it has a parabolic behaviour.The novel estimation method, reported in equation (13), has three coefficients, assigned as ATBFA CN:n, BTBFA CN:n, and CTBFA CN:n, and a correction factor, αTBCN:n, which depends on the number of unsaturation:(13)TBFA=αTBCN:n⋅AATBFA CN:nMW2+BTBFA CN:nMW+CTBFA CN:n.The values forATBFA CN:n, BTBFA CN:nCTBFA CN:n, and αTBCN:n were obtained by minimizing the relative deviations with the NIST database values, using the GRG nonlinear solving method to achieve the optimized result. The coefficients obtained are presented in Table 6.Table 6
Coefficients for the estimation of the normal boiling point according to equation (13).
Coefficients
Values
A
T
B
FA CN
:
n
−7.0670E − 04
B
T
B
FA CN
:
n
0.9173
C
T
B
FA CN
:
n
205.6120
α
T
B
CN
:
0
1.5697
α
T
B
CN
:
1
1.5501
α
T
B
CN
:
2
1.5323
α
T
B
CN
:
3
1.5329The proposed normal boiling point estimation method showed a lower estimation deviation when compared with the other methods previously presented. In particular, it was obtained:AAD=8.49K, RMSD=11.94K, AARD=1.3%, MINRD=−3.6%, and MAXRD=4.3%.


### 4.2. Normal Melting Point

The normal melting point for a pure substance is defined as the temperature in which the solid and the liquid form exist in equilibrium at atmospheric pressure. In general, it is expected that the melting point of fatty acids increases if the unsaturation is reduced. If the fatty acids are used for biodiesel production, the availability of the melting point is essential for the evaluation of the cold-flow properties.


#### 4.2.1. Estimation Methods

Several methods are available in the literature to determine the normal melting point. The most appropriate methods are gathered in Table7.Table 7
Methods for the normal melting point estimation.
Method
Parameter required
Joback and Reid [148]
Molecular structure
Constantinou and Gani [149]
Molecular structure
Marrero and Gani [151]
Molecular structureJoback and Reid proposed the estimation formula reported as follows [148]:(14)TMK=122.5+∑Ni⋅Tmi.According to the authors, however, estimations of the normal melting point are not accurate and should be considered only as approximate. The coefficients of the group contributions are presented in TableA1 of the Supplementary Material.The Constantinou and Gani method [149] is presented as follows:(15)TMK=102.425×ln∑iNi×Tm1i+W×∑jMj×Tm2j.The coefficients to determine the normal melting point can be found in TablesA2 and A3 of the Supplementary Material.Marrero and Gani [151] developed a method to determine the normal melting point using the following equation:(16)TMK=147.450×ln∑iNi×Tm1i+∑jMj×Tm2j+∑kOj×Tm3k.The coefficients to determine the normal melting point can be found in TablesA5 and A6 of the Supplementary Material.


#### 4.2.2. Normal Melting Point Estimation Results

Table8 presents the values of the estimated normal melting point together with the values obtained from the NIST database.Table 8
Comparison of experimental and estimated normal melting point (K).
CN : DB
NIST
Joback and Reid [148]
Constantinou and Gani [149]
Marrero and Gani [151]
C8 : 0
289.7
340.5
293.6
333.6
C10 : 0
304.5
363.1
303.8
341.2
C12 : 0
317.0
385.6
313.2
348.3
C14 : 0
327.3
408.1
321.7
355.1
C16 : 0
335.6
430.7
329.6
361.7
C18 : 0
342.6
453.2
336.9
367.9
C20 : 0
348.2
475.8
343.7
373.9
C22 : 0
353.8
498.3
350.1
379.7
C24 : 0
355.6
520.8
356.1
385.2
C16 : 1
274.8
425.6
326.2
365.9
C18 : 1
286.5
448.1
333.8
371.9
C20 : 1
297.3
470.7
340.8
377.8
C22 : 1
306.1
493.2
347.4
383.4
C24 : 1
—
515.8
353.6
388.8
C18 : 2
267.6
443.1
330.6
375.8
C18 : 3
260.0
438.0
327.2
379.7Figure5 reports the absolute relative deviation between the different methods when compared with the NIST values.Figure 5
Absolute relative deviation comparison for the normal melting point estimation.The Joback and Reid group contribution method overestimates the normal melting point. The overestimation regularly grows for saturated fatty acids with a consistent penalty for molecules with the highest molecular weight. The same trend is observed for the unsaturated compounds even if the increase of the absolute relative deviation is lower with respect to the case of saturated fatty acids.Table9 shows the deviation indexes for each method when compared with the data provided by NIST.Table 9
Deviations of different methods for the estimation of the normal melting point.
Joback and Reid [148]
Constantinou and Gani [149]
Marrero and Gani [151]
AAD (K)
132.98
24.10
56.68
RMSD (K)
141.25
34.02
65.14
AARD (%)
43.5
8.4
19.3
MIN RD (%)
18
−2
7
MAX RD (%)
69
26
46The method of Constantinou and Gani offers the best estimation with a very good prediction of the saturated compounds. However, for the unsaturated fatty acids, all the methods considered report a high estimation deviation. When comparing molecules with the same number of carbons, it is possible to observe that the melting point decreases when the unsaturation increases. This decrease is due to the molecular distortion induced by the double bonds that prevent the molecules from packing together easily. The difficulty in predicting the London dispersion forces between the oil molecules may be the cause of the high deviations observed for unsaturated fatty acids.


#### 4.2.3. Novel Normal Melting Point Estimation Method

It is possible to obtain the coefficients to determine the normal melting points applying the generalizedreduced gradient (GRG) method to minimize the deviations, with a similar procedure described in Section4.1.3 for the estimation of the normal boiling point.Equation (17) is the novel estimation method developed:(17)TMFA=αTMCN:n⋅ATMFAMW2+BTMFAMW+CTMFA.The main advantage of this method is the possibility of estimating the normal meting point, using the fatty acids’ molecular weight exclusively.The coefficientsATMFA CN:n, BTMFA CN:n, CTBFA CN:n, and αTMCN:n were obtained by minimizing the relative deviations with the NIST database values. The obtained coefficients are presented in Table 10.Table 10
Coefficients for the estimation of the normal melting point according to equation (17).
Coefficients
Values
A
T
M
FA CN
:
n
−4.1905E−04
B
T
M
FA CN
:
n
0.4863
C
T
M
FA CN
:
n
193.9418
α
T
M
CN
:
0
1.1380
α
T
M
CN
:
1
0.9752
α
T
M
CN
:
2
0.8999
α
T
M
CN
:
3
0.8759For the proposed method, the comparison indexes areAAD=3.15K, RMSD=4.09K, AARD=1.0%, MIN RD=−2.6%, and MAX RD=3.1%, giving a smaller deviation when compared to the other estimation methods.


### 4.3. Critical Parameters

The critical point is used to identify the fluid region where the liquid and gas phases can be no longer distinguished. Critical parameters are used for the estimation of other properties, such as the acentric factor, compressibility factor, liquid density, vapor pressure, and enthalpy of vaporization, among others.


#### 4.3.1. Estimation Methods

Several methods are available in the literature to determine the critical parameters such as critical temperature, critical pressure, and critical volume. The most appropriate methods are summarized in Table11.Table 11
Methods for the critical parameters’ estimation.
Method
Parameter(s) required
Joback and Reid [148]
Molecular structure andTB
Constantinou and Gani [149]
Molecular structure
Wilson and Jasperson [155]
Molecular structure andTB
Marrero-Marejón and Pardillo-Fontdevila [150]
Molecular structure andTB
Marrero and Gani [151]
Molecular structure
Nannoolalet al. [156]
Molecular structure,MW, and TBJoback and Reid proposed the relations reported in equations (18)–(20) to estimate the critical properties based on the group contribution methods [148]:(18)TC=TB0.584+0.965∑Ni⋅Tci−∑Ni⋅Tci2,(19)PC=10.113+0.0032NA−∑Ni⋅Pci2,(20)VC=17.5+∑Ni⋅Vci,where NA is the number of atoms in the molecule. The values Tci, Pci, and Vci are presented in Table A1.Constantinou and Gani [149] established a method to determine TC, PC, and VC using equations (21)–(23), respectively:(21)TCK=181.128⋅ln∑iNi⋅Tc1i+W⋅∑jMj⋅Tc2j,(22)PCbar=∑iNi⋅Pc1i+W×∑jMj⋅Pc2j+0.100220−2+1.3705,(23)VCcm3⋅mol−1=−0.004350+∑iNi⋅vc1i+W×∑jMj⋅vc2j.The coefficients to determine the critical properties can be found in TablesA2 and A3 in the Supplementary Material.Wilson and Jasperson suggested equation (24) to estimate the critical temperature and equation (25) to determine the critical pressure:(24)TCK=TB0.048271−0.019846Nr+∑kNk·tck+∑jMjΔtcj0.2,(25)PCbar=0.0186233⋅TC−0.96601+exp−0.0092295−0.0290403Nr+0.041∑kNk⋅Pcbk+∑jMjΔPcj.The coefficients to determine these critical properties can be found in TableA6 in the Supplementary Material.Marrero-Marejón and Pardillo-Fontdevila developed equations (26)–(28) to calculate the critical parameters [150]:(26)TCK=TB0.5851−0.9286∑kNk⋅tbck+∑jMj⋅Δtcbj2,(27)PCbar=0.1285−0.0059Natoms−∑kNk⋅Pcbk−2,(28)VCcm3⋅mol−1=−25.1+∑kNk⋅vcbk,where Natoms is the number of atoms in the compound and Nk is the number of atoms of type k with contributions tcbk, pcbk, and vcbk. The parameters to determine these critical parameters can be found at Table A4 in the Supplementary Material.Marrero and Gani [151] determined TC, PC, and VC using the following equations:(29)TCK=231.239⋅ln∑iNi⋅Tc1i+W⋅∑jMj⋅Tc2j+W⋅∑kOk⋅Tc3k,(30)PCbar=∑iNi⋅Pc1i+W×∑jMj⋅Pc2j+W×∑kOk⋅Pc3k+0.108998−2+5.9827,(31)VCcm3⋅mol−1=7.95+∑iNi⋅Vc1i+W×∑jMj⋅Vc2j+W×∑kOk⋅Vc3k.The coefficients to determine the critical properties can be found in TablesA5 and A6 in the Supplementary Material.The estimation of the critical properties proposed by Nannoolal et al. [156] is presented in the following equations:(32)TCK=TB0.6990+10.9889+∑iNiCi0.8607,(33)PCkPa=MW−0.14041g/mol0.00939+∑iNiCi2,(34)VCcm3⋅mol−1=∑iNiCin−0.2266+86.1539,where n is the number of atoms in the molecule (excluding hydrogen). Table A7 in the Supplementary Material presents the coefficients used.For the estimation methods that depend on the normal boiling point temperature, this value can be experimentally determined or estimated. The calculations were performed using the normal boiling point temperatures available from the NIST database, as reported in Table4.


