Estimation of thermal conductivity of liquid alcohols using finite element solution based on principle of minimum potential energy

doi:10.11949/j.issn.0438-1157.20181271

Abstract

Abstract:

Application of the finite element method of minimum potential energy principle, the displacement vector caused by each carbon - oxygen atom electronegativity difference in each molecule structure is obtained, and with the heavy atoms matrix weighted by Mulliken charge for the corresponding operation, molecular charge parameters was obtained. And then combining with molecular structure natural frequency, the temperatures, and 264 liquid thermal conductivity data of 23 alcohols such as alkane alcohols and naphthenic alcohols, aromatic alcohols, phytol, a 3-parameter nonlinear alcohols organic liquid thermal conductivity estimation model was developed . The model shows that the correlation coefficient between the experimental value and calculated value of the model r > 0.98, Standard error s < 3.98 mW/(m·K), F test value F > 2111. The model was used to calculate thermal conductivity of external prediction set which included 20 thermal conductivity data of tetranheptanol, tetradecanol and 2-octanol at different temperatures. For prediction set, the mean absolute error between the calculated results and the experimental values was 2.66 mW/(m·K) and the average relative error was 1.74%. The results show that the new method is significantly better than the Sastri and Latini estimation methods.

Key words: alcohol, thermal conductivity, thermodynamic properties, finite element method, principle of minimum potential energy

摘要：

应用势能极小原理的有限元解法，求解每个醇类化合物分子结构碳-氧原子电负性差所引起的位移矢量，并与每个分子结构中各重原子Mulliken电荷矩阵作相应的运算，可得到分子电荷参量，结合分子结构固有频率(基频、总频)、温度参量，利用链烷醇、环烷醇、芳醇、叶绿醇等23个醇不同温度下的264个液体热导率实验数据，建立了电荷参量、分子结构固有频率和温度参数的3参量非线性一元醇类液体热导率估算模型。该模型对训练集的计算值和实验值的相关系数r > 0.98，标准误差s < 3.98 mW/(m·K)，F检验值F > 2111；对外部预测集特庚醇、十四醇和2-辛醇在不同温度下的20个液体热导率进行估算，预测结果与实验数值的平均绝对误差为2.66 mW/(m·K)，平均相对误差为1.74%。结果表明新方法明显优于Sastri 和Latini估算方法。

关键词: 醇, 热导率, 热力学性质, 有限元方法, 势能极小原理

CLC Number:

Wanqiang LIU, Haixia LU, Fengping LIU, Guanfan CHEN, Tian HU, Ming YUE, Minghua QIU. Estimation of thermal conductivity of liquid alcohols using finite element solution based on principle of minimum potential energy[J]. CIESC Journal, 2019, 70(4): 1245-1254.

刘万强, 陆海霞, 刘凤萍, 陈冠凡, 胡田, 岳明, 仇明华. 应用势能极小原理有限元解法的一元醇液体热导率估算[J]. 化工学报, 2019, 70(4): 1245-1254.

Figures/Tables 12

Fig.1 Hidden hydrogen graph and Mulliken atomic charges of isopropyl alcohol optimized

Table 1 Standard orientation coordinates of heavy atoms in isopropanol molecule

Center number	Atomic type	Coordinate/?
Center number	Atomic type	x	y	z
1	C	?1.3175	?0.5703	?0.0906
2	C	?0.0015	0.0269	0.3712
3	C	1.2114	?0.7640	?0.1046
4	O	0.0282	1.3656	?0.1646

Table 2 Mulliken charges with hydrogens summed into heavy atoms in isopropanol molecule

Center number	Atomic type	Mulliken charge
1	C	0.05463
2	C	0.292
3	C	0.01081
4	O	?0.35745

Table 3 Elasticity coefficients of chemical bond space frame elements in alcohol molecules

Types	Elasticity coefficients
Types	E	G	A	I_x	I_y	J
C—C	0.8322	0.3329	3.1416	0.7854	0.7854	1.5708
C—O	0.8613	0.2871	2.7087	0.5839	0.5839	1.1678

