饱和气液相密度关联外推的新通用对比态方程

doi:10.11949/0438-1157.20251266

摘要/Abstract

摘要：

饱和密度是流体物质在气液相平衡时的重要热物性参数，在热力循环及化工过程中被广泛应用，但实验数据往往有限且离散，因此建立能够准确关联数据并能够可靠外推的饱和密度方程具有重要意义。基于饱和密度与饱和蒸气压的热力学特征相似性，此前提出的饱和蒸气压方程可推广至饱和密度，并以32种物质为对象与Zhang方程和Wagner型方程比较分析后证明，在关联性能上，新方程与Zhang方程和Wagner型方程偏差量级相当，足以在测量不确定度范围内关联实验数据；而在外推性能上，新方程总体优于Zhang方程和Wagner型方程，外推趋势更为稳定准确。如在外推拟合中将新方程部分参数固定后，外推趋势稳定性及准确性可进一步提升。

关键词: 热力学性质, 气液平衡, 模型, 预测, 饱和密度, Wagner型方程

Abstract:

Saturated density, a key thermophysical property of fluids at vapor–liquid equilibrium, is widely used in power cycles and chemical processes. However, experimental data of saturated density are often scarce and scattered; consequently, developing a correlation equation that accurately reproduces existing measurements while providing reliable extrapolation is essential. In this work, based on the thermodynamic similarity between saturated density and saturated vapor pressure, the previously proposed three-parameter saturated vapor pressure equation is extended to saturated density. The functional form is preserved, ensuring positive density and correct limiting behavior toward the critical point and triple point. Using thirty-two substances that possess abundant experimental data, the new equation is compared with the Zhang equation and Wagner equation. The results demonstrate that, with respect to correlation, the deviations of the new equation are of the same magnitude as those of the Zhang equation and Wagner equation and lie within the reported experimental uncertainties. In terms of extrapolation, the new equation is superior to or equivalent to the Zhang equation and Wagner equation, and the extrapolation trend is more stable and accurate. Furthermore, by fixing some parameters of the new equation in extrapolation fitting, the stability and accuracy of the extrapolation trend can be further improved, yielding a extrapolation that is both thermodynamically consistent and numerically sound.

Key words: thermodynamics properties, vapor–liquid equilibrium, model, prediction, saturated density, Wagner equation

中图分类号:

TK 12

卿康, 汪尔奇, 杨震, 段远源. 饱和气液相密度关联外推的新通用对比态方程[J]. 化工学报, DOI: 10.11949/0438-1157.20251266.

Kang QING, Erqi WANG, Zhen YANG, Yuanyuan DUAN. New universal corresponding state equation for correlation and extrapolation of saturated vapor-liquid density[J]. CIESC Journal, DOI: 10.11949/0438-1157.20251266.

图/表 9

图1 氮的pr，1/ρr′，ρr′′随Tr变化曲线

Fig. 1 The curve of nitrogen pr, 1/ρr′, ρr′′ as a function of Tr

表1 饱和蒸气压及饱和密度曲线的数学约束

Table 1 Mathematical constraints on saturated vapor pressure and saturated density curves

	$p r (T r)$	$1 / ρ r' (T r)$	$ρ r ″ (T r)$	$d p r d T r$	$d 1 / ρ r' d T r$	$d ρ r ″ d T r$	$d 2 p r d T r 2$	$d 2 1 / ρ r' d T r 2$	$d 2 ρ r ″ d T r 2$
$T r ∈ T t r, 1$	$p r > 0$	$1 / ρ r' > 0$	$ρ r ″ > 0$	$d p r d T r > 0$	$d 1 / ρ r' d T r > 0$	$d ρ r ″ d T r > 0$	$d 2 p r d T r 2 > 0$	$d 2 1 / ρ r' d T r 2 > 0$	$d 2 ρ r ″ d T r 2 > 0$
$T r → T t r$	$p r → p t r$	$1 / ρ r' > ρ c / ρ t'$	$ρ r ″ > ρ t ″ / ρ c$	$d p r d T r > 0$	$d 1 / ρ r' d T r > 0$	$d ρ r ″ d T r > 0$	$d 2 p r d T r 2 > 0$	$d 2 1 / ρ r' d T r 2 > 0$	$d 2 ρ r ″ d T r 2 > 0$
$T r → 1$	$p r → 1$	$1 / ρ r' → 1$	$ρ r ″ → 1$	$d p r d T r → F i n i t e$	$d 1 / ρ r' d T r → ∞$	$d ρ r ″ d T r → ∞$	$d 2 p r d T r 2 → ∞$	$d 2 1 / ρ r' d T r 2 → ∞$	$d 2 ρ r ″ d T r 2 → ∞$