#### 4.3.2. Critical Properties Estimation Results

(1) Critical Temperature Estimation Results. The estimated and experimental values for the critical temperatures are presented in Table 12.Table 12
Comparison of experimental and estimated critical temperature (K).
CN : DB
NIST
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
C8 : 0
693.7
692.1
694.6
695.0
692.3
738.2
694.9
C10 : 0
723.8
714.6
718.4
720.4
715.7
762.3
719.5
C12 : 0
743.0
736.6
740.5
742.7
739.1
784.0
742.0
C14 : 0
763.0
758.4
760.8
762.5
762.8
803.9
762.5
C16 : 0
785.0
773.5
772.7
780.4
780.4
822.2
774.3
C18 : 0
803.0
794.5
788.7
796.6
804.6
839.2
790.1
C20 : 0
820.0
826.2
813.0
811.6
840.4
855.0
814.1
C22 : 0
837.0
848.4
825.5
825.4
867.4
869.8
826.3
C24 : 0
825.0
863.1
828.2
838.2
887.6
883.7
828.6
C16 : 1
789.0
792.5
791.3
780.0
800.7
825.1
796.3
C18 : 1
775.0
787.9
782.4
796.3
798.1
841.8
786.9
C20 : 1
811.0
790.1
778.4
811.3
802.8
857.5
782.4
C22 : 1
806.0
815.6
795.3
825.1
831.9
872.1
798.8
C24 : 1
838.0
823.7
792.9
837.9
844.0
885.9
795.8
C18 : 2
787.0
775.8
770.4
796.0
786.6
844.5
778.0
C18 : 3
777.0
776.8
771.2
795.7
789.3
847.1
782.1Figure6 presents the absolute relative deviation between the methods considered and the NIST database values.Figure 6
Absolute relative deviation of estimation for the critical temperature.Except for the Marrero and Gani’s method, all the other methods have a general agreement until the C16 : 0. For the compounds with higher molecular weight, the method developed by Nannoolal et al. and by Constantinou and Gani give the best results. When one or more unsaturations are considered, it is not possible to define a method that consistently predicts the critical temperature. Table13 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by NIST.Table 13
Deviations of different methods for the estimation of the critical temperature.
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
AAD (K)
10.64
11.23
7.95
13.80
47.23
10.14
RMSD (K)
13.72
16.13
10.57
20.85
48.77
14.62
AARD (%)
1.3
1.4
1.0
1.7
6.0
1.3
MIN RD (%)
−3
−5
−1
−1
4
−5
MAX RD (%)
2
1
3
4
9
2From the analysis of Table13, it is possible to notice that the Constantinou and Gani’s method exhibits the best global performances for the critical temperature estimation.(2) Critical Pressure Estimation Results. The results for the estimated critical pressures and the experimental values for the fatty acids are presented in Table 14.Table 14
Comparison of experimental and estimated critical pressure (bar).
CN : DB
NIST
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
C8 : 0
28.57
27.79
26.91
27.67
31.07
26.87
26.27
C10 : 0
20.76
22.92
22.22
22.79
25.40
22.52
22.11
C12 : 0
19.31
19.22
18.79
19.14
21.16
19.40
18.87
C14 : 0
16.40
16.35
16.17
16.36
17.89
17.09
16.29
C16 : 0
14.90
14.08
13.96
14.18
15.33
15.33
14.21
C18 : 0
13.30
12.25
12.25
12.44
13.28
13.95
12.50
C20 : 0
12.00
10.76
10.97
11.03
11.62
12.86
11.07
C22 : 0
11.10
9.52
9.75
9.88
10.25
11.98
9.88
C24 : 0
8.49
8.49
8.61
8.92
9.11
11.25
8.86
C16 : 1
14.88
14.65
14.58
14.15
15.86
15.52
14.37
C18 : 1
13.90∗
12.71
12.39
12.42
13.71
14.10
12.63
C20 : 1
10.98
11.13
10.69
11.02
11.96
12.98
11.18
C22 : 1
9.83
9.83
9.55
9.87
10.53
12.07
9.97
C24 : 1
8.87
8.75
8.61
8.91
9.35
11.33
8.94
C18 : 2
14.10
13.19
12.43
12.40
14.15
14.25
12.76
C18 : 3
12.34∗
13.71
12.68
12.38
14.62
14.41
12.89
∗Values retrieved from the database APV88 PURE32 of Aspen Plus.Figure7 presents the relative deviation between the estimated values and the ones gathered from the NIST database. All the methods provide estimations included in a range of deviation of ±10%, except for the Marrero and Gani method when applied to saturated fatty acids with high molecular weight and unsaturated fatty acids with one and three double bonds. The Nannoolal et al. method gives the best estimation for the fatty acids at high molecular weight with one double bond. Regarding the saturated compounds, the Marrero-Marejón and Pardillo-Fontdevila method overestimates the critical pressure until C16 : 0. After this component, there is an underestimation that increases linearly with the molecular weight. Apart from the Marrero and Gani method, all the others underestimate the critical pressure in the range of C16 : 0–C22 : 0.Figure 7
Absolute relative deviation of estimation for the critical pressure.Table15 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by NIST.Table 15
Deviations of different methods for the estimation of the critical pressure.
NIST
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
AAD (bar)
0.73
0.83
0.71
1.15
1.23
0.77
RMSD (bar)
0.98
0.99
0.95
1.64
1.50
0.97
AARD (%)
5.1
5.7
4.9
7.5
10.4
5.1
MIN RD (%)
−14
−12
−12
−8
−6
−11
MAX RD (%)
11
7
10
22
33
7
Globally, the Nannoolal et al. method offers a better estimation together with the Constantinou and Gani approach.(3) Critical Volume Estimation Results. The estimated and experimental values for the fatty acids’ critical volume are presented in Table 16.Table 16
Comparison of experimental and estimated critical volume (m3/kmol).
CN : DB
NIST
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
C8 : 0
0.526
0.5075
0.5071
0.5107
0.5046
0.5267
C10 : 0
0.639
0.6195
0.6187
0.6239
0.6172
0.6579
C12 : 0
0.787
0.7315
0.7302
0.7371
0.7298
0.7948
C14 : 0
0.921
0.8435
0.8417
0.8503
0.8423
0.9367
C16 : 0
1.066
0.9555
0.9532
0.9635
0.9549
1.0831
C18 : 0
1.255
1.0675
1.0647
1.0767
1.0674
1.2335
C20 : 0
1.371
1.1795
1.1763
1.1899
1.1800
1.3877
C22 : 0
1.486
1.2915
1.2878
1.3031
1.2926
1.5453
C24 : 0
1.575
1.4035
1.3993
1.4163
1.4051
1.7062
C16 : 1
0.898
0.9355
0.9399
0.9429
0.9409
1.0525
C18 : 1
1.016
1.0475
1.0514
1.0561
1.0535
1.2022
C20 : 1
1.158
1.1595
1.1630
1.1693
1.1660
1.3557
C22 : 1
1.448
1.2715
1.2745
1.2825
1.2786
1.5127
C24 : 1
1.402
1.3835
1.3860
1.3957
1.3911
1.6729
C18 : 2
0.972
1.0275
1.0381
1.0355
1.0395
1.1708
C18 : 3
0.960
1.0075
1.0248
1.0149
1.0255
1.1395Figure8 presents the relative deviation distribution between the estimated values and the ones provided by NIST.Figure 8
Absolute relative deviations of estimation for the critical volume.The Nannoolal et al. method offers the best prediction for the saturated compounds with a peak of about 10% overestimation for C24 : 0. However, the overestimation increases to 20% when the unsaturated compounds are evaluated. All the other methods exhibit an opposite behaviour, with the tendency to underestimate the critical volume for saturated compounds, with a lower deviation for the unsaturated fatty acids.Table17 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by the NIST database.Table 17
Deviations of different methods for the estimation of the critical volume.
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
AAD (m3/kmol)
0.087
0.091
0.084
0.090
0.096
RMSD (m3/kmol)
0.112
0.114
0.106
0.112
0.130
AARD (%)
7.3
7.6
7.0
7.6
8.6
MIN RD (%)
−5
−15
−15
−14
−15
MAX RD (%)
31
6
7
7
7Except for the Nannoolal et al. method, that is only recommended for saturated compounds, all the other methods provide a general agreement in the critical volume estimation.


#### 4.3.3. Novel Critical Parameters Estimation Method

(1) Novel Critical Temperature Estimation Method. Anand et al. [154] proposed a critical temperature estimation method for FAMEs, according to the following:(35)TCFAME=ATCFAMEMW2+BTCFAMEMW+CTCFAME.Using a similar equation, it is possible to obtain the coefficients to determine the critical temperature for the fatty acids applying the generalized reduced gradients method to minimize the deviations. Equation (36) is the novel estimation method proposed to estimate the fatty acids critical temperature:(36)TCFAK=αTCCN:n⋅ATCFAMW2+BTCFAMW+CTCFA.The coefficientsATCFACN:n, BTCFA CN:n, CTCFA CN:n, and αTCCN:n were obtained by minimizing the relative deviations with the NIST database values. The coefficients obtained are presented in Table 18.Table 18
Coefficients for the estimation of the critical temperature according to equation (36).
Coefficients
Values
A
T
C
FA CN
:
n
−1.8345E − 03
B
T
C
FA CN
:
n
1.6239
C
T
C
FA CN
:
n
550.5882
α
T
C
CN
:
0
0.9286
α
T
C
CN
:
1
0.9185
α
T
C
CN
:
2
0.9134
α
T
C
CN
:
3
0.9028For the proposed method, the critical temperature deviations resulted inAAD=5.88K, RMSD=8.01K, AARD=0.7%, MIN  RD=−1.6%, and MAX  RD=2.3% compared with NIST database. Thus, AAD, RMSD, and AARD significantly outperform the estimation methods considered.(2) Novel Critical Pressure Evaluation Method. Applying the same procedure, it is possible to optimize equation (37) in order to obtain a functional equation to estimate the critical pressure:(37)PCFAbar=αPCCN:n⋅APCFAMW2+BPCFAMW+CPCFA.The coefficientsAPCFA CN:n, BPCFA CN:n, CPCFA CN:n, and αPCCN:n were obtained by minimizing the relative deviations with the NIST database values, using the GRG nonlinear solving method to find the optimized result. The obtained coefficients are presented in Table 19.Table 19
Coefficients for the estimation of the critical pressure according to equation (37).
Coefficients
Values
A
P
C
FA CN
:
n
7.0357E − 04
B
P
C
FA CN
:
n
−0.5776
C
P
C
FA CN
:
n
143.4049
α
P
C
CN
:
0
0.3515
α
P
C
CN
:
1
0.3556
α
P
C
CN
:
2
0.3836
α
P
C
CN
:
3
0.3323For the proposed method, the critical pressure deviations resulted inAAD=0.69bar, RMSD=0.94bar, AARD=4.6%, MIN  RD=−10.4%, and MAX RD=9.7%. Again, AAD, RMSD, and AARD show an improvement in accuracy, when compared with the considered estimation methods.(3) Novel Critical Volume Estimation Method. Applying the same procedure as it was suggested before, it is possible to derive equation (38) to estimate the critical volume:(38)VCFAmL/mol=αVCCN:n⋅AVCFAMW2+BVCFAMW+CVCFA.The coefficientsAVCFA CN:n, BVCFA CN:n, CVFA CN:n, and αVCCN:n were obtained by minimizing the relative deviations with the NIST database values, using the GRG nonlinear solving method to achieve the optimized result. The obtained coefficients are presented in Table 20.Table 20
Coefficients for the estimation of the critical volume according to equation (38).
Coefficients
Values
A
V
C
FA CN
:
n
−1.0023E−06
B
V
C
FA CN
:
n
0.0206
C
V
C
FA CN
:
n
−0.9068
α
V
C
CN
:
0
0.2485
α
V
C
CN
:
1
0.2106
α
V
C
CN
:
2
0.2033
α
V
C
CN
:
3
0.2025The proposed method for the critical volume yield deviations ofAAD=0.03m3/kmol, RMSD=0.05m3/kmol, AARD=2.1%, MIN RD=−13.6%, and MAX  RD=3.1%. The first three indicators significantly outperform the ones computed for the considered estimation methods.


### 4.4. Acentric Factor

The acentric factor (ω) is a pure component constant that indicates the acentricity or nonsphericity of a molecule. Monoatomic molecules are expected to have acentric factors equal to zero, and their values increase with the molecular weight and polarity [155].This property is used in the estimation of transport and thermodynamic properties for gases and liquids, such as compressibility factor, heat capacity, enthalpy of vaporization, and saturated density [155, 157].


#### 4.4.1. Estimation Method

Several methods are available in the literature to determine the acentric factor. The most appropriate methods are summarized in Table21.Table 21
Methods for the estimation of the acentric factor.
Method
Parameters required
Pitzer et al. [158]
Vapor pressure andPC
Lee and Kesler [159]
T
B
,
T
C
,
P
C
Watanasiri et al. [160]
Specific gravity,MW,TB
Ambrose and Walton [161]
T
B
,
T
C
,
P
C
Chen et al. [157]
T
B
,
T
C
,
P
C
Constantinou et al. [162]
Molecular structureThe acentric factor was initially proposed by Pitzer et al., and it is given by the following equation [158]:(39)ω=−log10Pr−1.0,where Pr is the reduced vapor pressure, defined as Pr=Pvap/PC, and Pvap is the vapor pressure at temperature T, defined at the reduced temperature T/TC=0.7. The vapor pressure can be determined by the NIST Wagner Equation, given as follows:(40)lnPvap=C5+C1×1−T/C6+C2×1−T/C61.5+C3×1−T/C62.5+C4×1−T/C65T/C6,where the pressure is given in Pa. The coefficients C1, C2, C3, C4, C5, and C6 can be found in Table A7 of the Supplementary Material.The parameter proposed by Pitzer et al. is, however, limited to reduced temperatures above 0.8.Lee and Kesler proposed an analytical correlation for the acentric factor employing equations of state, based on a wide variety of compounds. Their goal was to improve the representation of the property at low temperatures and near the critical region [159]. The proposed equation (41) is given as follows:(41)ω=−lnPC−5.92714+6.09648/TBr+1.28862lnTBr+0.169347TBr615.2518−15.6875/TBr−13.4721lnTBr+0.43577TBr6,where TBr=TB/TC is the reduced temperature at the boiling point.Watanasiri et al. developed a correlation to estimate the acentric factor based on heterocyclic compounds and other hydrocarbons and their derivatives [160]. According to their model, the acentric factor is given by the following equation:(42)ω=0.92217×10−3TB+0.507288TBMW+382.904MW+0.242×10−5TBSG2−0.2165×10−4TB×MW+0.1261×10−2SG×MW+0.1265×10−4MW2+0.2016×10−4SG×MW2−80.6495TB1/3MW−0.3738×10−2TB2/3SG2TBMW,where SG is the specific gravity of the component. Values of specific gravity were obtained from the NIST database.According to Poling et al. [155], the most accurate technique to obtain an unknown acentric factor is from the critical temperature and pressure combined with the normal boiling point. They have found that the most reliable method is given as follows:(43)ω=−lnPC/1.01325+f0f1,where the critical pressure is in bar and the functions f0 and f1 are defined as functions of TBr. These functions are obtained according to the equations proposed by Ambrose and Walton [161]:(44)f0=−5.976161−TBr+1.298741−TBr1.5−0.603941−TBr2.5−1.068411−TBr5TBr,(45)f1=−5.033651−TBr+1.115051−TBr1.5−5.412171−TBr2.5−7.466281−TBr5TBr.The method proposed by Chen et al. requires the same properties for the estimation of the acentric factor. However, this work suggested a correlation based on the Antoine Equation [157]. Based on 217 compounds over the broadest possible range of TBr, the acentric factor can be estimated as follows:(46)ω=0.30.2803+0.4789TBrlogPC1−TBr0.9803−0.5211TBr−1.Furthermore, Constantinou et al. proposed a method for the estimation of the acentric factor using group contribution [162]. The equation for the property estimation is reported as follows:(47)expω0.40850.5050=∑iNiω1i+∑iMjω2j+1.1507,where ω1i is the contribution of the first-order group type i occurring Ni times and ω2j is the contribution of the second-order group type j occurring Mj times. The coefficients to determine the acentric factor can be found in Tables A2 and A3 of the Supplementary Material.


