Research and evaluation on generalized alpha functions based on PR EoS

doi:10.11949/0438-1157.20191532

Abstract

Abstract:

17 generalized alpha functions for Peng-Robinson equation of state (PR EoS) were evaluated using the vapor pressures of 11 kinds of non-polar, weakly polar and polar compounds. The estimative abilities of the generalized alpha functions were compared according to the average absolute deviation of each kind of compounds. The results indicated that the vapor pressures of alkanes, aromatic hydrocarbons, gases and halogenated hydrocarbons can be accurately predicted with Robinson-Peng (1978), Wang (2004) and Li-Yang (2011) alpha function. However, the predictions of vapor pressures for weakly polar and polar compounds, such as alcohols, ethers, esters, acids, and water, with Robinson-Peng (1978), Wang (2004) and Li-Yang (2011) alpha function were less accurate than that predicted with Forero (2016) alpha function. In conclusion, the generalized alpha functions with acentric factor and polar factor as variables are more accurate than that generalized only with acentric factor for the prediction of vapor pressures of polar compounds, such as alcohols, acids and water.

Key words: thermodynamics properties, equation of state, Peng-Robinson, generalized alpha functions, prediction, evaluation

摘要：

用11类70种非极性、弱极性和极性物质的实验蒸汽压数据对适用于Peng-Robinson状态方程(PR EoS)的17种普遍化温度函数对蒸汽压的预测能力进行了评价。根据各类物质蒸汽压预测结果平均相对偏差评价普遍化温度函数的预测能力。结果表明，Robinson-Peng (1978)、汪萍(2004)和Li-Yang (2011)温度函数能够准确预测烷烃、芳烃、气体和卤代烃类物质的蒸汽压，但对弱极性和极性的醇、醚、酯、酸、水等物质的预测结果不如Forero (2016)温度函数准确。以偏心因子和极性因子普遍化的温度函数对醇、酸、水等极性物质蒸汽压的预测结果明显优于仅以偏心因子普遍化的温度函数。

关键词: 热力学性质, 状态方程, Peng-Robinson, 普遍化温度函数, 预测, 评价

CLC Number:

TQ 013.1

Wenying ZHAO, Wenwen LI, Xiaoyan SUN, Xiaorong CAO, Shuguang XIANG. Research and evaluation on generalized alpha functions based on PR EoS[J]. CIESC Journal, 2020, 71(3): 1234-1245.

赵文英, 李文文, 孙晓岩, 曹晓荣, 项曙光. 基于PR立方型状态方程普遍化温度函数的研究与评价[J]. 化工学报, 2020, 71(3): 1234-1245.