表1 饱和蒸气压及饱和密度曲线的数学约束

Table 1 Mathematical constraints on saturated vapor pressure and saturated density curves

	$p r (T r)$	$1 / ρ r' (T r)$	$ρ r ″ (T r)$	$d p r d T r$	$d 1 / ρ r' d T r$	$d ρ r ″ d T r$	$d 2 p r d T r 2$	$d 2 1 / ρ r' d T r 2$	$d 2 ρ r ″ d T r 2$
$T r ∈ T t r, 1$	$p r > 0$	$1 / ρ r' > 0$	$ρ r ″ > 0$	$d p r d T r > 0$	$d 1 / ρ r' d T r > 0$	$d ρ r ″ d T r > 0$	$d 2 p r d T r 2 > 0$	$d 2 1 / ρ r' d T r 2 > 0$	$d 2 ρ r ″ d T r 2 > 0$
$T r → T t r$	$p r → p t r$	$1 / ρ r' > ρ c / ρ t'$	$ρ r ″ > ρ t ″ / ρ c$	$d p r d T r > 0$	$d 1 / ρ r' d T r > 0$	$d ρ r ″ d T r > 0$	$d 2 p r d T r 2 > 0$	$d 2 1 / ρ r' d T r 2 > 0$	$d 2 ρ r ″ d T r 2 > 0$
$T r → 1$	$p r → 1$	$1 / ρ r' → 1$	$ρ r ″ → 1$	$d p r d T r → F i n i t e$	$d 1 / ρ r' d T r → ∞$	$d ρ r ″ d T r → ∞$	$d 2 p r d T r 2 → ∞$	$d 2 1 / ρ r' d T r 2 → ∞$	$d 2 ρ r ″ d T r 2 → ∞$

图2 式(10)和(11)对氮实验数据的拟合

Fig. 2 The fitting of nitrogen experimental data by Eq. (10) and (11)

表2 所研究物质的临界温度（Tc）、临界密度（ρc）及实验数据信息（数量（N）、对比温度（Tr）范围、对比密度（ρr′或ρr′′）范围及其平均不确定度（uave（ρr′或ρr′′）））

Table 2 The critical temperature （Tc）， critical density （ρc） and information on experimental data （number （N）， reduced temperature （Tr） range， reduced density （ρr′ or ρr′′） range and average uncertainty of reduced density uave（ρr′ or ρr′′）） of the studied substance