#### 4.4.2. Acentric Factor Estimation Results

Estimated values of the acentric factor are given in Table22.Table 22
Comparison of experimental and estimated acentric factor.
CN : DB
NIST
Pitzer et al. [158]
Lee and Kesler [159]
Watanasiri [160]
Ambrose and Walton [161]
Chen et al. [157]
Constantinou et al. [162]
C8 : 0
0.782
0.782
0.790
0.493
0.776
0.776
0.780
C10 : 0
0.749
0.749
0.725
0.495
0.709
0.712
0.861
C12 : 0
0.879
0.879
0.889
0.532
0.869
0.872
0.940
C14 : 0
0.940
0.940
0.965
0.605
0.940
0.946
1.017
C16 : 0
0.970
0.970
0.942
0.714
0.917
0.925
1.093
C18 : 0
1.029
1.029
1.001
0.863
0.973
0.984
1.167
C20 : 0
1.184
1.184
1.206
1.031
1.171
1.187
1.239
C22 : 0
1.233
1.233
1.252
1.243
1.215
1.234
1.310
C24 : 0
0.810
0.810
1.449
1.508
1.403
1.437
1.379
C16 : 1
1.088
1.088
1.105
0.751
1.075
1.084
1.079
C18 : 1
1.247
1.345
1.342
0.856
1.304
1.315
1.153
C20 : 1
0.725
—
0.742
1.184
0.719
0.736
1.226
C22 : 1
1.080
1.080
1.102
1.294
1.068
1.091
1.297
C24 : 1
0.676
—
0.694
1.766
0.671
0.694
1.366
C18 : 2
0.996
0.956
0.945
1.043
0.920
0.929
1.140
C18 : 3
1.022
0.976
0.950
1.054
0.922
0.936
1.127Figure9 presents the relative deviation between the different methods when compared to the NIST values, while Table 23 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by NIST.Figure 9
Absolute relative deviation of estimation of the acentric factor.Table 23
Deviations of different methods for the prediction of the acentric factor.
Pitzer et al. [158]
Lee and Kesler [159]
Watanasiri [160]
Ambrose and Walton [161]
Chen et al. [157]
Constantinou et al. [162]
AAD (K)
0.01
0.07
0.32
0.07
0.07
0.19
RMSD (K)
0.03
0.16
0.41
0.15
0.16
0.27
AARD (%)
1.2
7.8
37.7
7.5
7.5
22.6
MIN RD (%)
−4
−7
−40
−10
−8
−7
MAX RD (%)
8
79
161
73
77
102The acentric factor provided by NIST is obtained from the Pitzer correlation. The relative deviation concerning the components C20 : 1 and C24 : 1 is higher for the methods proposed by Watanasiri et al. and Constantinou et al. This occurs since the estimation of the vapor pressure was not available in the NIST database. Therefore, the NIST values of the acentric factor for these two components were probably estimated using another method. In addition, the acentric factor of the component C24 : 0 is inconsistent when compared to the other saturated fatty acids, which makes this value doubtful. Once the Pitzer equation was used for the estimation of the values from NIST, the absolute deviation of Pitzer regarding the NIST acentric value is negligible.The method proposed by Watanasiri et al. reported a higher deviation with respect to NIST. This is due to the fact that, in their study, the correlations were built using a different set of components that did not include fatty acids. Followed by this method, Constantinou et al. reported the second-highest deviation since their estimation was made by using the group contributions instead of the physical properties of the components. Conversely, the methods by Lee and Kesler, Ambrose and Walton, and Chen et al. have reported low deviation compared to NIST.The acentric factor has a definition based on vapor pressure. In this way, the property is indirectly measured, which can result in uncertainties in the estimation. The estimation of the acentric factor for the unsaturated fatty acids as well as the component C24 : 0 reported some inconsistencies. Disregarding those fatty acids, the average absolute relative deviation of the methods reduced to 0% for Pitzer et al., 2.1% to Lee and Kesler method, 25.3% for Watanasiri, 2.6% for Ambrose and Walton, 2.1% for Chen et al., and 8.4% for the method proposed by Constantinou et al. Therefore, these estimations are more reliable for the saturated fatty acids from C8 : 0 to C22 : 0.


#### 4.4.3. Novel Acentric Factor Estimation Method

The novel method proposed to estimate the acentric factor is given as follows:(48)ω=αωCN:n×AωFAMW2+BωFAMW+CωFA.The coefficientsAPCFA CN:n, BPCFA CN:n, CPCFA  CN:n, and αPMCN:n were obtained by minimizing the relative deviations between the values estimated by the novel method and the NIST database values, using the GRG nonlinear solving method to achieve the optimized result.As the variability of the NIST database is very high, mainly for the unsaturated fatty acids, the novel method will be only valid to estimate the acentric factors for fatty acids within the range between C8 : 0 and C22 : 0. The coefficients to determine the acentric factor are provided in Table24.Table 24
Coefficients for the estimation of the acentric factor (valid from C8 : 0 to C22 : 0) according to equation (48).
Coefficients
Values
A
ω
FA CN
:
n
5.4382E−05
B
ω
FA CN
:
n
−0.0032
C
ω
FA CN
:
n
6.4600
α
ω
CN
:
0
0.1066For the novel method for the acentric factor,AAD=0.03K, RMSD=0.03K, AARD=3.0%, MIN RD=−3.0%, and MAX RD=7.1%.


## 4.1. Normal Boiling Point

The normal boiling point (TB) is defined as the temperature in which the vapor pressure equals the value of 1 atmosphere. The normal boiling point is used to compare different liquids and represents an essential input for the estimation of other properties like the critical parameters.


### 4.1.1. Estimation Methods

To determine the normal boiling point, several methods are available in the literature. The most relevant ones are summarized in Table3.Table 3
Methods for the normal boiling point estimation.
Method
Parameter(s) required
Joback and Reid [148]
Molecular structure
Constantinou and Gani [149]
Molecular structure
Marrero-Marejón and Pardillo-Fontdevila [150]
Molecular structure andMW
Marrero and Gani [151]
Molecular structure
Nannoolal et al. [152]
Molecular structureEach of these methods has been applied for the set of fatty acids identified in Table2.The Joback and Reid method [148] estimates the normal boiling point considering the group incremental value (TBi) multiplied by the number of times the group appears in the compound (Ni). The estimation formula is reported in the following equation:(7)TB=198.2+∑iNi⋅Tbi.The coefficients used for the estimation of the normal boiling point are reported as Supplementary Material in TableA1.Constantinou and Gani [149] established the two-level evaluation method reported as follows:(8)TBK=204.359⋅ln∑iNi⋅Tb1i+W⋅∑jMj⋅Tb2j.The method is based on the first- and second-order group contribution. In the former, each group has a single contribution independently of the type of compound involved (TB1i). The latter (TB2j) has the function to provide more structural information about the portions of the structure where the description through the first-order groups is insufficient. The model constant W assumes the value zero if only the first-order group estimation is taken into account, or one if the second order is also considered. Ni and Mj are the group occurrences in the compound. The coefficients used to determine the normal boiling point are reported in Tables A2 and A3 of the Supplementary Material.Marrero-Marejón and Pardillo-Fontdevila proposed a family of models to estimate the normal boiling point of pure organic compounds based on the compound chemical structure and molecular weight [150]. The normal boiling point is evaluated according to equation (9), where Ni is the number of times that type i occurs with contributions tbbk:(9)TB=MW−0.404∑iNi⋅tbbki+156.0.In this case, the contributions are based on group interaction, taking into account the interactions between bonding groups in the molecule. TableA4, in the Supplementary Material, presents the coefficients used.Marrero and Gani [151] defined an estimation of the normal boiling point based on three levels, as follows:(10)TBK=222.543⋅ln∑iNi⋅Tb1i+∑jMj⋅Tb2j+∑kOj⋅Tb3k.The first level includes the simple groups used for describing a wide variety of organic compounds. The second level involves groups that permit a better description of polyfunctional compounds and differentiation among isomers. Finally, the third level provides more structural information about molecular fragments of compounds whose description is insufficient through the first- and second-level groups. The coefficients to determine the normal boiling point can be found in TablesA5 and A6 of the Supplementary Material.The estimation of the boiling point proposed by Nannoolal et al. [152] is reported as follows:(11)TBK=∑iNiCin0.6583+1.6868+84.3395,where Ci is the group contribution of group i and n is the number of atoms in the molecule (excluding the hydrogen atoms). The method is valid for nonelectrolyte organic compounds and was developed using experimental data for about 2850 compounds stored in the Dortmund data bank. Table A7 of the Supplementary Material presents the coefficients correspondent to this method.


### 4.1.2. Normal Boiling Point Estimation Results

The results for the normal boiling point estimated using the methods described are summarized in Table4. The values extracted from the NIST database were accessed through the Aspen Property software, and they were used as a reference for the comparison with the estimated values. The values obtained were checked for consistency, and it was noticed that the normal boiling point for the C24 : 0 (673.7 K) was lower than the value reported for C22 : 0 (685.0 K). For this reason, the value of 704.4 K reported by Yaws [153] was used.Table 4
Comparison of experimental and estimated normal boiling temperature (K).
CN : DB
NIST
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [152]
C8 : 0
512.6
528.2
512.2
518.2
517.8
516.1
C10 : 0
543.2
573.9
540.8
554.3
546.8
549.6
C12 : 0
572.1
619.7
565.9
588.3
572.5
580.0
C14 : 0
599.5
665.4
588.3
620.6
595.6
607.9
C16 : 0
619.8
711.2
608.5
651.4
616.4
633.7
C18 : 0
643.0
757.0
626.8
681.0
635.5
657.9
C20 : 0
672.9
802.7
643.7
709.5
653.1
680.6
C22 : 0
692.9
848.5
659.2
737.1
669.4
702.1
C24 : 0
704.4∗
894.2
673.7
763.8
684.5
722.4
C16 : 1
632.5
715.4
607.0
653.4
617.5
625.8
C18 : 1
635.9
761.1
625.5
682.9
636.5
650.4
C20 : 1
642.4
806.9
642.4
711.4
654.0
673.5
C22 : 1
665.7
852.6
658.1
738.8
670.2
695.3
C24 : 1
672.6
898.4
672.6
765.5
685.3
715.9
C18 : 2
624.1
765.3
624.1
684.8
637.5
643.0
C18 : 3
622.7
769.4
622.7
686.7
638.5
635.5
∗Value reported by Yaws [153].Figure4 indicates the absolute relative deviation between the different methods when compared to the NIST values.Figure 4
Absolute relative deviation comparison for the normal boiling point estimation.From Figure4, it is possible to notice that the Joback and Raid method always overestimates the normal boiling point, and the overestimation increases with the molecular weight. In general, for the same number of atoms of carbon in the molecule, the overestimation increases with the unsaturation. The same trend, but with a reduced absolute relative deviation, is observed for the Marrero-Marejon and Pardillo-Fontdevila method. The opposite tendency is observed for the method developed by Constantinou and Gani, where the normal boiling point is underestimated for the saturated fatty acids with the highest molecular weight. A similar behaviour, but with a reduced margin of deviations, is observed for the Marrero and Gani method. The Nannoolal et al. method has a positive deviation for the unsaturated fatty acids that seem to be independent from the molecular weight. However, for unsaturated fatty acids, the overestimation increases with the molecular weight.Table5 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by the NIST database.Table 5
Deviations of the different methods for the normal boiling point estimation.
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [152]
AAD (K)
119.60
11.55
43.23
10.05
15.43
RMSD (K)
133.05
16.40
49.56
12.33
18.65
AARD (%)
18.5
1.8
6.7
1.6
2.4
MIN RD (%)
3
−5
1
−3
−11
MAX RD (%)
34
0
14
3
6Among all the estimation methods explored, the two-level and the three-level approach proposed by Marrero and Gani provided a more reliable estimation of the normal boiling point for the FAs considered.


### 4.1.3. Novel Normal Boiling Point Estimation Method

Anand et al. [154] defined the normal boiling point for fatty acids methyl esters (FAMEs), as a function of molecular weight (equation (12)). The same approach was followed to propose an alternative correlation for the fatty acids:(12)TBFAME=AFAMEMW2+BFAMEMW+CFAME.Using a similar equation, it was possible to obtain the coefficients to determine the normal boiling points for the fatty acids considered by applying the generalized reduced gradient method to minimize the deviations. In fact, observing the normal boiling point values, as a function of the molecular weight, it is possible to conclude that it has a parabolic behaviour.The novel estimation method, reported in equation (13), has three coefficients, assigned as ATBFA CN:n, BTBFA CN:n, and CTBFA CN:n, and a correction factor, αTBCN:n, which depends on the number of unsaturation:(13)TBFA=αTBCN:n⋅AATBFA CN:nMW2+BTBFA CN:nMW+CTBFA CN:n.The values forATBFA CN:n, BTBFA CN:nCTBFA CN:n, and αTBCN:n were obtained by minimizing the relative deviations with the NIST database values, using the GRG nonlinear solving method to achieve the optimized result. The coefficients obtained are presented in Table 6.Table 6
Coefficients for the estimation of the normal boiling point according to equation (13).
Coefficients
Values
A
T
B
FA CN
:
n
−7.0670E − 04
B
T
B
FA CN
:
n
0.9173
C
T
B
FA CN
:
n
205.6120
α
T
B
CN
:
0
1.5697
α
T
B
CN
:
1
1.5501
α
T
B
CN
:
2
1.5323
α
T
B
CN
:
3
1.5329The proposed normal boiling point estimation method showed a lower estimation deviation when compared with the other methods previously presented. In particular, it was obtained:AAD=8.49K, RMSD=11.94K, AARD=1.3%, MINRD=−3.6%, and MAXRD=4.3%.


## 4.1.1. Estimation Methods

To determine the normal boiling point, several methods are available in the literature. The most relevant ones are summarized in Table3.Table 3
Methods for the normal boiling point estimation.
Method
Parameter(s) required
Joback and Reid [148]
Molecular structure
Constantinou and Gani [149]
Molecular structure
Marrero-Marejón and Pardillo-Fontdevila [150]
Molecular structure andMW
Marrero and Gani [151]
Molecular structure
Nannoolal et al. [152]
Molecular structureEach of these methods has been applied for the set of fatty acids identified in Table2.The Joback and Reid method [148] estimates the normal boiling point considering the group incremental value (TBi) multiplied by the number of times the group appears in the compound (Ni). The estimation formula is reported in the following equation:(7)TB=198.2+∑iNi⋅Tbi.The coefficients used for the estimation of the normal boiling point are reported as Supplementary Material in TableA1.Constantinou and Gani [149] established the two-level evaluation method reported as follows:(8)TBK=204.359⋅ln∑iNi⋅Tb1i+W⋅∑jMj⋅Tb2j.The method is based on the first- and second-order group contribution. In the former, each group has a single contribution independently of the type of compound involved (TB1i). The latter (TB2j) has the function to provide more structural information about the portions of the structure where the description through the first-order groups is insufficient. The model constant W assumes the value zero if only the first-order group estimation is taken into account, or one if the second order is also considered. Ni and Mj are the group occurrences in the compound. The coefficients used to determine the normal boiling point are reported in Tables A2 and A3 of the Supplementary Material.Marrero-Marejón and Pardillo-Fontdevila proposed a family of models to estimate the normal boiling point of pure organic compounds based on the compound chemical structure and molecular weight [150]. The normal boiling point is evaluated according to equation (9), where Ni is the number of times that type i occurs with contributions tbbk:(9)TB=MW−0.404∑iNi⋅tbbki+156.0.In this case, the contributions are based on group interaction, taking into account the interactions between bonding groups in the molecule. TableA4, in the Supplementary Material, presents the coefficients used.Marrero and Gani [151] defined an estimation of the normal boiling point based on three levels, as follows:(10)TBK=222.543⋅ln∑iNi⋅Tb1i+∑jMj⋅Tb2j+∑kOj⋅Tb3k.The first level includes the simple groups used for describing a wide variety of organic compounds. The second level involves groups that permit a better description of polyfunctional compounds and differentiation among isomers. Finally, the third level provides more structural information about molecular fragments of compounds whose description is insufficient through the first- and second-level groups. The coefficients to determine the normal boiling point can be found in TablesA5 and A6 of the Supplementary Material.The estimation of the boiling point proposed by Nannoolal et al. [152] is reported as follows:(11)TBK=∑iNiCin0.6583+1.6868+84.3395,where Ci is the group contribution of group i and n is the number of atoms in the molecule (excluding the hydrogen atoms). The method is valid for nonelectrolyte organic compounds and was developed using experimental data for about 2850 compounds stored in the Dortmund data bank. Table A7 of the Supplementary Material presents the coefficients correspondent to this method.