Table 4 Calculated values of charge parameter of monohydric alcohols

No.	Compound	Q _e	No.	Compound	Q _e
1	methanol	0.0167	13	n-octadecanol	0.0474
2	ethanol	0.0311	14	2-hexanol	0.0651
3	n-proanol	0.0488	15	isoproanol	0.0556
4	n-butanol	0.0322	16	isobutanol	0.0443
5	n-pentanol	0.0480	17	2-methyl-2-propanol	0.0554
6	n-hexanol	0.0320	18	3-methyl-1-butanol	0.0484
7	n-heptanol	0.0477	19	2-methyl-2-butanol	0.0684
8	n-octanol	0.0474	20	isoheptanol	0.0477
9	n-nonanol	0.0318	21	phytol	0.0455
10	n-decanol	0.0319	22	cyclohexanol	0.0590
11	n-undecylalcohol	0.0475	23	benzyl alcohol	0.0465
12	n-dodecanol	0.0319

Table 5 Values of basic frequency and sum-frequency from natural frequencies of molecular structures

No.	Compound	ω ₀	$∑ ω i$	No.	Compound	ω ₀	$∑ ω i$
1	methanol	0.0167	2.2879	13	n-octadecanol	0.0474	43.3410
2	ethanol	0.0311	4.7659	14	2-hexanol	0.0651	14.4188
3	n-proanol	0.0488	7.1973	15	isoproanol	0.0556	7.1875
4	n-butanol	0.0322	9.6188	16	isobutanol	0.0443	9.5870
5	n-pentanol	0.0480	12.0349	17	2-methoxy-2-propanol	0.0554	9.4219
6	n-hexanol	0.0320	14.4466	18	3-methoxy-1-butanol	0.0484	11.5553
7	n-heptanol	0.0477	16.8574	19	2-methoxy-2-butanol	0.0684	11.9710
8	n-octanol	0.0474	19.2668	20	isoheptanol	0.0477	16.8190
9	n-nonanol	0.0318	21.7656	21	phytol	0.0455	48.6498
10	n-decanol	0.0319	24.0841	22	cyclohexanol	0.0590	16.3980
11	n-undecylalcohol	0.0475	26.4919	23	benzyl alcohol	0.0465	24.1498
12	n-dodecanol	0.0319	28.8955

Table 5 Values of basic frequency and sum-frequency from natural frequencies of molecular structures

No.	Compound	ω ₀	$∑ ω i$	No.	Compound	ω ₀	$∑ ω i$
1	methanol	0.0167	2.2879	13	n-octadecanol	0.0474	43.3410
2	ethanol	0.0311	4.7659	14	2-hexanol	0.0651	14.4188
3	n-proanol	0.0488	7.1973	15	isoproanol	0.0556	7.1875
4	n-butanol	0.0322	9.6188	16	isobutanol	0.0443	9.5870
5	n-pentanol	0.0480	12.0349	17	2-methoxy-2-propanol	0.0554	9.4219
6	n-hexanol	0.0320	14.4466	18	3-methoxy-1-butanol	0.0484	11.5553
7	n-heptanol	0.0477	16.8574	19	2-methoxy-2-butanol	0.0684	11.9710
8	n-octanol	0.0474	19.2668	20	isoheptanol	0.0477	16.8190
9	n-nonanol	0.0318	21.7656	21	phytol	0.0455	48.6498
10	n-decanol	0.0319	24.0841	22	cyclohexanol	0.0590	16.3980
11	n-undecylalcohol	0.0475	26.4919	23	benzyl alcohol	0.0465	24.1498
12	n-dodecanol	0.0319	28.8955

Table 6 Average optimum values of charge parameter E to power exponent m in temperature range

No.	Compound	m	m－1	No.	Compound	m	m－1
1	methanol	1.26	0.26	13	n-octadecanol	0.97	?0.03
2	ethanol	1.06	0.06	14	2-hexanol	1.02	0.02
3	n-proanol	1.14	0.14	15	isoproanol	1.01	0.01
4	n-butanol	1.04	0.04	16	isobutanol	0.96	?0.04
5	n-pentanol	1.07	0.07	17	2-methoxy-2-propanol	0.95	?0.05
6	n-hexanol	1.02	0.02	18	3-methoxy-1-butanol	0.97	?0.03
7	n-heptanol	1.06	0.06	19	2-methoxy-2-butanol	0.95	?0.05
8	n-octanol	1.05	0.05	20	isoheptanol	0.98	?0.02
9	n-nonanol	1.02	0.02	21	phytol	0.99	?0.01
10	n-decanol	1.04	0.04	22	cyclohexanol	0.95	?0.05
11	n-undecylalcohol	1.08	0.08	23	benzyl alcohol	1.03	0.03
12	n-dodecanol	1.05	0.05