物质	T_c / K	ρ_c / kg/m³	ρ_r′				ρ_r′′
物质	T_c / K	ρ_c / kg/m³	N	T_r范围	1/ρ_r′范围	u_ave(ρ_r′)	N	T_r范围	ρ_r′′范围	u_ave(ρ_r′′)
氖（Neon）	44.4	486.31	36	0.555~1.0	0.39~0.984	0.062	51	0.567~1.0	0.011~0.884	0.015
氩（Argon）	150.687	535.60	234	0.557~1.0	0.377~0.999	0.108	100	0.557~1.0	0.008~0.99	0.056
氪（Krypton）	209.48	909.21	75	0.553~1.0	0.371~0.947	0.016	17	0.601~0.999	0.015~0.888	0.001
氙（Xenon）	289.733	1102.86	94	0.557~1.0	0.37~0.998	0.005	43	0.712~1.0	0.053~0.99	0.048
甲烷（Methane）	190.564	162.66	348	0.466~1.0	0.358~0.939	0.006	102	0.478~1.0	0.002~0.955	0.001
氧气（Oxygen）	154.581	436.14	317	0.352~1.0	0.333~0.945	0.059	194	0.352~1.0	0.0~0.945	0.015
氮气（Nitrogen）	126.192	313.30	270	0.496~1.0	0.359~0.898	0.019	56	0.513~1.0	0.003~0.888	0.001
一氧化碳（Carbon monoxide）	132.86	303.91	57	0.513~0.995	0.359~0.72	0.003	56	0.513~0.998	0.003~0.837	0.001
乙烯（Ethene）	282.35	214.24	221	0.368~1.0	0.327~0.916	0.010	81	0.561~1.0	0.005~0.928	0.003
乙烯（Ethylene）	282.35	214.24	221	0.368~1.0	0.327~0.916	0.010	81	0.561~1.0	0.005~0.928	0.003
乙烷（Ethane）	305.322	206.18	437	0.291~1.0	0.316~0.993	0.058	126	0.605~1.0	0.01~0.965	0.007
丙烯（Propylene）	364.211	229.63	135	0.256~1.0	0.302~0.849	0.038	50	0.632~1.0	0.012~0.839	0.001
丙烷（Propane）	369.89	220.48	385	0.234~1.0	0.301~0.984	0.010	99	0.622~1.0	0.01~0.9	0.003
异丁烷（Isobutane）	407.81	225.50	262	0.282~1.0	0.305~0.983	0.012	53	0.638~1.0	0.012~1.0	0.011
新戊烷（Neopentane）	433.74	235.93	94	0.592~0.999	0.374~0.851	0.037	44	0.791~0.999	0.074~0.795	0.001
正丁烷（n-Butane）	425.125	228.00	359	0.209~1.0	0.293~0.887	0.005	125	0.317~1.0	0.0~0.874	0.001
苯（Benzene）	562.02	304.71	548	0.486~0.999	0.338~0.911	0.042	140	0.513~0.999	0.001~0.88	0.007
六氟化硫（Sulfur hexafluoride）	318.7232	742.30	64	0.703~1.0	0.403~0.988	0.003	56	0.703~1.0	0.027~1.0	0.001
二氧化碳（Carbon dioxide）	304.1282	467.60	170	0.714~1.0	0.397~0.93	0.025	133	0.714~1.0	0.03~0.915	0.011
2,3-二甲基丁烷（2,3-Dimethylbutane）	500.6	241.29	69	0.546~0.999	0.355~0.813	0.009	55	0.645~0.999	0.011~0.767	0.003
正戊烷（n-Pentane）	469.7	231.60	250	0.317~1.0	0.306~0.923	0.020	79	0.645~1.0	0.011~0.925	0.008
二氧化硫（Sulfur dioxide）	430.64	517.51	66	0.518~1.0	0.332~0.872	0.006	38	0.651~0.999	0.012~0.884	0.000
三氟甲烷（Trifluoromethane）	299	522.40	194	0.417~1.0	0.311~0.994	0.006	50	0.793~1.0	0.066~0.909	0.001
甲苯（Toluene）	591.75	291.99	278	0.313~0.994	0.305~0.706	0.029	39	0.504~1.0	0.0~0.92	0.002
1,1-二氟乙烷（1,1-Difluoroethane）	386.43	368.33	129	0.406~1.0	0.31~0.985	0.015	70	0.705~1.0	0.017~0.982	0.005
二氟甲烷（Difluoromethane）	351.25	424.80	184	0.395~1.0	0.299~0.911	0.012	69	0.72~1.0	0.027~0.877	0.009
正己烷（n-Hexane）	507.82	233.19	97	0.361~0.999	0.31~0.809	0.013	21	0.656~0.999	0.011~0.775	0.007
异辛烷（Isooctane）	544	242.16	111	0.428~1.0	0.327~0.997	0.011	42	0.535~0.997	0.001~0.722	0.003
丙酮（Acetone）	508.1	272.97	128	0.361~0.998	0.3~0.834	0.049	30	0.537~0.997	0.001~0.741	0.010
正庚烷（n-Heptane）	540.2	233.47	195	0.342~1.0	0.302~0.949	0.018	52	0.635~1.0	0.006~0.994	0.014
正辛烷（n-Octane）	568.74	232.00	124	0.427~1.0	0.312~0.999	0.011	71	0.691~1.0	0.014~0.97	0.003
甲醇（Methanol）	513.38	281.50	258	0.4~0.999	0.322~0.997	0.016	45	0.581~0.999	0.001~0.976	0.007