## 4.1.2. Normal Boiling Point Estimation Results

The results for the normal boiling point estimated using the methods described are summarized in Table4. The values extracted from the NIST database were accessed through the Aspen Property software, and they were used as a reference for the comparison with the estimated values. The values obtained were checked for consistency, and it was noticed that the normal boiling point for the C24 : 0 (673.7 K) was lower than the value reported for C22 : 0 (685.0 K). For this reason, the value of 704.4 K reported by Yaws [153] was used.Table 4
Comparison of experimental and estimated normal boiling temperature (K).
CN : DB
NIST
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [152]
C8 : 0
512.6
528.2
512.2
518.2
517.8
516.1
C10 : 0
543.2
573.9
540.8
554.3
546.8
549.6
C12 : 0
572.1
619.7
565.9
588.3
572.5
580.0
C14 : 0
599.5
665.4
588.3
620.6
595.6
607.9
C16 : 0
619.8
711.2
608.5
651.4
616.4
633.7
C18 : 0
643.0
757.0
626.8
681.0
635.5
657.9
C20 : 0
672.9
802.7
643.7
709.5
653.1
680.6
C22 : 0
692.9
848.5
659.2
737.1
669.4
702.1
C24 : 0
704.4∗
894.2
673.7
763.8
684.5
722.4
C16 : 1
632.5
715.4
607.0
653.4
617.5
625.8
C18 : 1
635.9
761.1
625.5
682.9
636.5
650.4
C20 : 1
642.4
806.9
642.4
711.4
654.0
673.5
C22 : 1
665.7
852.6
658.1
738.8
670.2
695.3
C24 : 1
672.6
898.4
672.6
765.5
685.3
715.9
C18 : 2
624.1
765.3
624.1
684.8
637.5
643.0
C18 : 3
622.7
769.4
622.7
686.7
638.5
635.5
∗Value reported by Yaws [153].Figure4 indicates the absolute relative deviation between the different methods when compared to the NIST values.Figure 4
Absolute relative deviation comparison for the normal boiling point estimation.From Figure4, it is possible to notice that the Joback and Raid method always overestimates the normal boiling point, and the overestimation increases with the molecular weight. In general, for the same number of atoms of carbon in the molecule, the overestimation increases with the unsaturation. The same trend, but with a reduced absolute relative deviation, is observed for the Marrero-Marejon and Pardillo-Fontdevila method. The opposite tendency is observed for the method developed by Constantinou and Gani, where the normal boiling point is underestimated for the saturated fatty acids with the highest molecular weight. A similar behaviour, but with a reduced margin of deviations, is observed for the Marrero and Gani method. The Nannoolal et al. method has a positive deviation for the unsaturated fatty acids that seem to be independent from the molecular weight. However, for unsaturated fatty acids, the overestimation increases with the molecular weight.Table5 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by the NIST database.Table 5
Deviations of the different methods for the normal boiling point estimation.
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [152]
AAD (K)
119.60
11.55
43.23
10.05
15.43
RMSD (K)
133.05
16.40
49.56
12.33
18.65
AARD (%)
18.5
1.8
6.7
1.6
2.4
MIN RD (%)
3
−5
1
−3
−11
MAX RD (%)
34
0
14
3
6Among all the estimation methods explored, the two-level and the three-level approach proposed by Marrero and Gani provided a more reliable estimation of the normal boiling point for the FAs considered.


## 4.1.3. Novel Normal Boiling Point Estimation Method

Anand et al. [154] defined the normal boiling point for fatty acids methyl esters (FAMEs), as a function of molecular weight (equation (12)). The same approach was followed to propose an alternative correlation for the fatty acids:(12)TBFAME=AFAMEMW2+BFAMEMW+CFAME.Using a similar equation, it was possible to obtain the coefficients to determine the normal boiling points for the fatty acids considered by applying the generalized reduced gradient method to minimize the deviations. In fact, observing the normal boiling point values, as a function of the molecular weight, it is possible to conclude that it has a parabolic behaviour.The novel estimation method, reported in equation (13), has three coefficients, assigned as ATBFA CN:n, BTBFA CN:n, and CTBFA CN:n, and a correction factor, αTBCN:n, which depends on the number of unsaturation:(13)TBFA=αTBCN:n⋅AATBFA CN:nMW2+BTBFA CN:nMW+CTBFA CN:n.The values forATBFA CN:n, BTBFA CN:nCTBFA CN:n, and αTBCN:n were obtained by minimizing the relative deviations with the NIST database values, using the GRG nonlinear solving method to achieve the optimized result. The coefficients obtained are presented in Table 6.Table 6
Coefficients for the estimation of the normal boiling point according to equation (13).
Coefficients
Values
A
T
B
FA CN
:
n
−7.0670E − 04
B
T
B
FA CN
:
n
0.9173
C
T
B
FA CN
:
n
205.6120
α
T
B
CN
:
0
1.5697
α
T
B
CN
:
1
1.5501
α
T
B
CN
:
2
1.5323
α
T
B
CN
:
3
1.5329The proposed normal boiling point estimation method showed a lower estimation deviation when compared with the other methods previously presented. In particular, it was obtained:AAD=8.49K, RMSD=11.94K, AARD=1.3%, MINRD=−3.6%, and MAXRD=4.3%.


## 4.2. Normal Melting Point

The normal melting point for a pure substance is defined as the temperature in which the solid and the liquid form exist in equilibrium at atmospheric pressure. In general, it is expected that the melting point of fatty acids increases if the unsaturation is reduced. If the fatty acids are used for biodiesel production, the availability of the melting point is essential for the evaluation of the cold-flow properties.


### 4.2.1. Estimation Methods

Several methods are available in the literature to determine the normal melting point. The most appropriate methods are gathered in Table7.Table 7
Methods for the normal melting point estimation.
Method
Parameter required
Joback and Reid [148]
Molecular structure
Constantinou and Gani [149]
Molecular structure
Marrero and Gani [151]
Molecular structureJoback and Reid proposed the estimation formula reported as follows [148]:(14)TMK=122.5+∑Ni⋅Tmi.According to the authors, however, estimations of the normal melting point are not accurate and should be considered only as approximate. The coefficients of the group contributions are presented in TableA1 of the Supplementary Material.The Constantinou and Gani method [149] is presented as follows:(15)TMK=102.425×ln∑iNi×Tm1i+W×∑jMj×Tm2j.The coefficients to determine the normal melting point can be found in TablesA2 and A3 of the Supplementary Material.Marrero and Gani [151] developed a method to determine the normal melting point using the following equation:(16)TMK=147.450×ln∑iNi×Tm1i+∑jMj×Tm2j+∑kOj×Tm3k.The coefficients to determine the normal melting point can be found in TablesA5 and A6 of the Supplementary Material.


### 4.2.2. Normal Melting Point Estimation Results

Table8 presents the values of the estimated normal melting point together with the values obtained from the NIST database.Table 8
Comparison of experimental and estimated normal melting point (K).
CN : DB
NIST
Joback and Reid [148]
Constantinou and Gani [149]
Marrero and Gani [151]
C8 : 0
289.7
340.5
293.6
333.6
C10 : 0
304.5
363.1
303.8
341.2
C12 : 0
317.0
385.6
313.2
348.3
C14 : 0
327.3
408.1
321.7
355.1
C16 : 0
335.6
430.7
329.6
361.7
C18 : 0
342.6
453.2
336.9
367.9
C20 : 0
348.2
475.8
343.7
373.9
C22 : 0
353.8
498.3
350.1
379.7
C24 : 0
355.6
520.8
356.1
385.2
C16 : 1
274.8
425.6
326.2
365.9
C18 : 1
286.5
448.1
333.8
371.9
C20 : 1
297.3
470.7
340.8
377.8
C22 : 1
306.1
493.2
347.4
383.4
C24 : 1
—
515.8
353.6
388.8
C18 : 2
267.6
443.1
330.6
375.8
C18 : 3
260.0
438.0
327.2
379.7Figure5 reports the absolute relative deviation between the different methods when compared with the NIST values.Figure 5
Absolute relative deviation comparison for the normal melting point estimation.The Joback and Reid group contribution method overestimates the normal melting point. The overestimation regularly grows for saturated fatty acids with a consistent penalty for molecules with the highest molecular weight. The same trend is observed for the unsaturated compounds even if the increase of the absolute relative deviation is lower with respect to the case of saturated fatty acids.Table9 shows the deviation indexes for each method when compared with the data provided by NIST.Table 9
Deviations of different methods for the estimation of the normal melting point.
Joback and Reid [148]
Constantinou and Gani [149]
Marrero and Gani [151]
AAD (K)
132.98
24.10
56.68
RMSD (K)
141.25
34.02
65.14
AARD (%)
43.5
8.4
19.3
MIN RD (%)
18
−2
7
MAX RD (%)
69
26
46The method of Constantinou and Gani offers the best estimation with a very good prediction of the saturated compounds. However, for the unsaturated fatty acids, all the methods considered report a high estimation deviation. When comparing molecules with the same number of carbons, it is possible to observe that the melting point decreases when the unsaturation increases. This decrease is due to the molecular distortion induced by the double bonds that prevent the molecules from packing together easily. The difficulty in predicting the London dispersion forces between the oil molecules may be the cause of the high deviations observed for unsaturated fatty acids.


### 4.2.3. Novel Normal Melting Point Estimation Method

It is possible to obtain the coefficients to determine the normal melting points applying the generalizedreduced gradient (GRG) method to minimize the deviations, with a similar procedure described in Section4.1.3 for the estimation of the normal boiling point.Equation (17) is the novel estimation method developed:(17)TMFA=αTMCN:n⋅ATMFAMW2+BTMFAMW+CTMFA.The main advantage of this method is the possibility of estimating the normal meting point, using the fatty acids’ molecular weight exclusively.The coefficientsATMFA CN:n, BTMFA CN:n, CTBFA CN:n, and αTMCN:n were obtained by minimizing the relative deviations with the NIST database values. The obtained coefficients are presented in Table 10.Table 10
Coefficients for the estimation of the normal melting point according to equation (17).
Coefficients
Values
A
T
M
FA CN
:
n
−4.1905E−04
B
T
M
FA CN
:
n
0.4863
C
T
M
FA CN
:
n
193.9418
α
T
M
CN
:
0
1.1380
α
T
M
CN
:
1
0.9752
α
T
M
CN
:
2
0.8999
α
T
M
CN
:
3
0.8759For the proposed method, the comparison indexes areAAD=3.15K, RMSD=4.09K, AARD=1.0%, MIN RD=−2.6%, and MAX RD=3.1%, giving a smaller deviation when compared to the other estimation methods.


## 4.2.1. Estimation Methods

Several methods are available in the literature to determine the normal melting point. The most appropriate methods are gathered in Table7.Table 7
Methods for the normal melting point estimation.
Method
Parameter required
Joback and Reid [148]
Molecular structure
Constantinou and Gani [149]
Molecular structure
Marrero and Gani [151]
Molecular structureJoback and Reid proposed the estimation formula reported as follows [148]:(14)TMK=122.5+∑Ni⋅Tmi.According to the authors, however, estimations of the normal melting point are not accurate and should be considered only as approximate. The coefficients of the group contributions are presented in TableA1 of the Supplementary Material.The Constantinou and Gani method [149] is presented as follows:(15)TMK=102.425×ln∑iNi×Tm1i+W×∑jMj×Tm2j.The coefficients to determine the normal melting point can be found in TablesA2 and A3 of the Supplementary Material.Marrero and Gani [151] developed a method to determine the normal melting point using the following equation:(16)TMK=147.450×ln∑iNi×Tm1i+∑jMj×Tm2j+∑kOj×Tm3k.The coefficients to determine the normal melting point can be found in TablesA5 and A6 of the Supplementary Material.


## 4.2.2. Normal Melting Point Estimation Results

Table8 presents the values of the estimated normal melting point together with the values obtained from the NIST database.Table 8
Comparison of experimental and estimated normal melting point (K).
CN : DB
NIST
Joback and Reid [148]
Constantinou and Gani [149]
Marrero and Gani [151]
C8 : 0
289.7
340.5
293.6
333.6
C10 : 0
304.5
363.1
303.8
341.2
C12 : 0
317.0
385.6
313.2
348.3
C14 : 0
327.3
408.1
321.7
355.1
C16 : 0
335.6
430.7
329.6
361.7
C18 : 0
342.6
453.2
336.9
367.9
C20 : 0
348.2
475.8
343.7
373.9
C22 : 0
353.8
498.3
350.1
379.7
C24 : 0
355.6
520.8
356.1
385.2
C16 : 1
274.8
425.6
326.2
365.9
C18 : 1
286.5
448.1
333.8
371.9
C20 : 1
297.3
470.7
340.8
377.8
C22 : 1
306.1
493.2
347.4
383.4
C24 : 1
—
515.8
353.6
388.8
C18 : 2
267.6
443.1
330.6
375.8
C18 : 3
260.0
438.0
327.2
379.7Figure5 reports the absolute relative deviation between the different methods when compared with the NIST values.Figure 5
Absolute relative deviation comparison for the normal melting point estimation.The Joback and Reid group contribution method overestimates the normal melting point. The overestimation regularly grows for saturated fatty acids with a consistent penalty for molecules with the highest molecular weight. The same trend is observed for the unsaturated compounds even if the increase of the absolute relative deviation is lower with respect to the case of saturated fatty acids.Table9 shows the deviation indexes for each method when compared with the data provided by NIST.Table 9
Deviations of different methods for the estimation of the normal melting point.
Joback and Reid [148]
Constantinou and Gani [149]
Marrero and Gani [151]
AAD (K)
132.98
24.10
56.68
RMSD (K)
141.25
34.02
65.14
AARD (%)
43.5
8.4
19.3
MIN RD (%)
18
−2
7
MAX RD (%)
69
26
46The method of Constantinou and Gani offers the best estimation with a very good prediction of the saturated compounds. However, for the unsaturated fatty acids, all the methods considered report a high estimation deviation. When comparing molecules with the same number of carbons, it is possible to observe that the melting point decreases when the unsaturation increases. This decrease is due to the molecular distortion induced by the double bonds that prevent the molecules from packing together easily. The difficulty in predicting the London dispersion forces between the oil molecules may be the cause of the high deviations observed for unsaturated fatty acids.