Table 7 Calculated values of charge parameter E to power exponent m

No.	Compound	n	(m－1)_calc	m _calc	No.	Compound	n	(m－1)_calc	m _calc
1	methanol	1	0.26	1.26	7	n-heptanol	7	0.04	1.04
2	ethanol	2	0.06	1.06	8	n-octanol	8	0.05	1.05
3	n-proanol	3	0.14	1.14	9	n-nonanol	9	0.02	1.02
4	n-butanol	4	0.03	1.03	10	n-decanol	10	0.04	1.04
5	n-pentanol	5	0.07	1.07	11	n-dodecanol	12	0.05	1.05
6	n-hexanol	6	0.03	1.03	12	n-octadecanol	18	-0.03	0.97

Table 8 Characteristic group number and calculated values of power exponent m in no straight molecular structure

Group number	Isoproanol	Isobutanol	2-Methoxy-2-propanol	3-Methoxy -1-butanol	2-Methoxy-2-butanol	Isoheptanol	Phytol	Cyclohexanol	Benzyl alcohol
$n C H 3$	2	2	3	2	3	2	5	0	0
$n C H 2$	0	1	0	2	1	4	10	0	0
$n C H$	1	1	0	1	0	1	4	0	0
n _O	0	0	0	0	0	0	0	1	0
n _BO	0	0	0	0	0	0	0	0	1
n _H	1	2	0	2	0	2	2	1	2
m?1	0.01	?0.04	?0.05	?0.03	?0.05	?0.02	?0.01	?0.05	0.04

Table 8 Characteristic group number and calculated values of power exponent m in no straight molecular structure

Group number	Isoproanol	Isobutanol	2-Methoxy-2-propanol	3-Methoxy -1-butanol	2-Methoxy-2-butanol	Isoheptanol	Phytol	Cyclohexanol	Benzyl alcohol
$n C H 3$	2	2	3	2	3	2	5	0	0
$n C H 2$	0	1	0	2	1	4	10	0	0
$n C H$	1	1	0	1	0	1	4	0	0
n _O	0	0	0	0	0	0	0	1	0
n _BO	0	0	0	0	0	0	0	0	1
n _H	1	2	0	2	0	2	2	1	2
m?1	0.01	?0.04	?0.05	?0.03	?0.05	?0.02	?0.01	?0.05	0.04

Fig.2 Scatter diagram between experimental and calculated values

Table 9 Temperature range, parameter, mean absolute deviation and relative deviation between experiment[4] and calculated values

