Predicting Temperature-Dependent Viscosity of Amazonian Vegetable Oils and Their Mixtures from Fatty Acid Composition

Figures & data

TABLE 1 Compositional design of the mixtures

Samples	<Proportions (%)
	^aX1	X2	X3
Buriti	100	0	0
Pataua	0	100	0
Brazil nut	0	0	100
Mixture 1	50	0	50
Mixture 2	0	50	50
Mixture 3	50	50	0
Mixture 4	33.33	33.33	33.33
Mixture 5	16.67	16.67	66.66
Mixture 6	66.66	16.67	16.67
Mixture 7	16.67	66.66	16.67

TABLE 2 Mathematical models used to correlate viscosity data

Model name	Equation	References
Andrade	$ln\eta =A+\frac{B}{T}$(1)	(23)
Modified Andrade	$ln\eta =A+BlnT$ (2)	(35)
Arrhenius	$ln\eta =lnA+\frac{E_a}{RT}$ (3)	(17)
Power law	$ln\eta =A(T-T_{ref}{)}^B$ (4)	(17)
William-Landel-Ferry (WLF)	$ln\eta =\frac{AT}{B+T}$ (5)	(17)
Kendall and Monroe	${\eta }_m={\left[\sum _{i=1}^n\left(x_i{\eta }_i^{0.5}\right)\right]}^2$ (6)	(36)
Grunberg-Nissan	$ln{\eta }_m=\sum _{i=1}^n\left(x_{i}ln{\eta }_i^{ }\right)$ (7)	(37)

FIGURE 1 Relationship between shear stress and shear rate for the buriti oil at various temperatures.

FIGURE 2 Relationship between shear stress and shear rate for the pataua oil at various temperatures.

FIGURE 3 Relationship between shear stress and shear rate for the Brazil nut oil at various temperatures.

TABLE 3 Viscosities^a (mPa.s) of vegetable oils at different temperatures

	Sample temperature (°C)
Source of oil	15	20	30	40	50	60	70	80
Buriti	110.03	86.98	53.89	36.85	26.01	18.07	13.86	11.10
Pataua	105.96	81.93	52.86	35.96	26.01	19.04	13.87	11.10
Brazil nut	92.08	71.92	47.21	32.05	22.84	17.02	13.07	10.01
Mixture 1	106.90	83.34	53.54	36.14	25.43	18.81	14.01	10.84
Mixture 2	98.60	77.20	50.34	33.79	24.03	17.67	13.55	10.27
Mixture 3	100.14	78.41	50.86	34.25	24.33	17.82	13.43	10.50
Mixture 4	102.30	78.03	51.34	34.60	24.78	17.81	13.62	10.57
Mixture 5	103.29	81.26	52.00	35.18	24.79	18.01	13.82	10.38
Mixture 6	105.80	83.07	52.48	35.41	24.88	18.34	13.62	10.56
Mixture 7	96.63	75.20	49.06	33.39	23.88	17.74	13.24	10.24

TABLE 4 Values of constants A (^a(mPa.s) ×10⁶) and B obtained from the five models applied to the various samples of vegetable oils

Models	Parameters	Buriti	Pataua	Brazil nut	Mixture 1	Mixture 2	Mixture 3	Mixture 4	Mixture 5	Mixture 6	Mixture 7
Andrade	A	–7.97	–7.65	–7.53	–7.80	–7.70	–7.72	–7.72	-7.85	-7.92	-7.60
	B	3636.9	3532.4	3457.4	3579.5	3527.5	3538.3	3540.7	3585.2	3611.7	3493.3
	x^2	0.003	0.002	0.002	0.001	0.001	0.002	0.002	0.001	0.002	0.001
	RMSE	0.045	0.036	0.035	0.032	0.033	0.035	0.036	0.032	0.036	0.032
	R^2	0.9963	0.9975	0.9975	0.9979	0.9978	0.9976	0.9975	0.9980	0.9975	0.9979
Modified Andrade	A	69.31	67.42	65.95	68.28	67.27	67.48	67.53	68.35	68.83	66.64
	B	–11.42	–11.10	–10.86	–11.25	–11.08	–11.11	–11.12	-11.26	-11.35	-10.97
	x^2	0.006	0.004	0.004	0.004	0.004	0.004	0.004	0.004	0.004	0.004
	RMSE	0.066	0.056	0.055	0.054	0.054	0.056	0.057	0.054	0.058	0.053
	R^2	0.9919	0.9938	0.9936	0.9944	0.9941	0.9938	0.9936	0.9945	0.9937	0.9943
Arrhenius	A^a	345.7	475.3	537.9	409.3	453.7	443.4	444.2	390.3	362.9	498.1
	Ea	30.2	29.4	28.7	29.8	29.3	29.4	29.4	29.8	30.0	29.0
	x^2	0.0026	0.0017	0.0016	0.0014	0.0015	0.0016	0.0017	0.0014	0.0017	0.0014
	RMSE	0.045	0.036	0.035	0.032	0.033	0.035	0.036	0.032	0.036	0.032
	R^2	0.9963	0.9975	0.9975	0.9979	0.9978	0.9976	0.9975	0.9980	0.9975	0.9979
Power Law	A	13.41	12.93	12.69	13.12	12.94	12.99	13.01	13.17	13.30	12.81
	B	–0.372	–0.362	–0.366	–0.366	–0.368	–0.368	–0.367	-0.370	-0.372	-0.366
	x^2	0.035	0.033	0.032	0.035	0.034	0.034	0.034	0.036	0.035	0.033
	RMSE	0.162	0.158	0.155	0.163	0.161	0.159	0.159	0.165	0.163	0.158
	R^2	0.9510	0.9503	0.9505	0.9487	0.9488	0.9495	0.9504	0.9476	0.9497	0.9492
WLF	A	0.8285	0.8426	0.8083	0.8347	0.8182	0.8206	0.8237	0.8219	0.8209	0.8181
	B	–238.54	–237.12	–237.73	–237.77	–237.94	–237.96	–237.88	-238.26	-238.56	-237.65
	x^2	0.013	0.012	0.012	0.014	0.013	0.013	0.013	0.014	0.014	0.013
	RMSE	0.350	0.350	0.350	0.350	0.350	0.350	0.350	0.350	0.350	0.350
	R^2	0.9818	0.9815	0.9816	0.9804	0.9804	0.9810	0.9813	0.9795	0.9808	0.9806