## 4.2.3. Novel Normal Melting Point Estimation Method

It is possible to obtain the coefficients to determine the normal melting points applying the generalizedreduced gradient (GRG) method to minimize the deviations, with a similar procedure described in Section4.1.3 for the estimation of the normal boiling point.Equation (17) is the novel estimation method developed:(17)TMFA=αTMCN:n⋅ATMFAMW2+BTMFAMW+CTMFA.The main advantage of this method is the possibility of estimating the normal meting point, using the fatty acids’ molecular weight exclusively.The coefficientsATMFA CN:n, BTMFA CN:n, CTBFA CN:n, and αTMCN:n were obtained by minimizing the relative deviations with the NIST database values. The obtained coefficients are presented in Table 10.Table 10
Coefficients for the estimation of the normal melting point according to equation (17).
Coefficients
Values
A
T
M
FA CN
:
n
−4.1905E−04
B
T
M
FA CN
:
n
0.4863
C
T
M
FA CN
:
n
193.9418
α
T
M
CN
:
0
1.1380
α
T
M
CN
:
1
0.9752
α
T
M
CN
:
2
0.8999
α
T
M
CN
:
3
0.8759For the proposed method, the comparison indexes areAAD=3.15K, RMSD=4.09K, AARD=1.0%, MIN RD=−2.6%, and MAX RD=3.1%, giving a smaller deviation when compared to the other estimation methods.


## 4.3. Critical Parameters

The critical point is used to identify the fluid region where the liquid and gas phases can be no longer distinguished. Critical parameters are used for the estimation of other properties, such as the acentric factor, compressibility factor, liquid density, vapor pressure, and enthalpy of vaporization, among others.


### 4.3.1. Estimation Methods

Several methods are available in the literature to determine the critical parameters such as critical temperature, critical pressure, and critical volume. The most appropriate methods are summarized in Table11.Table 11
Methods for the critical parameters’ estimation.
Method
Parameter(s) required
Joback and Reid [148]
Molecular structure andTB
Constantinou and Gani [149]
Molecular structure
Wilson and Jasperson [155]
Molecular structure andTB
Marrero-Marejón and Pardillo-Fontdevila [150]
Molecular structure andTB
Marrero and Gani [151]
Molecular structure
Nannoolalet al. [156]
Molecular structure,MW, and TBJoback and Reid proposed the relations reported in equations (18)–(20) to estimate the critical properties based on the group contribution methods [148]:(18)TC=TB0.584+0.965∑Ni⋅Tci−∑Ni⋅Tci2,(19)PC=10.113+0.0032NA−∑Ni⋅Pci2,(20)VC=17.5+∑Ni⋅Vci,where NA is the number of atoms in the molecule. The values Tci, Pci, and Vci are presented in Table A1.Constantinou and Gani [149] established a method to determine TC, PC, and VC using equations (21)–(23), respectively:(21)TCK=181.128⋅ln∑iNi⋅Tc1i+W⋅∑jMj⋅Tc2j,(22)PCbar=∑iNi⋅Pc1i+W×∑jMj⋅Pc2j+0.100220−2+1.3705,(23)VCcm3⋅mol−1=−0.004350+∑iNi⋅vc1i+W×∑jMj⋅vc2j.The coefficients to determine the critical properties can be found in TablesA2 and A3 in the Supplementary Material.Wilson and Jasperson suggested equation (24) to estimate the critical temperature and equation (25) to determine the critical pressure:(24)TCK=TB0.048271−0.019846Nr+∑kNk·tck+∑jMjΔtcj0.2,(25)PCbar=0.0186233⋅TC−0.96601+exp−0.0092295−0.0290403Nr+0.041∑kNk⋅Pcbk+∑jMjΔPcj.The coefficients to determine these critical properties can be found in TableA6 in the Supplementary Material.Marrero-Marejón and Pardillo-Fontdevila developed equations (26)–(28) to calculate the critical parameters [150]:(26)TCK=TB0.5851−0.9286∑kNk⋅tbck+∑jMj⋅Δtcbj2,(27)PCbar=0.1285−0.0059Natoms−∑kNk⋅Pcbk−2,(28)VCcm3⋅mol−1=−25.1+∑kNk⋅vcbk,where Natoms is the number of atoms in the compound and Nk is the number of atoms of type k with contributions tcbk, pcbk, and vcbk. The parameters to determine these critical parameters can be found at Table A4 in the Supplementary Material.Marrero and Gani [151] determined TC, PC, and VC using the following equations:(29)TCK=231.239⋅ln∑iNi⋅Tc1i+W⋅∑jMj⋅Tc2j+W⋅∑kOk⋅Tc3k,(30)PCbar=∑iNi⋅Pc1i+W×∑jMj⋅Pc2j+W×∑kOk⋅Pc3k+0.108998−2+5.9827,(31)VCcm3⋅mol−1=7.95+∑iNi⋅Vc1i+W×∑jMj⋅Vc2j+W×∑kOk⋅Vc3k.The coefficients to determine the critical properties can be found in TablesA5 and A6 in the Supplementary Material.The estimation of the critical properties proposed by Nannoolal et al. [156] is presented in the following equations:(32)TCK=TB0.6990+10.9889+∑iNiCi0.8607,(33)PCkPa=MW−0.14041g/mol0.00939+∑iNiCi2,(34)VCcm3⋅mol−1=∑iNiCin−0.2266+86.1539,where n is the number of atoms in the molecule (excluding hydrogen). Table A7 in the Supplementary Material presents the coefficients used.For the estimation methods that depend on the normal boiling point temperature, this value can be experimentally determined or estimated. The calculations were performed using the normal boiling point temperatures available from the NIST database, as reported in Table4.


### 4.3.2. Critical Properties Estimation Results

(1) Critical Temperature Estimation Results. The estimated and experimental values for the critical temperatures are presented in Table 12.Table 12
Comparison of experimental and estimated critical temperature (K).
CN : DB
NIST
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
C8 : 0
693.7
692.1
694.6
695.0
692.3
738.2
694.9
C10 : 0
723.8
714.6
718.4
720.4
715.7
762.3
719.5
C12 : 0
743.0
736.6
740.5
742.7
739.1
784.0
742.0
C14 : 0
763.0
758.4
760.8
762.5
762.8
803.9
762.5
C16 : 0
785.0
773.5
772.7
780.4
780.4
822.2
774.3
C18 : 0
803.0
794.5
788.7
796.6
804.6
839.2
790.1
C20 : 0
820.0
826.2
813.0
811.6
840.4
855.0
814.1
C22 : 0
837.0
848.4
825.5
825.4
867.4
869.8
826.3
C24 : 0
825.0
863.1
828.2
838.2
887.6
883.7
828.6
C16 : 1
789.0
792.5
791.3
780.0
800.7
825.1
796.3
C18 : 1
775.0
787.9
782.4
796.3
798.1
841.8
786.9
C20 : 1
811.0
790.1
778.4
811.3
802.8
857.5
782.4
C22 : 1
806.0
815.6
795.3
825.1
831.9
872.1
798.8
C24 : 1
838.0
823.7
792.9
837.9
844.0
885.9
795.8
C18 : 2
787.0
775.8
770.4
796.0
786.6
844.5
778.0
C18 : 3
777.0
776.8
771.2
795.7
789.3
847.1
782.1Figure6 presents the absolute relative deviation between the methods considered and the NIST database values.Figure 6
Absolute relative deviation of estimation for the critical temperature.Except for the Marrero and Gani’s method, all the other methods have a general agreement until the C16 : 0. For the compounds with higher molecular weight, the method developed by Nannoolal et al. and by Constantinou and Gani give the best results. When one or more unsaturations are considered, it is not possible to define a method that consistently predicts the critical temperature. Table13 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by NIST.Table 13
Deviations of different methods for the estimation of the critical temperature.
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
AAD (K)
10.64
11.23
7.95
13.80
47.23
10.14
RMSD (K)
13.72
16.13
10.57
20.85
48.77
14.62
AARD (%)
1.3
1.4
1.0
1.7
6.0
1.3
MIN RD (%)
−3
−5
−1
−1
4
−5
MAX RD (%)
2
1
3
4
9
2From the analysis of Table13, it is possible to notice that the Constantinou and Gani’s method exhibits the best global performances for the critical temperature estimation.(2) Critical Pressure Estimation Results. The results for the estimated critical pressures and the experimental values for the fatty acids are presented in Table 14.Table 14
Comparison of experimental and estimated critical pressure (bar).
CN : DB
NIST
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
C8 : 0
28.57
27.79
26.91
27.67
31.07
26.87
26.27
C10 : 0
20.76
22.92
22.22
22.79
25.40
22.52
22.11
C12 : 0
19.31
19.22
18.79
19.14
21.16
19.40
18.87
C14 : 0
16.40
16.35
16.17
16.36
17.89
17.09
16.29
C16 : 0
14.90
14.08
13.96
14.18
15.33
15.33
14.21
C18 : 0
13.30
12.25
12.25
12.44
13.28
13.95
12.50
C20 : 0
12.00
10.76
10.97
11.03
11.62
12.86
11.07
C22 : 0
11.10
9.52
9.75
9.88
10.25
11.98
9.88
C24 : 0
8.49
8.49
8.61
8.92
9.11
11.25
8.86
C16 : 1
14.88
14.65
14.58
14.15
15.86
15.52
14.37
C18 : 1
13.90∗
12.71
12.39
12.42
13.71
14.10
12.63
C20 : 1
10.98
11.13
10.69
11.02
11.96
12.98
11.18
C22 : 1
9.83
9.83
9.55
9.87
10.53
12.07
9.97
C24 : 1
8.87
8.75
8.61
8.91
9.35
11.33
8.94
C18 : 2
14.10
13.19
12.43
12.40
14.15
14.25
12.76
C18 : 3
12.34∗
13.71
12.68
12.38
14.62
14.41
12.89
∗Values retrieved from the database APV88 PURE32 of Aspen Plus.Figure7 presents the relative deviation between the estimated values and the ones gathered from the NIST database. All the methods provide estimations included in a range of deviation of ±10%, except for the Marrero and Gani method when applied to saturated fatty acids with high molecular weight and unsaturated fatty acids with one and three double bonds. The Nannoolal et al. method gives the best estimation for the fatty acids at high molecular weight with one double bond. Regarding the saturated compounds, the Marrero-Marejón and Pardillo-Fontdevila method overestimates the critical pressure until C16 : 0. After this component, there is an underestimation that increases linearly with the molecular weight. Apart from the Marrero and Gani method, all the others underestimate the critical pressure in the range of C16 : 0–C22 : 0.Figure 7
Absolute relative deviation of estimation for the critical pressure.Table15 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by NIST.Table 15
Deviations of different methods for the estimation of the critical pressure.
NIST
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
AAD (bar)
0.73
0.83
0.71
1.15
1.23
0.77
RMSD (bar)
0.98
0.99
0.95
1.64
1.50
0.97
AARD (%)
5.1
5.7
4.9
7.5
10.4
5.1
MIN RD (%)
−14
−12
−12
−8
−6
−11
MAX RD (%)
11
7
10
22
33
7
Globally, the Nannoolal et al. method offers a better estimation together with the Constantinou and Gani approach.(3) Critical Volume Estimation Results. The estimated and experimental values for the fatty acids’ critical volume are presented in Table 16.Table 16
Comparison of experimental and estimated critical volume (m3/kmol).
CN : DB
NIST
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
C8 : 0
0.526
0.5075
0.5071
0.5107
0.5046
0.5267
C10 : 0
0.639
0.6195
0.6187
0.6239
0.6172
0.6579
C12 : 0
0.787
0.7315
0.7302
0.7371
0.7298
0.7948
C14 : 0
0.921
0.8435
0.8417
0.8503
0.8423
0.9367
C16 : 0
1.066
0.9555
0.9532
0.9635
0.9549
1.0831
C18 : 0
1.255
1.0675
1.0647
1.0767
1.0674
1.2335
C20 : 0
1.371
1.1795
1.1763
1.1899
1.1800
1.3877
C22 : 0
1.486
1.2915
1.2878
1.3031
1.2926
1.5453
C24 : 0
1.575
1.4035
1.3993
1.4163
1.4051
1.7062
C16 : 1
0.898
0.9355
0.9399
0.9429
0.9409
1.0525
C18 : 1
1.016
1.0475
1.0514
1.0561
1.0535
1.2022
C20 : 1
1.158
1.1595
1.1630
1.1693
1.1660
1.3557
C22 : 1
1.448
1.2715
1.2745
1.2825
1.2786
1.5127
C24 : 1
1.402
1.3835
1.3860
1.3957
1.3911
1.6729
C18 : 2
0.972
1.0275
1.0381
1.0355
1.0395
1.1708
C18 : 3
0.960
1.0075
1.0248
1.0149
1.0255
1.1395Figure8 presents the relative deviation distribution between the estimated values and the ones provided by NIST.Figure 8
Absolute relative deviations of estimation for the critical volume.The Nannoolal et al. method offers the best prediction for the saturated compounds with a peak of about 10% overestimation for C24 : 0. However, the overestimation increases to 20% when the unsaturated compounds are evaluated. All the other methods exhibit an opposite behaviour, with the tendency to underestimate the critical volume for saturated compounds, with a lower deviation for the unsaturated fatty acids.Table17 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by the NIST database.Table 17
Deviations of different methods for the estimation of the critical volume.
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
AAD (m3/kmol)
0.087
0.091
0.084
0.090
0.096
RMSD (m3/kmol)
0.112
0.114
0.106
0.112
0.130
AARD (%)
7.3
7.6
7.0
7.6
8.6
MIN RD (%)
−5
−15
−15
−14
−15
MAX RD (%)
31
6
7
7
7Except for the Nannoolal et al. method, that is only recommended for saturated compounds, all the other methods provide a general agreement in the critical volume estimation.