物质	T_c / K	ρ_c / kg/m³	ρ_r′				ρ_r′′
物质	T_c / K	ρ_c / kg/m³	N	T_r范围	1/ρ_r′范围	u_ave(ρ_r′)	N	T_r范围	ρ_r′′范围	u_ave(ρ_r′′)
氖（Neon）	44.4	486.31	36	0.555~1.0	0.39~0.984	0.062	51	0.567~1.0	0.011~0.884	0.015
氩（Argon）	150.687	535.60	234	0.557~1.0	0.377~0.999	0.108	100	0.557~1.0	0.008~0.99	0.056
氪（Krypton）	209.48	909.21	75	0.553~1.0	0.371~0.947	0.016	17	0.601~0.999	0.015~0.888	0.001
氙（Xenon）	289.733	1102.86	94	0.557~1.0	0.37~0.998	0.005	43	0.712~1.0	0.053~0.99	0.048
甲烷（Methane）	190.564	162.66	348	0.466~1.0	0.358~0.939	0.006	102	0.478~1.0	0.002~0.955	0.001
氧气（Oxygen）	154.581	436.14	317	0.352~1.0	0.333~0.945	0.059	194	0.352~1.0	0.0~0.945	0.015
氮气（Nitrogen）	126.192	313.30	270	0.496~1.0	0.359~0.898	0.019	56	0.513~1.0	0.003~0.888	0.001
一氧化碳（Carbon monoxide）	132.86	303.91	57	0.513~0.995	0.359~0.72	0.003	56	0.513~0.998	0.003~0.837	0.001
乙烯（Ethene）	282.35	214.24	221	0.368~1.0	0.327~0.916	0.010	81	0.561~1.0	0.005~0.928	0.003
乙烯（Ethylene）	282.35	214.24	221	0.368~1.0	0.327~0.916	0.010	81	0.561~1.0	0.005~0.928	0.003
乙烷（Ethane）	305.322	206.18	437	0.291~1.0	0.316~0.993	0.058	126	0.605~1.0	0.01~0.965	0.007
丙烯（Propylene）	364.211	229.63	135	0.256~1.0	0.302~0.849	0.038	50	0.632~1.0	0.012~0.839	0.001
丙烷（Propane）	369.89	220.48	385	0.234~1.0	0.301~0.984	0.010	99	0.622~1.0	0.01~0.9	0.003
异丁烷（Isobutane）	407.81	225.50	262	0.282~1.0	0.305~0.983	0.012	53	0.638~1.0	0.012~1.0	0.011
新戊烷（Neopentane）	433.74	235.93	94	0.592~0.999	0.374~0.851	0.037	44	0.791~0.999	0.074~0.795	0.001
正丁烷（n-Butane）	425.125	228.00	359	0.209~1.0	0.293~0.887	0.005	125	0.317~1.0	0.0~0.874	0.001
苯（Benzene）	562.02	304.71	548	0.486~0.999	0.338~0.911	0.042	140	0.513~0.999	0.001~0.88	0.007
六氟化硫（Sulfur hexafluoride）	318.7232	742.30	64	0.703~1.0	0.403~0.988	0.003	56	0.703~1.0	0.027~1.0	0.001
二氧化碳（Carbon dioxide）	304.1282	467.60	170	0.714~1.0	0.397~0.93	0.025	133	0.714~1.0	0.03~0.915	0.011
2,3-二甲基丁烷（2,3-Dimethylbutane）	500.6	241.29	69	0.546~0.999	0.355~0.813	0.009	55	0.645~0.999	0.011~0.767	0.003
正戊烷（n-Pentane）	469.7	231.60	250	0.317~1.0	0.306~0.923	0.020	79	0.645~1.0	0.011~0.925	0.008
二氧化硫（Sulfur dioxide）	430.64	517.51	66	0.518~1.0	0.332~0.872	0.006	38	0.651~0.999	0.012~0.884	0.000
三氟甲烷（Trifluoromethane）	299	522.40	194	0.417~1.0	0.311~0.994	0.006	50	0.793~1.0	0.066~0.909	0.001
甲苯（Toluene）	591.75	291.99	278	0.313~0.994	0.305~0.706	0.029	39	0.504~1.0	0.0~0.92	0.002
1,1-二氟乙烷（1,1-Difluoroethane）	386.43	368.33	129	0.406~1.0	0.31~0.985	0.015	70	0.705~1.0	0.017~0.982	0.005
二氟甲烷（Difluoromethane）	351.25	424.80	184	0.395~1.0	0.299~0.911	0.012	69	0.72~1.0	0.027~0.877	0.009
正己烷（n-Hexane）	507.82	233.19	97	0.361~0.999	0.31~0.809	0.013	21	0.656~0.999	0.011~0.775	0.007
异辛烷（Isooctane）	544	242.16	111	0.428~1.0	0.327~0.997	0.011	42	0.535~0.997	0.001~0.722	0.003
丙酮（Acetone）	508.1	272.97	128	0.361~0.998	0.3~0.834	0.049	30	0.537~0.997	0.001~0.741	0.010
正庚烷（n-Heptane）	540.2	233.47	195	0.342~1.0	0.302~0.949	0.018	52	0.635~1.0	0.006~0.994	0.014
正辛烷（n-Octane）	568.74	232.00	124	0.427~1.0	0.312~0.999	0.011	71	0.691~1.0	0.014~0.97	0.003
甲醇（Methanol）	513.38	281.50	258	0.4~0.999	0.322~0.997	0.016	45	0.581~0.999	0.001~0.976	0.007