TABLE 5 Viscosity values (mPa.s) of the mixtures predicted by Equations 6 and 7 and statistical parameters

	Temperature (°C)								Stat. parameter
Source	15	20	30	40	50	60	70	80	χ^2	RMSE
^a
Mixture 1	107.99	84.44	53.37	36.40	26.01	18.55	13.86	11.10	0.1454	0.3302
Mixture 2	98.90	76.84	50.00	33.98	24.40	18.02	13.47	10.55	0.0993	0.2729
Mixture 3	100.86	79.27	50.49	34.41	24.40	17.54	13.46	10.55	0.1004	0.2744
Mixture 4	102.54	80.15	51.28	34.92	24.93	18.03	13.60	10.73	0.4527	0.5827
Mixture 5	104.24	81.04	52.07	35.44	25.47	18.53	13.73	10.91	0.1868	0.3743
Mixture 6	106.25	83.53	52.58	35.88	25.47	18.05	13.73	10.91	0.0838	0.2508
Mixture 7	96.63	75.2	49.06	33.39	23.88	17.74	13.24	10.24	0.0338	0.1591
^b
Mixture 1	107.98	84.42	53.37	36.40	26.01	18.55	13.86	11.10	0.1426	0.3271
Mixture 2	98.78	76.76	49.96	33.95	24.37	18.00	13.46	10.54	0.0950	0.2670
Mixture 3	100.66	79.09	50.44	34.37	24.37	17.54	13.46	10.54	0.0893	0.2589
Mixture 4	102.39	80.03	51.23	34.89	24.91	18.02	13.59	10.72	0.4440	0.5771
Mixture 5	104.16	80.97	52.04	35.42	25.45	18.53	13.73	10.91	0.1869	0.3744
Mixture 6	106.14	83.43	52.55	35.86	25.45	18.05	13.73	10.91	0.0821	0.2482
Mixture 7	97.10	75.87	49.18	33.44	23.85	17.52	13.33	10.36	0.0311	0.1527

TABLE 6 FA composition (percentage by weight of total FA) of the vegetable oil samples

Source of oil	Saturated	Unsaturated	Monounsaturated	Polyunsaturated
Buriti	22.00	78.00	75.75	2.25
Pataua	18.50	81.45	77.42	4.03
Brazil Nut	25.26	74.74	36.17	38.57
Mixture 1	19.13	80.87	78.26	2.61
Mixture 2	21.11	78.89	57.68	21.21
Mixture 3	22.67	77.33	21.48	21.34
Mixture 4	20.94	79.06	63.69	15.37
Mixture 5	19.63	80.37	71.34	9.03
Mixture 6	21.16	78.84	68.51	10.33
Mixture 7	22.81	77.19	50.09	27.1

TABLE 7 Values of constants (Eq. 9) that can be used to predict vegetable oil viscosity based on either monounsaturated or polyunsaturated FA

Parameter	Monounsaturated	Polyunsaturated
P1	7.45	25.06
P2	5.01x10^–5	9.60x10^–5
P3	34725.30	33619.26
RMSE	1.0737	0.9701
R^2	0.9987	0.9986