### 4.3.3. Novel Critical Parameters Estimation Method

(1) Novel Critical Temperature Estimation Method. Anand et al. [154] proposed a critical temperature estimation method for FAMEs, according to the following:(35)TCFAME=ATCFAMEMW2+BTCFAMEMW+CTCFAME.Using a similar equation, it is possible to obtain the coefficients to determine the critical temperature for the fatty acids applying the generalized reduced gradients method to minimize the deviations. Equation (36) is the novel estimation method proposed to estimate the fatty acids critical temperature:(36)TCFAK=αTCCN:n⋅ATCFAMW2+BTCFAMW+CTCFA.The coefficientsATCFACN:n, BTCFA CN:n, CTCFA CN:n, and αTCCN:n were obtained by minimizing the relative deviations with the NIST database values. The coefficients obtained are presented in Table 18.Table 18
Coefficients for the estimation of the critical temperature according to equation (36).
Coefficients
Values
A
T
C
FA CN
:
n
−1.8345E − 03
B
T
C
FA CN
:
n
1.6239
C
T
C
FA CN
:
n
550.5882
α
T
C
CN
:
0
0.9286
α
T
C
CN
:
1
0.9185
α
T
C
CN
:
2
0.9134
α
T
C
CN
:
3
0.9028For the proposed method, the critical temperature deviations resulted inAAD=5.88K, RMSD=8.01K, AARD=0.7%, MIN  RD=−1.6%, and MAX  RD=2.3% compared with NIST database. Thus, AAD, RMSD, and AARD significantly outperform the estimation methods considered.(2) Novel Critical Pressure Evaluation Method. Applying the same procedure, it is possible to optimize equation (37) in order to obtain a functional equation to estimate the critical pressure:(37)PCFAbar=αPCCN:n⋅APCFAMW2+BPCFAMW+CPCFA.The coefficientsAPCFA CN:n, BPCFA CN:n, CPCFA CN:n, and αPCCN:n were obtained by minimizing the relative deviations with the NIST database values, using the GRG nonlinear solving method to find the optimized result. The obtained coefficients are presented in Table 19.Table 19
Coefficients for the estimation of the critical pressure according to equation (37).
Coefficients
Values
A
P
C
FA CN
:
n
7.0357E − 04
B
P
C
FA CN
:
n
−0.5776
C
P
C
FA CN
:
n
143.4049
α
P
C
CN
:
0
0.3515
α
P
C
CN
:
1
0.3556
α
P
C
CN
:
2
0.3836
α
P
C
CN
:
3
0.3323For the proposed method, the critical pressure deviations resulted inAAD=0.69bar, RMSD=0.94bar, AARD=4.6%, MIN  RD=−10.4%, and MAX RD=9.7%. Again, AAD, RMSD, and AARD show an improvement in accuracy, when compared with the considered estimation methods.(3) Novel Critical Volume Estimation Method. Applying the same procedure as it was suggested before, it is possible to derive equation (38) to estimate the critical volume:(38)VCFAmL/mol=αVCCN:n⋅AVCFAMW2+BVCFAMW+CVCFA.The coefficientsAVCFA CN:n, BVCFA CN:n, CVFA CN:n, and αVCCN:n were obtained by minimizing the relative deviations with the NIST database values, using the GRG nonlinear solving method to achieve the optimized result. The obtained coefficients are presented in Table 20.Table 20
Coefficients for the estimation of the critical volume according to equation (38).
Coefficients
Values
A
V
C
FA CN
:
n
−1.0023E−06
B
V
C
FA CN
:
n
0.0206
C
V
C
FA CN
:
n
−0.9068
α
V
C
CN
:
0
0.2485
α
V
C
CN
:
1
0.2106
α
V
C
CN
:
2
0.2033
α
V
C
CN
:
3
0.2025The proposed method for the critical volume yield deviations ofAAD=0.03m3/kmol, RMSD=0.05m3/kmol, AARD=2.1%, MIN RD=−13.6%, and MAX  RD=3.1%. The first three indicators significantly outperform the ones computed for the considered estimation methods.


## 4.3.1. Estimation Methods

Several methods are available in the literature to determine the critical parameters such as critical temperature, critical pressure, and critical volume. The most appropriate methods are summarized in Table11.Table 11
Methods for the critical parameters’ estimation.
Method
Parameter(s) required
Joback and Reid [148]
Molecular structure andTB
Constantinou and Gani [149]
Molecular structure
Wilson and Jasperson [155]
Molecular structure andTB
Marrero-Marejón and Pardillo-Fontdevila [150]
Molecular structure andTB
Marrero and Gani [151]
Molecular structure
Nannoolalet al. [156]
Molecular structure,MW, and TBJoback and Reid proposed the relations reported in equations (18)–(20) to estimate the critical properties based on the group contribution methods [148]:(18)TC=TB0.584+0.965∑Ni⋅Tci−∑Ni⋅Tci2,(19)PC=10.113+0.0032NA−∑Ni⋅Pci2,(20)VC=17.5+∑Ni⋅Vci,where NA is the number of atoms in the molecule. The values Tci, Pci, and Vci are presented in Table A1.Constantinou and Gani [149] established a method to determine TC, PC, and VC using equations (21)–(23), respectively:(21)TCK=181.128⋅ln∑iNi⋅Tc1i+W⋅∑jMj⋅Tc2j,(22)PCbar=∑iNi⋅Pc1i+W×∑jMj⋅Pc2j+0.100220−2+1.3705,(23)VCcm3⋅mol−1=−0.004350+∑iNi⋅vc1i+W×∑jMj⋅vc2j.The coefficients to determine the critical properties can be found in TablesA2 and A3 in the Supplementary Material.Wilson and Jasperson suggested equation (24) to estimate the critical temperature and equation (25) to determine the critical pressure:(24)TCK=TB0.048271−0.019846Nr+∑kNk·tck+∑jMjΔtcj0.2,(25)PCbar=0.0186233⋅TC−0.96601+exp−0.0092295−0.0290403Nr+0.041∑kNk⋅Pcbk+∑jMjΔPcj.The coefficients to determine these critical properties can be found in TableA6 in the Supplementary Material.Marrero-Marejón and Pardillo-Fontdevila developed equations (26)–(28) to calculate the critical parameters [150]:(26)TCK=TB0.5851−0.9286∑kNk⋅tbck+∑jMj⋅Δtcbj2,(27)PCbar=0.1285−0.0059Natoms−∑kNk⋅Pcbk−2,(28)VCcm3⋅mol−1=−25.1+∑kNk⋅vcbk,where Natoms is the number of atoms in the compound and Nk is the number of atoms of type k with contributions tcbk, pcbk, and vcbk. The parameters to determine these critical parameters can be found at Table A4 in the Supplementary Material.Marrero and Gani [151] determined TC, PC, and VC using the following equations:(29)TCK=231.239⋅ln∑iNi⋅Tc1i+W⋅∑jMj⋅Tc2j+W⋅∑kOk⋅Tc3k,(30)PCbar=∑iNi⋅Pc1i+W×∑jMj⋅Pc2j+W×∑kOk⋅Pc3k+0.108998−2+5.9827,(31)VCcm3⋅mol−1=7.95+∑iNi⋅Vc1i+W×∑jMj⋅Vc2j+W×∑kOk⋅Vc3k.The coefficients to determine the critical properties can be found in TablesA5 and A6 in the Supplementary Material.The estimation of the critical properties proposed by Nannoolal et al. [156] is presented in the following equations:(32)TCK=TB0.6990+10.9889+∑iNiCi0.8607,(33)PCkPa=MW−0.14041g/mol0.00939+∑iNiCi2,(34)VCcm3⋅mol−1=∑iNiCin−0.2266+86.1539,where n is the number of atoms in the molecule (excluding hydrogen). Table A7 in the Supplementary Material presents the coefficients used.For the estimation methods that depend on the normal boiling point temperature, this value can be experimentally determined or estimated. The calculations were performed using the normal boiling point temperatures available from the NIST database, as reported in Table4.


## 4.3.2. Critical Properties Estimation Results

(1) Critical Temperature Estimation Results. The estimated and experimental values for the critical temperatures are presented in Table 12.Table 12
Comparison of experimental and estimated critical temperature (K).
CN : DB
NIST
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
C8 : 0
693.7
692.1
694.6
695.0
692.3
738.2
694.9
C10 : 0
723.8
714.6
718.4
720.4
715.7
762.3
719.5
C12 : 0
743.0
736.6
740.5
742.7
739.1
784.0
742.0
C14 : 0
763.0
758.4
760.8
762.5
762.8
803.9
762.5
C16 : 0
785.0
773.5
772.7
780.4
780.4
822.2
774.3
C18 : 0
803.0
794.5
788.7
796.6
804.6
839.2
790.1
C20 : 0
820.0
826.2
813.0
811.6
840.4
855.0
814.1
C22 : 0
837.0
848.4
825.5
825.4
867.4
869.8
826.3
C24 : 0
825.0
863.1
828.2
838.2
887.6
883.7
828.6
C16 : 1
789.0
792.5
791.3
780.0
800.7
825.1
796.3
C18 : 1
775.0
787.9
782.4
796.3
798.1
841.8
786.9
C20 : 1
811.0
790.1
778.4
811.3
802.8
857.5
782.4
C22 : 1
806.0
815.6
795.3
825.1
831.9
872.1
798.8
C24 : 1
838.0
823.7
792.9
837.9
844.0
885.9
795.8
C18 : 2
787.0
775.8
770.4
796.0
786.6
844.5
778.0
C18 : 3
777.0
776.8
771.2
795.7
789.3
847.1
782.1Figure6 presents the absolute relative deviation between the methods considered and the NIST database values.Figure 6
Absolute relative deviation of estimation for the critical temperature.Except for the Marrero and Gani’s method, all the other methods have a general agreement until the C16 : 0. For the compounds with higher molecular weight, the method developed by Nannoolal et al. and by Constantinou and Gani give the best results. When one or more unsaturations are considered, it is not possible to define a method that consistently predicts the critical temperature. Table13 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by NIST.Table 13
Deviations of different methods for the estimation of the critical temperature.
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
AAD (K)
10.64
11.23
7.95
13.80
47.23
10.14
RMSD (K)
13.72
16.13
10.57
20.85
48.77
14.62
AARD (%)
1.3
1.4
1.0
1.7
6.0
1.3
MIN RD (%)
−3
−5
−1
−1
4
−5
MAX RD (%)
2
1
3
4
9
2From the analysis of Table13, it is possible to notice that the Constantinou and Gani’s method exhibits the best global performances for the critical temperature estimation.(2) Critical Pressure Estimation Results. The results for the estimated critical pressures and the experimental values for the fatty acids are presented in Table 14.Table 14
Comparison of experimental and estimated critical pressure (bar).
CN : DB
NIST
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
C8 : 0
28.57
27.79
26.91
27.67
31.07
26.87
26.27
C10 : 0
20.76
22.92
22.22
22.79
25.40
22.52
22.11
C12 : 0
19.31
19.22
18.79
19.14
21.16
19.40
18.87
C14 : 0
16.40
16.35
16.17
16.36
17.89
17.09
16.29
C16 : 0
14.90
14.08
13.96
14.18
15.33
15.33
14.21
C18 : 0
13.30
12.25
12.25
12.44
13.28
13.95
12.50
C20 : 0
12.00
10.76
10.97
11.03
11.62
12.86
11.07
C22 : 0
11.10
9.52
9.75
9.88
10.25
11.98
9.88
C24 : 0
8.49
8.49
8.61
8.92
9.11
11.25
8.86
C16 : 1
14.88
14.65
14.58
14.15
15.86
15.52
14.37
C18 : 1
13.90∗
12.71
12.39
12.42
13.71
14.10
12.63
C20 : 1
10.98
11.13
10.69
11.02
11.96
12.98
11.18
C22 : 1
9.83
9.83
9.55
9.87
10.53
12.07
9.97
C24 : 1
8.87
8.75
8.61
8.91
9.35
11.33
8.94
C18 : 2
14.10
13.19
12.43
12.40
14.15
14.25
12.76
C18 : 3
12.34∗
13.71
12.68
12.38
14.62
14.41
12.89
∗Values retrieved from the database APV88 PURE32 of Aspen Plus.Figure7 presents the relative deviation between the estimated values and the ones gathered from the NIST database. All the methods provide estimations included in a range of deviation of ±10%, except for the Marrero and Gani method when applied to saturated fatty acids with high molecular weight and unsaturated fatty acids with one and three double bonds. The Nannoolal et al. method gives the best estimation for the fatty acids at high molecular weight with one double bond. Regarding the saturated compounds, the Marrero-Marejón and Pardillo-Fontdevila method overestimates the critical pressure until C16 : 0. After this component, there is an underestimation that increases linearly with the molecular weight. Apart from the Marrero and Gani method, all the others underestimate the critical pressure in the range of C16 : 0–C22 : 0.Figure 7
Absolute relative deviation of estimation for the critical pressure.Table15 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by NIST.Table 15
Deviations of different methods for the estimation of the critical pressure.
NIST
Joback and Reid [148]
Wilson and Jasperson [155]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
AAD (bar)
0.73
0.83
0.71
1.15
1.23
0.77
RMSD (bar)
0.98
0.99
0.95
1.64
1.50
0.97
AARD (%)
5.1
5.7
4.9
7.5
10.4
5.1
MIN RD (%)
−14
−12
−12
−8
−6
−11
MAX RD (%)
11
7
10
22
33
7
Globally, the Nannoolal et al. method offers a better estimation together with the Constantinou and Gani approach.(3) Critical Volume Estimation Results. The estimated and experimental values for the fatty acids’ critical volume are presented in Table 16.Table 16
Comparison of experimental and estimated critical volume (m3/kmol).
CN : DB
NIST
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
C8 : 0
0.526
0.5075
0.5071
0.5107
0.5046
0.5267
C10 : 0
0.639
0.6195
0.6187
0.6239
0.6172
0.6579
C12 : 0
0.787
0.7315
0.7302
0.7371
0.7298
0.7948
C14 : 0
0.921
0.8435
0.8417
0.8503
0.8423
0.9367
C16 : 0
1.066
0.9555
0.9532
0.9635
0.9549
1.0831
C18 : 0
1.255
1.0675
1.0647
1.0767
1.0674
1.2335
C20 : 0
1.371
1.1795
1.1763
1.1899
1.1800
1.3877
C22 : 0
1.486
1.2915
1.2878
1.3031
1.2926
1.5453
C24 : 0
1.575
1.4035
1.3993
1.4163
1.4051
1.7062
C16 : 1
0.898
0.9355
0.9399
0.9429
0.9409
1.0525
C18 : 1
1.016
1.0475
1.0514
1.0561
1.0535
1.2022
C20 : 1
1.158
1.1595
1.1630
1.1693
1.1660
1.3557
C22 : 1
1.448
1.2715
1.2745
1.2825
1.2786
1.5127
C24 : 1
1.402
1.3835
1.3860
1.3957
1.3911
1.6729
C18 : 2
0.972
1.0275
1.0381
1.0355
1.0395
1.1708
C18 : 3
0.960
1.0075
1.0248
1.0149
1.0255
1.1395Figure8 presents the relative deviation distribution between the estimated values and the ones provided by NIST.Figure 8
Absolute relative deviations of estimation for the critical volume.The Nannoolal et al. method offers the best prediction for the saturated compounds with a peak of about 10% overestimation for C24 : 0. However, the overestimation increases to 20% when the unsaturated compounds are evaluated. All the other methods exhibit an opposite behaviour, with the tendency to underestimate the critical volume for saturated compounds, with a lower deviation for the unsaturated fatty acids.Table17 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by the NIST database.Table 17
Deviations of different methods for the estimation of the critical volume.
Joback and Reid [148]
Constantinou and Gani [149]
Marrero-Marejón and Pardillo-Fontdevila [150]
Marrero and Gani [151]
Nannoolal et al. [156]
AAD (m3/kmol)
0.087
0.091
0.084
0.090
0.096
RMSD (m3/kmol)
0.112
0.114
0.106
0.112
0.130
AARD (%)
7.3
7.6
7.0
7.6
8.6
MIN RD (%)
−5
−15
−15
−14
−15
MAX RD (%)
31
6
7
7
7Except for the Nannoolal et al. method, that is only recommended for saturated compounds, all the other methods provide a general agreement in the critical volume estimation.