表3 各方程关联及外推性能比较

Table 3 Comparison of correlation and extrapolation performance of various equations

拟合数据范围	相态	100·OAAD^①				100·OMAD^②
拟合数据范围	相态	Zhang	Wagner	式(10)和式(11)	式(10)和式(11) (固定参数)^③	Zhang	Wagner	式(10)和式(11)	式(10)和式(11) (固定参数)^③
关联 T_tr < T_r ≤ 1	饱和液相	0.86	0.80	0.93	-	8.11	8.43	8.73	-
关联 T_tr < T_r ≤ 1	饱和气相	0.78	0.63	0.72	-	5.63	5.08	4.99	-
向临界点方向外推 T_tr < T_r ≤ 0.7	饱和液相	11.71 (30)	6.24 (31)	5.34 (31)	1.34 (32)	29.94 (27)	18.18 (31)	15.48 (31)	9.15 (32)
向临界点方向外推 T_tr < T_r ≤ 0.7	饱和气相	2.33 (32)	32.24 (28)	5.30 (32)	1.88 (32)	10.37 (32)	56.84 (21)	19.67 (32)	7.03 (32)
向三相点与临界点方向同时外推 0.8 < T_r ≤ 0.9	饱和液相	3.55 (32)	41.47 (24)	5.46 (32)	1.78 (32)	15.12 (32)	70.19 (15)	27.52 (32)	9.56 (32)
向三相点与临界点方向同时外推 0.8 < T_r ≤ 0.9	饱和气相	1.69 (32)	24.29 (26)	3.51 (32)	1.08 (32)	9.33 (32)	39.00 (26)	16.43 (32)	6.83 (32)
向三相点方向外推 0.9 < T_r ≤ 1	饱和液相	12.49 (32)	87.33 (7)	29.25 (31)	5.73 (32)	33.20 (32)	95.57 (3)	62.05 (22)	15.13 (32)
向三相点方向外推 0.9 < T_r ≤ 1	饱和气相	5.02 (32)	24.15 (26)	1.67 (32)	0.89 (32)	21.38 (31)	38.63 (22)	8.15 (31)	5.25 (32)

图3 式(10)和(11)、Zhang方程和Wagner型方程对实验数据的关联偏差比较

Fig. 3 Comparison of correlation bias between Eq. (10) and (11), Zhang equation and Wagner equation on experimental data

图4 以二氧化碳和1,1,2-三氟三氯乙烷为例的各方程关联效果比较

Fig. 4 Comparison of correlation effect of various equations using carbon dioxide and 1,1,2-trifluorotrichloroethane as examples

图5 以正戊烷、1,1,1,2-四氟乙烷、二氧化硫、二氟一氯甲烷、2,3-二甲基丁烷和六氟乙烷为例的各方程外推效果比较

Fig. 5 Comparison of extrapolation effects for various equations, using n-pentane, 1,1,1,2-tetrafluoroethane, sulfur dioxide, difluorochloromethane, 2,3-dimethylbutane, and hexafluoroethane as examples

表4 式（10）和（11）在不同方向外推时固定参数设定

Table 4 Fixed parameter settings when Eq. （10） and （11） are extrapolated in different directions

		式(10)	式(11)
拟合初始值设定^①		a₀′ = 0.04, b₀′ = 0.30, c₀′ = 1.34	a₀′′ = 5.23, b₀′′ = 0.36, c₀′′ = 2.44
外推时固定参数设定^②	向临界点方向外推	b′ = 0.30, c′ = 1.34	b′′ = 0.36, c′′ = 2.44
	向三相点与临界点方向同时外推	a′ = 0.04, b′ = 0.30	a′′ = 5.23, b′′ = 0.36
	向三相点方向外推	a′ = 0.04	a′′ = 5.23

参考文献 32

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