No.	Compound	Temperature range/K	n	ω ₀	$Q e ∑ ω i$	$Q e m$ T	MAD×10³ /(W/(m·K))	MRD/%
1	methanol	290—320	3	0.1712	0.0382	0.0167^1.30 T	4.35	2.14
2	ethanol	290—340	4	0.1248	0.1482	0.0311^1.06 T	3.25	2.02
3	n-proanol	273—353	5	0.0964	0.3512	0.0488^1.15 T	1.98	1.31
4	n-butanol	193—380	24	0.0762	0.3097	0.0322^0.99 T	3.52	2.46
5	n-pentanol	290—400	7	0.0585	0.5777	0.0480^1.08 T	1.67	1.21
6	n-hexanol	240—420	10	0.0470	0.4623	0.0320^1.02 T	3.77	2.68
7	n-heptanol	250—480	13	0.0374	0.8041	0.0477^1.04 T	3.25	2.24
8	n-octanol	280—520	13	0.0310	0.9132	0.0474^1.05 T	3.21	2.48
9	n-nonanol	280—460	10	0.0255	0.6921	0.0318^1.02 T	4.34	3.12
10	n-decanol	290—423	14	0.0184	0.7683	0.0319^1.04 T	3.04	2.05
11	n-undecylalcohol	300—440	8	0.0184	1.2584	0.0475^1.09 T	1.72	1.10
12	n-dodecanol	310—440	8	0.0159	0.9215	0.0319^1.05 T	3.10	2.07
13	n-octadecanol	337—440	15	0.0077	2.0544	0.0474^0.97 T	1.61	0.93
14	2-hexanol	200—370	10	0.0517	0.9387	0.0651^1.02 T	3.91	2.87
15	isoproanol	190—470	21	0.1192	0.3996	0.0556^1.01 T	2.68	2.12
16	isobutanol	290—470	22	0.0825	0.4247	0.0443^0.96 T	3.85	3.75
17	2-methoxy-2-propanol	310—400	9	0.0834	0.5220	0.0554^0.95 T	4.70	4.34
18	3-methoxy-1-butanol	160—400	13	0.0616	0.5593	0.0484^0.97 T	4.63	3.47
19	2-methoxy-2-butanol	280—360	6	0.0792	0.7793	0.0311^1.06 T	4.40	3.88
20	isoheptanol	230—430	21	0.0412	0.8023	0.0477^0.98 T	2.96	2.27
21	phytol	250—310	7	0.0084	2.2136	0.0455^0.99 T	1.58	0.78
22	cyclohexanol	300—360	7	0.0942	0.9675	0.0590^0.95 T	1.76	1.32
23	benzyl alcohol	290—420	14	0.0698	0.9370	0.0465^1.04 T	3.16	2.03

Table 9 Temperature range, parameter, mean absolute deviation and relative deviation between experiment[4] and calculated values

No.	Compound	Temperature range/K	n	ω ₀	$Q e ∑ ω i$	$Q e m$ T	MAD×10³ /(W/(m·K))	MRD/%
1	methanol	290—320	3	0.1712	0.0382	0.0167^1.30 T	4.35	2.14
2	ethanol	290—340	4	0.1248	0.1482	0.0311^1.06 T	3.25	2.02
3	n-proanol	273—353	5	0.0964	0.3512	0.0488^1.15 T	1.98	1.31
4	n-butanol	193—380	24	0.0762	0.3097	0.0322^0.99 T	3.52	2.46
5	n-pentanol	290—400	7	0.0585	0.5777	0.0480^1.08 T	1.67	1.21
6	n-hexanol	240—420	10	0.0470	0.4623	0.0320^1.02 T	3.77	2.68
7	n-heptanol	250—480	13	0.0374	0.8041	0.0477^1.04 T	3.25	2.24
8	n-octanol	280—520	13	0.0310	0.9132	0.0474^1.05 T	3.21	2.48
9	n-nonanol	280—460	10	0.0255	0.6921	0.0318^1.02 T	4.34	3.12
10	n-decanol	290—423	14	0.0184	0.7683	0.0319^1.04 T	3.04	2.05
11	n-undecylalcohol	300—440	8	0.0184	1.2584	0.0475^1.09 T	1.72	1.10
12	n-dodecanol	310—440	8	0.0159	0.9215	0.0319^1.05 T	3.10	2.07
13	n-octadecanol	337—440	15	0.0077	2.0544	0.0474^0.97 T	1.61	0.93
14	2-hexanol	200—370	10	0.0517	0.9387	0.0651^1.02 T	3.91	2.87
15	isoproanol	190—470	21	0.1192	0.3996	0.0556^1.01 T	2.68	2.12
16	isobutanol	290—470	22	0.0825	0.4247	0.0443^0.96 T	3.85	3.75
17	2-methoxy-2-propanol	310—400	9	0.0834	0.5220	0.0554^0.95 T	4.70	4.34
18	3-methoxy-1-butanol	160—400	13	0.0616	0.5593	0.0484^0.97 T	4.63	3.47
19	2-methoxy-2-butanol	280—360	6	0.0792	0.7793	0.0311^1.06 T	4.40	3.88
20	isoheptanol	230—430	21	0.0412	0.8023	0.0477^0.98 T	2.96	2.27
21	phytol	250—310	7	0.0084	2.2136	0.0455^0.99 T	1.58	0.78
22	cyclohexanol	300—360	7	0.0942	0.9675	0.0590^0.95 T	1.76	1.32
23	benzyl alcohol	290—420	14	0.0698	0.9370	0.0465^1.04 T	3.16	2.03