## 4.3.3. Novel Critical Parameters Estimation Method

(1) Novel Critical Temperature Estimation Method. Anand et al. [154] proposed a critical temperature estimation method for FAMEs, according to the following:(35)TCFAME=ATCFAMEMW2+BTCFAMEMW+CTCFAME.Using a similar equation, it is possible to obtain the coefficients to determine the critical temperature for the fatty acids applying the generalized reduced gradients method to minimize the deviations. Equation (36) is the novel estimation method proposed to estimate the fatty acids critical temperature:(36)TCFAK=αTCCN:n⋅ATCFAMW2+BTCFAMW+CTCFA.The coefficientsATCFACN:n, BTCFA CN:n, CTCFA CN:n, and αTCCN:n were obtained by minimizing the relative deviations with the NIST database values. The coefficients obtained are presented in Table 18.Table 18
Coefficients for the estimation of the critical temperature according to equation (36).
Coefficients
Values
A
T
C
FA CN
:
n
−1.8345E − 03
B
T
C
FA CN
:
n
1.6239
C
T
C
FA CN
:
n
550.5882
α
T
C
CN
:
0
0.9286
α
T
C
CN
:
1
0.9185
α
T
C
CN
:
2
0.9134
α
T
C
CN
:
3
0.9028For the proposed method, the critical temperature deviations resulted inAAD=5.88K, RMSD=8.01K, AARD=0.7%, MIN  RD=−1.6%, and MAX  RD=2.3% compared with NIST database. Thus, AAD, RMSD, and AARD significantly outperform the estimation methods considered.(2) Novel Critical Pressure Evaluation Method. Applying the same procedure, it is possible to optimize equation (37) in order to obtain a functional equation to estimate the critical pressure:(37)PCFAbar=αPCCN:n⋅APCFAMW2+BPCFAMW+CPCFA.The coefficientsAPCFA CN:n, BPCFA CN:n, CPCFA CN:n, and αPCCN:n were obtained by minimizing the relative deviations with the NIST database values, using the GRG nonlinear solving method to find the optimized result. The obtained coefficients are presented in Table 19.Table 19
Coefficients for the estimation of the critical pressure according to equation (37).
Coefficients
Values
A
P
C
FA CN
:
n
7.0357E − 04
B
P
C
FA CN
:
n
−0.5776
C
P
C
FA CN
:
n
143.4049
α
P
C
CN
:
0
0.3515
α
P
C
CN
:
1
0.3556
α
P
C
CN
:
2
0.3836
α
P
C
CN
:
3
0.3323For the proposed method, the critical pressure deviations resulted inAAD=0.69bar, RMSD=0.94bar, AARD=4.6%, MIN  RD=−10.4%, and MAX RD=9.7%. Again, AAD, RMSD, and AARD show an improvement in accuracy, when compared with the considered estimation methods.(3) Novel Critical Volume Estimation Method. Applying the same procedure as it was suggested before, it is possible to derive equation (38) to estimate the critical volume:(38)VCFAmL/mol=αVCCN:n⋅AVCFAMW2+BVCFAMW+CVCFA.The coefficientsAVCFA CN:n, BVCFA CN:n, CVFA CN:n, and αVCCN:n were obtained by minimizing the relative deviations with the NIST database values, using the GRG nonlinear solving method to achieve the optimized result. The obtained coefficients are presented in Table 20.Table 20
Coefficients for the estimation of the critical volume according to equation (38).
Coefficients
Values
A
V
C
FA CN
:
n
−1.0023E−06
B
V
C
FA CN
:
n
0.0206
C
V
C
FA CN
:
n
−0.9068
α
V
C
CN
:
0
0.2485
α
V
C
CN
:
1
0.2106
α
V
C
CN
:
2
0.2033
α
V
C
CN
:
3
0.2025The proposed method for the critical volume yield deviations ofAAD=0.03m3/kmol, RMSD=0.05m3/kmol, AARD=2.1%, MIN RD=−13.6%, and MAX  RD=3.1%. The first three indicators significantly outperform the ones computed for the considered estimation methods.


## 4.4. Acentric Factor

The acentric factor (ω) is a pure component constant that indicates the acentricity or nonsphericity of a molecule. Monoatomic molecules are expected to have acentric factors equal to zero, and their values increase with the molecular weight and polarity [155].This property is used in the estimation of transport and thermodynamic properties for gases and liquids, such as compressibility factor, heat capacity, enthalpy of vaporization, and saturated density [155, 157].


### 4.4.1. Estimation Method

Several methods are available in the literature to determine the acentric factor. The most appropriate methods are summarized in Table21.Table 21
Methods for the estimation of the acentric factor.
Method
Parameters required
Pitzer et al. [158]
Vapor pressure andPC
Lee and Kesler [159]
T
B
,
T
C
,
P
C
Watanasiri et al. [160]
Specific gravity,MW,TB
Ambrose and Walton [161]
T
B
,
T
C
,
P
C
Chen et al. [157]
T
B
,
T
C
,
P
C
Constantinou et al. [162]
Molecular structureThe acentric factor was initially proposed by Pitzer et al., and it is given by the following equation [158]:(39)ω=−log10Pr−1.0,where Pr is the reduced vapor pressure, defined as Pr=Pvap/PC, and Pvap is the vapor pressure at temperature T, defined at the reduced temperature T/TC=0.7. The vapor pressure can be determined by the NIST Wagner Equation, given as follows:(40)lnPvap=C5+C1×1−T/C6+C2×1−T/C61.5+C3×1−T/C62.5+C4×1−T/C65T/C6,where the pressure is given in Pa. The coefficients C1, C2, C3, C4, C5, and C6 can be found in Table A7 of the Supplementary Material.The parameter proposed by Pitzer et al. is, however, limited to reduced temperatures above 0.8.Lee and Kesler proposed an analytical correlation for the acentric factor employing equations of state, based on a wide variety of compounds. Their goal was to improve the representation of the property at low temperatures and near the critical region [159]. The proposed equation (41) is given as follows:(41)ω=−lnPC−5.92714+6.09648/TBr+1.28862lnTBr+0.169347TBr615.2518−15.6875/TBr−13.4721lnTBr+0.43577TBr6,where TBr=TB/TC is the reduced temperature at the boiling point.Watanasiri et al. developed a correlation to estimate the acentric factor based on heterocyclic compounds and other hydrocarbons and their derivatives [160]. According to their model, the acentric factor is given by the following equation:(42)ω=0.92217×10−3TB+0.507288TBMW+382.904MW+0.242×10−5TBSG2−0.2165×10−4TB×MW+0.1261×10−2SG×MW+0.1265×10−4MW2+0.2016×10−4SG×MW2−80.6495TB1/3MW−0.3738×10−2TB2/3SG2TBMW,where SG is the specific gravity of the component. Values of specific gravity were obtained from the NIST database.According to Poling et al. [155], the most accurate technique to obtain an unknown acentric factor is from the critical temperature and pressure combined with the normal boiling point. They have found that the most reliable method is given as follows:(43)ω=−lnPC/1.01325+f0f1,where the critical pressure is in bar and the functions f0 and f1 are defined as functions of TBr. These functions are obtained according to the equations proposed by Ambrose and Walton [161]:(44)f0=−5.976161−TBr+1.298741−TBr1.5−0.603941−TBr2.5−1.068411−TBr5TBr,(45)f1=−5.033651−TBr+1.115051−TBr1.5−5.412171−TBr2.5−7.466281−TBr5TBr.The method proposed by Chen et al. requires the same properties for the estimation of the acentric factor. However, this work suggested a correlation based on the Antoine Equation [157]. Based on 217 compounds over the broadest possible range of TBr, the acentric factor can be estimated as follows:(46)ω=0.30.2803+0.4789TBrlogPC1−TBr0.9803−0.5211TBr−1.Furthermore, Constantinou et al. proposed a method for the estimation of the acentric factor using group contribution [162]. The equation for the property estimation is reported as follows:(47)expω0.40850.5050=∑iNiω1i+∑iMjω2j+1.1507,where ω1i is the contribution of the first-order group type i occurring Ni times and ω2j is the contribution of the second-order group type j occurring Mj times. The coefficients to determine the acentric factor can be found in Tables A2 and A3 of the Supplementary Material.


### 4.4.2. Acentric Factor Estimation Results

Estimated values of the acentric factor are given in Table22.Table 22
Comparison of experimental and estimated acentric factor.
CN : DB
NIST
Pitzer et al. [158]
Lee and Kesler [159]
Watanasiri [160]
Ambrose and Walton [161]
Chen et al. [157]
Constantinou et al. [162]
C8 : 0
0.782
0.782
0.790
0.493
0.776
0.776
0.780
C10 : 0
0.749
0.749
0.725
0.495
0.709
0.712
0.861
C12 : 0
0.879
0.879
0.889
0.532
0.869
0.872
0.940
C14 : 0
0.940
0.940
0.965
0.605
0.940
0.946
1.017
C16 : 0
0.970
0.970
0.942
0.714
0.917
0.925
1.093
C18 : 0
1.029
1.029
1.001
0.863
0.973
0.984
1.167
C20 : 0
1.184
1.184
1.206
1.031
1.171
1.187
1.239
C22 : 0
1.233
1.233
1.252
1.243
1.215
1.234
1.310
C24 : 0
0.810
0.810
1.449
1.508
1.403
1.437
1.379
C16 : 1
1.088
1.088
1.105
0.751
1.075
1.084
1.079
C18 : 1
1.247
1.345
1.342
0.856
1.304
1.315
1.153
C20 : 1
0.725
—
0.742
1.184
0.719
0.736
1.226
C22 : 1
1.080
1.080
1.102
1.294
1.068
1.091
1.297
C24 : 1
0.676
—
0.694
1.766
0.671
0.694
1.366
C18 : 2
0.996
0.956
0.945
1.043
0.920
0.929
1.140
C18 : 3
1.022
0.976
0.950
1.054
0.922
0.936
1.127Figure9 presents the relative deviation between the different methods when compared to the NIST values, while Table 23 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by NIST.Figure 9
Absolute relative deviation of estimation of the acentric factor.Table 23
Deviations of different methods for the prediction of the acentric factor.
Pitzer et al. [158]
Lee and Kesler [159]
Watanasiri [160]
Ambrose and Walton [161]
Chen et al. [157]
Constantinou et al. [162]
AAD (K)
0.01
0.07
0.32
0.07
0.07
0.19
RMSD (K)
0.03
0.16
0.41
0.15
0.16
0.27
AARD (%)
1.2
7.8
37.7
7.5
7.5
22.6
MIN RD (%)
−4
−7
−40
−10
−8
−7
MAX RD (%)
8
79
161
73
77
102The acentric factor provided by NIST is obtained from the Pitzer correlation. The relative deviation concerning the components C20 : 1 and C24 : 1 is higher for the methods proposed by Watanasiri et al. and Constantinou et al. This occurs since the estimation of the vapor pressure was not available in the NIST database. Therefore, the NIST values of the acentric factor for these two components were probably estimated using another method. In addition, the acentric factor of the component C24 : 0 is inconsistent when compared to the other saturated fatty acids, which makes this value doubtful. Once the Pitzer equation was used for the estimation of the values from NIST, the absolute deviation of Pitzer regarding the NIST acentric value is negligible.The method proposed by Watanasiri et al. reported a higher deviation with respect to NIST. This is due to the fact that, in their study, the correlations were built using a different set of components that did not include fatty acids. Followed by this method, Constantinou et al. reported the second-highest deviation since their estimation was made by using the group contributions instead of the physical properties of the components. Conversely, the methods by Lee and Kesler, Ambrose and Walton, and Chen et al. have reported low deviation compared to NIST.The acentric factor has a definition based on vapor pressure. In this way, the property is indirectly measured, which can result in uncertainties in the estimation. The estimation of the acentric factor for the unsaturated fatty acids as well as the component C24 : 0 reported some inconsistencies. Disregarding those fatty acids, the average absolute relative deviation of the methods reduced to 0% for Pitzer et al., 2.1% to Lee and Kesler method, 25.3% for Watanasiri, 2.6% for Ambrose and Walton, 2.1% for Chen et al., and 8.4% for the method proposed by Constantinou et al. Therefore, these estimations are more reliable for the saturated fatty acids from C8 : 0 to C22 : 0.


### 4.4.3. Novel Acentric Factor Estimation Method

The novel method proposed to estimate the acentric factor is given as follows:(48)ω=αωCN:n×AωFAMW2+BωFAMW+CωFA.The coefficientsAPCFA CN:n, BPCFA CN:n, CPCFA  CN:n, and αPMCN:n were obtained by minimizing the relative deviations between the values estimated by the novel method and the NIST database values, using the GRG nonlinear solving method to achieve the optimized result.As the variability of the NIST database is very high, mainly for the unsaturated fatty acids, the novel method will be only valid to estimate the acentric factors for fatty acids within the range between C8 : 0 and C22 : 0. The coefficients to determine the acentric factor are provided in Table24.Table 24
Coefficients for the estimation of the acentric factor (valid from C8 : 0 to C22 : 0) according to equation (48).
Coefficients
Values
A
ω
FA CN
:
n
5.4382E−05
B
ω
FA CN
:
n
−0.0032
C
ω
FA CN
:
n
6.4600
α
ω
CN
:
0
0.1066For the novel method for the acentric factor,AAD=0.03K, RMSD=0.03K, AARD=3.0%, MIN RD=−3.0%, and MAX RD=7.1%.


## 4.4.1. Estimation Method

Several methods are available in the literature to determine the acentric factor. The most appropriate methods are summarized in Table21.Table 21
Methods for the estimation of the acentric factor.
Method
Parameters required
Pitzer et al. [158]
Vapor pressure andPC
Lee and Kesler [159]
T
B
,
T
C
,
P
C
Watanasiri et al. [160]
Specific gravity,MW,TB
Ambrose and Walton [161]
T
B
,
T
C
,
P
C
Chen et al. [157]
T
B
,
T
C
,
P
C
Constantinou et al. [162]
Molecular structureThe acentric factor was initially proposed by Pitzer et al., and it is given by the following equation [158]:(39)ω=−log10Pr−1.0,where Pr is the reduced vapor pressure, defined as Pr=Pvap/PC, and Pvap is the vapor pressure at temperature T, defined at the reduced temperature T/TC=0.7. The vapor pressure can be determined by the NIST Wagner Equation, given as follows:(40)lnPvap=C5+C1×1−T/C6+C2×1−T/C61.5+C3×1−T/C62.5+C4×1−T/C65T/C6,where the pressure is given in Pa. The coefficients C1, C2, C3, C4, C5, and C6 can be found in Table A7 of the Supplementary Material.The parameter proposed by Pitzer et al. is, however, limited to reduced temperatures above 0.8.Lee and Kesler proposed an analytical correlation for the acentric factor employing equations of state, based on a wide variety of compounds. Their goal was to improve the representation of the property at low temperatures and near the critical region [159]. The proposed equation (41) is given as follows:(41)ω=−lnPC−5.92714+6.09648/TBr+1.28862lnTBr+0.169347TBr615.2518−15.6875/TBr−13.4721lnTBr+0.43577TBr6,where TBr=TB/TC is the reduced temperature at the boiling point.Watanasiri et al. developed a correlation to estimate the acentric factor based on heterocyclic compounds and other hydrocarbons and their derivatives [160]. According to their model, the acentric factor is given by the following equation:(42)ω=0.92217×10−3TB+0.507288TBMW+382.904MW+0.242×10−5TBSG2−0.2165×10−4TB×MW+0.1261×10−2SG×MW+0.1265×10−4MW2+0.2016×10−4SG×MW2−80.6495TB1/3MW−0.3738×10−2TB2/3SG2TBMW,where SG is the specific gravity of the component. Values of specific gravity were obtained from the NIST database.According to Poling et al. [155], the most accurate technique to obtain an unknown acentric factor is from the critical temperature and pressure combined with the normal boiling point. They have found that the most reliable method is given as follows:(43)ω=−lnPC/1.01325+f0f1,where the critical pressure is in bar and the functions f0 and f1 are defined as functions of TBr. These functions are obtained according to the equations proposed by Ambrose and Walton [161]:(44)f0=−5.976161−TBr+1.298741−TBr1.5−0.603941−TBr2.5−1.068411−TBr5TBr,(45)f1=−5.033651−TBr+1.115051−TBr1.5−5.412171−TBr2.5−7.466281−TBr5TBr.The method proposed by Chen et al. requires the same properties for the estimation of the acentric factor. However, this work suggested a correlation based on the Antoine Equation [157]. Based on 217 compounds over the broadest possible range of TBr, the acentric factor can be estimated as follows:(46)ω=0.30.2803+0.4789TBrlogPC1−TBr0.9803−0.5211TBr−1.Furthermore, Constantinou et al. proposed a method for the estimation of the acentric factor using group contribution [162]. The equation for the property estimation is reported as follows:(47)expω0.40850.5050=∑iNiω1i+∑iMjω2j+1.1507,where ω1i is the contribution of the first-order group type i occurring Ni times and ω2j is the contribution of the second-order group type j occurring Mj times. The coefficients to determine the acentric factor can be found in Tables A2 and A3 of the Supplementary Material.


## 4.4.2. Acentric Factor Estimation Results

Estimated values of the acentric factor are given in Table22.Table 22
Comparison of experimental and estimated acentric factor.
CN : DB
NIST
Pitzer et al. [158]
Lee and Kesler [159]
Watanasiri [160]
Ambrose and Walton [161]
Chen et al. [157]
Constantinou et al. [162]
C8 : 0
0.782
0.782
0.790
0.493
0.776
0.776
0.780
C10 : 0
0.749
0.749
0.725
0.495
0.709
0.712
0.861
C12 : 0
0.879
0.879
0.889
0.532
0.869
0.872
0.940
C14 : 0
0.940
0.940
0.965
0.605
0.940
0.946
1.017
C16 : 0
0.970
0.970
0.942
0.714
0.917
0.925
1.093
C18 : 0
1.029
1.029
1.001
0.863
0.973
0.984
1.167
C20 : 0
1.184
1.184
1.206
1.031
1.171
1.187
1.239
C22 : 0
1.233
1.233
1.252
1.243
1.215
1.234
1.310
C24 : 0
0.810
0.810
1.449
1.508
1.403
1.437
1.379
C16 : 1
1.088
1.088
1.105
0.751
1.075
1.084
1.079
C18 : 1
1.247
1.345
1.342
0.856
1.304
1.315
1.153
C20 : 1
0.725
—
0.742
1.184
0.719
0.736
1.226
C22 : 1
1.080
1.080
1.102
1.294
1.068
1.091
1.297
C24 : 1
0.676
—
0.694
1.766
0.671
0.694
1.366
C18 : 2
0.996
0.956
0.945
1.043
0.920
0.929
1.140
C18 : 3
1.022
0.976
0.950
1.054
0.922
0.936
1.127Figure9 presents the relative deviation between the different methods when compared to the NIST values, while Table 23 shows the minimum and maximum absolute relative deviation for each method when compared with the data provided by NIST.Figure 9
Absolute relative deviation of estimation of the acentric factor.Table 23
Deviations of different methods for the prediction of the acentric factor.
Pitzer et al. [158]
Lee and Kesler [159]
Watanasiri [160]
Ambrose and Walton [161]
Chen et al. [157]
Constantinou et al. [162]
AAD (K)
0.01
0.07
0.32
0.07
0.07
0.19
RMSD (K)
0.03
0.16
0.41
0.15
0.16
0.27
AARD (%)
1.2
7.8
37.7
7.5
7.5
22.6
MIN RD (%)
−4
−7
−40
−10
−8
−7
MAX RD (%)
8
79
161
73
77
102The acentric factor provided by NIST is obtained from the Pitzer correlation. The relative deviation concerning the components C20 : 1 and C24 : 1 is higher for the methods proposed by Watanasiri et al. and Constantinou et al. This occurs since the estimation of the vapor pressure was not available in the NIST database. Therefore, the NIST values of the acentric factor for these two components were probably estimated using another method. In addition, the acentric factor of the component C24 : 0 is inconsistent when compared to the other saturated fatty acids, which makes this value doubtful. Once the Pitzer equation was used for the estimation of the values from NIST, the absolute deviation of Pitzer regarding the NIST acentric value is negligible.The method proposed by Watanasiri et al. reported a higher deviation with respect to NIST. This is due to the fact that, in their study, the correlations were built using a different set of components that did not include fatty acids. Followed by this method, Constantinou et al. reported the second-highest deviation since their estimation was made by using the group contributions instead of the physical properties of the components. Conversely, the methods by Lee and Kesler, Ambrose and Walton, and Chen et al. have reported low deviation compared to NIST.The acentric factor has a definition based on vapor pressure. In this way, the property is indirectly measured, which can result in uncertainties in the estimation. The estimation of the acentric factor for the unsaturated fatty acids as well as the component C24 : 0 reported some inconsistencies. Disregarding those fatty acids, the average absolute relative deviation of the methods reduced to 0% for Pitzer et al., 2.1% to Lee and Kesler method, 25.3% for Watanasiri, 2.6% for Ambrose and Walton, 2.1% for Chen et al., and 8.4% for the method proposed by Constantinou et al. Therefore, these estimations are more reliable for the saturated fatty acids from C8 : 0 to C22 : 0.


## 4.4.3. Novel Acentric Factor Estimation Method

The novel method proposed to estimate the acentric factor is given as follows:(48)ω=αωCN:n×AωFAMW2+BωFAMW+CωFA.The coefficientsAPCFA CN:n, BPCFA CN:n, CPCFA  CN:n, and αPMCN:n were obtained by minimizing the relative deviations between the values estimated by the novel method and the NIST database values, using the GRG nonlinear solving method to achieve the optimized result.As the variability of the NIST database is very high, mainly for the unsaturated fatty acids, the novel method will be only valid to estimate the acentric factors for fatty acids within the range between C8 : 0 and C22 : 0. The coefficients to determine the acentric factor are provided in Table24.Table 24
Coefficients for the estimation of the acentric factor (valid from C8 : 0 to C22 : 0) according to equation (48).
Coefficients
Values
A
ω
FA CN
:
n
5.4382E−05
B
ω
FA CN
:
n
−0.0032
C
ω
FA CN
:
n
6.4600
α
ω
CN
:
0
0.1066For the novel method for the acentric factor,AAD=0.03K, RMSD=0.03K, AARD=3.0%, MIN RD=−3.0%, and MAX RD=7.1%.


## 5. Conclusions

Over one hundred biomass sources were reviewed, and 16 saturated, unsaturated, and polyunsaturated fatty acids were selected, based on their frequency and amount. The selected saturated fatty acids ranged from C8 to C24, in agreement with Kenar et al. [163], which states that “saturated fatty acids having alkyl chain lengths greater than 18 carbon atoms are often negligible in most seed oils and are only found at useful levels in a few unusual seed oils.” Different unsaturated fatty acids with 1, 2, or 3 double bonds were considered.Among the saturated fatty acids, the highest average yields were found for palmitic (C16 : 0) and stearic (C18 : 0) acids, with a value of around 15 and 5%, respectively. The highest yields for the unsaturated fatty acids were found for oleic (C18 : 1), linoleic (C18 : 2), and linolenic (C18 : 3) acids, at around 32, 27, and 7%, respectively. The frequency of palmitic (C16 : 0), stearic (C18 : 0), oleic (C18 : 1), and linoleic (C18 : 2) acids is around 100% since almost all the selected biomasses have these four fatty acids.Available estimation methods for the temperature nondependent properties, such as normal boiling point, normal melting point, critical properties, and acentric factor, were compared, and novel estimation methods were developed to enhance their predictions. The novel proposed method is simple to use, and it is based on the molecular weight and the maximum number of double bonds in the molecule. This method is generalized in equation (49), where PP is the temperature nondependent physical property:(49)PP=αCN:n⋅AMW2+BMW+C.The three coefficients, assigned asA, B, and C, and the correction factor αTBCN:n, which depends on the number of unsaturation, are summarized in Table 25.Table 25
Coefficients for the estimation the temperature nondependent properties according to equation (49).
Coefficients
T
B
T
M
T
C
P
C
V
C
ω
A
−7.0670E − 04
−4.1905E − 04
−1.8345E − 03
7.0357E − 04
−1.0023E − 06
5.4382E − 05
B
0.9173
0.4863
1.6239
−0.5776
0.0206
−0.0032
C
205.6120
193.9418
550.5882
143.4049
−0.9068
6.4600
α
CN
:
0
1.5697
1.1380
0.9286
0.3515
0.2485
0.1066
α
CN
:
1
1.5501
0.9752
0.9185
0.3556
0.2106
—
α
CN
:
2
1.5323
0.8999
0.9134
0.3836
0.2033
—
α
CN
:
3
1.5329
0.8759
0.9028
0.3323
0.2025
—For all the properties, new equations provide a better estimation with an absolute average deviation equal to or lower than 4.6%, compared to the values of the NIST database used as reference. This approach allows a better fatty acid property prediction compared to other previously published methods.It should be highlighted that the classic group contribution methods have been developed to be applied to a wide range of chemical compounds and different research groups implemented various corrections to take into account the possible interaction between different molecule fragments. However, the validity of the equations developed in this work is limited for the fatty acids selected.The equations proposed represent an improvement in the modelling of processes related to the recovery and purification of fatty acids from biomasses. Indeed, these equations have the potential to become a stepping stone for investigations of fatty acid mixtures or triglycerides [4], taking into account that the first step in process modelling relies on accurately predicting their physical properties.



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*Source: 2430234-2019-11-23.xml*