Table 10 Comparison of predicted values of liquid thermal conductivity of 3 alcohols compounds

Compound	T/K	λ/(mW/(m·K))	ω ₀	$Q e ∑ ω i$	TQ _e ^m	Δ/%
Compound	T/K	λ/(mW/(m·K))	ω ₀	$Q e ∑ ω i$	TQ _e ^m	This work	Sastri^[6]	Latini^[7]
tert-heptanol	310	139	0.0483	1.3046	22.3098	?1.13	—	—
	320	136	0.0483	1.306	23.0295	?1.25	—	—
	330	133	0.0483	1.3046	23.7491	?0.92	—	—
	340	130	0.0483	1.3046	24.4688	?0.46	—	—
2-octanol	300	138	0.0324	1.0050	16.6126	1.74	20.76	0.10
	320	135	0.0324	1.0050	17.7201	0.95	25.30	?1.12
	340	132	0.0324	1.0050	18.8276	0.12	30.53	?2.37
	360	129	0.0324	1.0050	19.9352	?0.74	36.67	?3.67
n-tetradecanol	340	166	0.0039	0.5680	4.0452	?4.23	?12.37	?22.37
350	164	0.0039	0.5680	4.1642	?3.33	?10.31	?22.82
360	163	0.0039	0.5680	4.2832	?3.01	?8.60	?23.76
370	161	0.0039	0.5680	4.4021	?2.08	?6.13	?24.25
380	160	0.0039	0.5680	4.5211	?1.74	?3.97	?25.22
390	158	0.0039	0.5680	4.6401	?0.78	?0.87	?25.75
400	156	0.0039	0.5680	4.7591	0.21	2.69	?26.30
410	154	0.0039	0.5680	4.8780	1.23	6.85	?26.88
420	153	0.0039	0.5680	4.9970	1.60	?24.81	?27.97
430	151	0.0039	0.5680	5.1160	2.65	?23.97	?28.61
440	150	0.0039	0.5680	5.2350	3.04	?23.63	?29.77
450	148	0.0039	0.5680	5.3540	4.13	?22.76	?30.51

Table 10 Comparison of predicted values of liquid thermal conductivity of 3 alcohols compounds

Compound	T/K	λ/(mW/(m·K))	ω ₀	$Q e ∑ ω i$	TQ _e ^m	Δ/%
Compound	T/K	λ/(mW/(m·K))	ω ₀	$Q e ∑ ω i$	TQ _e ^m	This work	Sastri^[6]	Latini^[7]
tert-heptanol	310	139	0.0483	1.3046	22.3098	?1.13	—	—
	320	136	0.0483	1.306	23.0295	?1.25	—	—
	330	133	0.0483	1.3046	23.7491	?0.92	—	—
	340	130	0.0483	1.3046	24.4688	?0.46	—	—
2-octanol	300	138	0.0324	1.0050	16.6126	1.74	20.76	0.10
	320	135	0.0324	1.0050	17.7201	0.95	25.30	?1.12
	340	132	0.0324	1.0050	18.8276	0.12	30.53	?2.37
	360	129	0.0324	1.0050	19.9352	?0.74	36.67	?3.67
n-tetradecanol	340	166	0.0039	0.5680	4.0452	?4.23	?12.37	?22.37
350	164	0.0039	0.5680	4.1642	?3.33	?10.31	?22.82
360	163	0.0039	0.5680	4.2832	?3.01	?8.60	?23.76
370	161	0.0039	0.5680	4.4021	?2.08	?6.13	?24.25
380	160	0.0039	0.5680	4.5211	?1.74	?3.97	?25.22
390	158	0.0039	0.5680	4.6401	?0.78	?0.87	?25.75
400	156	0.0039	0.5680	4.7591	0.21	2.69	?26.30
410	154	0.0039	0.5680	4.8780	1.23	6.85	?26.88
420	153	0.0039	0.5680	4.9970	1.60	?24.81	?27.97
430	151	0.0039	0.5680	5.1160	2.65	?23.97	?28.61
440	150	0.0039	0.5680	5.2350	3.04	?23.63	?29.77
450	148	0.0039	0.5680	5.3540	4.13	?22.76	?30.51

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