Figures/Tables 7

Table 1 Generalized alpha functions for evaluation

函数名称	参考文献	温度函数及参数关联式
Peng-Robinson (1976)	[4]	$α T r = 1 + k 1 - T r 2$ $k = 0.37464 + 1.54226 ω - 0.26992 ω 2$
Robinson-Peng (1978)	[20]	$α T r = 1 + k 1 - T r 2$ $k = 0.37464 + 1.54226 ω - 0.26992 ω 2$ as $ω ≤ 0.491$ $k = 0.379642 + 1.487503 ω - 0.164423 ω 2 + 0.016666 ω 3$ as $ω > 0.491$
Gasem (2001)	[9]	$α (T r) = e x p A + B T r 1 - T r m$ $A = 2, ?? B = 0.836, ?$ $m = 0.134 + 0.508 ω - 0.0467 ω 2$
Coquelet (2004)	[16]	$a T r = 1 + c 1 1 - T r + c 2 1 - T r 2 ? + c 3 1 - T r 3 2$ $c 1 = 0.1316 ω 2 + 1.4031 ω + 0.3906$ $c 2 = - 1.3127 ω 2 + 0.3015 ω - 0.1213$ $c 3 = 0.7661 ω + 0.3041$
汪萍(2004)	[17]	$a T r = 1 + k 1 - T r 2$ $k = k 0 + k 1 1 + T r 0.7 - T r ?$ as $T r ≤ 0.7$ $k 0 = 0.3784 + 1.5182 ω - 0.2663 ω 2 + 0.0923 ω 3$ $k 1 = - 0.0035 + 0.0028 ω + 0.8369 ω 2 - 1.7359 ω 3 + 0.8325 ω 4$ $k = k 0$ as $T r > 0.7$
Joshipura (2009)	[22]	$α (T r) = e x p m 1 - T r$ $m = 1.2294 ω + 0.4887$
Li-Yang (2011)	[18]	$α T r = e x p ? k 1 1 - T r + l l n ? 1 + ? k 2 1 - T r 2$ l= $0.81769$ $k 1 = 0.13280 - 0.05052 ω + 0.25948 ω 2$ $k 2 = 0.31355 + 1.86745 ω - 0.52604 ω 2$ $k 3 = 0.38856 + 1.40137 ω - 0.05387 ω 2 - 0.01824 ω 3$
Haghtalab (2011)	[23]	$α T r = e x p 1 - n l n T r$ $n = 1.7309 + 1.6571 ω + 0.1554 ω 2$
Saffari-Zahedi (2013)	[24]	$α T r = e x p ? k 1 T r + k 2 l n T r + k 3 1 - T r$ $k 1 = 0.003091 + 0.013145 ω$ $k 2 = - 0.006487 + 0.482173 ω$ $k 3 = 0.721306 + 3.586161 ω$
Hou (2015)	[25]	$α T r = e x p ? k 1 T r + k 2 l n T r + k 3 1 - T r$ $k 1 = 0.0033 + 0.0113 ω$ $k 2 = - 0.0085 + 0.4106 ω$ $k 3 = 0.7532 + 3.541 ω$
Valiollahi (2016)	[21]	$α T r = 1 + k (1 - T r) 2 1 + F (T) × F (P)$ $k = 0.3746 + 1.54226 ω - 0.26992 ω 2$ $F T = T r - 1 0.5$ $F P = f 1 f 2 + f 3 P r + f 4 P r 2$ $f 1 = 1.03 × e x p - e x p 9.3890 - 0.8524 ρ c$ $f 2 = 3744.556 × ρ c - 4.2525$ $f 3 = - 568313.250 × ρ c - 5.9531$ $f 4 = 1166215.400 × ρ c - 6.2409$
Le Guennec (2016)	[10]	$α (T r) = T r 2 M - 1 e x p L 1 - T r 2 M$ $L = 0.0877 + 0.6039 ω + 0.1290 ω 2$ $M = 0.8884 - 0.2600 ω + 0.1760 ω 2$
Forero (2016)	[19]	$α (T r) = e x p ? m 1 - T r n$ $n = γ β - 1 + β$ , $m = β - 1 n$ $β = 1.3469 + 1.4923 ω + 1.4252 χ - 0.1204 ω 2 + 2.8166 χ 2 + 0.3127 ω χ$ $γ = - 0.3759 - 1.4419 ω + 6.2044 χ - 0.6747 ω 2 + 15.586 χ 2 - 0.7058 ω χ$ $χ = l g P r T r = 0.6 + 1.7 ω + 1.552$
Twu-Mahmoodi (2017)	[11]	$α T r = α 0 + ω α 1 - α 0$ $α 0 = T r 2.8848 (0.92291 - 1) e x p 0.06683 (1 - T r 2.8848 × 0.92291)$ $α 0 = T r 1.4636 (0.87704 - 1) e x p 1.1744 (1 - T r 1.4636 × 0.87704)$
Coquelet-Mahmoodi (2017)	[11]	$α (T r) = e x p ? m 1 1 - T r 1 + m 2 1 - T r 2 + m 3 1 - T r 3 2$ $m 1 = 0.40464 + 1.3361 ω - 0.07987 ω 2$ $m 2 = - 0.08139 + 0.096493 ω - 0.85454 ω 2$ $m 3 = 0.31953 + 0.001007 ω - 0.88858 ω 2$
PR-PM (2017)	[11]	$α T r = e x p 2 c 1 1 - T r - c 2 1 - T r 2 + 23 c 3 1 - T r 3$ ， $0 ≤ c 3 ≤ 1.25 c 1$ $c 1 = 0.36818 + 1.4801 ω - 0.14407 ω 2$ $c 2 = 0.19422 + 1.9061 ω - 0.46577 ω 2$ $c 3 = 1.514 e x p - ω - 1.6645 1.5474 4$
PR-PM2 (2017)	[11]	$α T r = e x p 2 c 1 1 - T r - c 2 1 - T r 2 + 23 c 1 1 - T r 3$ $c 1 = 0.3464 + 1.4886 ω - 0.13347 ω 2$ $c 2 = 0.18339 + 1.8435 ω - 0.26929 ω 2$

Table 1 Generalized alpha functions for evaluation

函数名称	参考文献	温度函数及参数关联式
Peng-Robinson (1976)	[4]	$α T r = 1 + k 1 - T r 2$ $k = 0.37464 + 1.54226 ω - 0.26992 ω 2$
Robinson-Peng (1978)	[20]	$α T r = 1 + k 1 - T r 2$ $k = 0.37464 + 1.54226 ω - 0.26992 ω 2$ as $ω ≤ 0.491$ $k = 0.379642 + 1.487503 ω - 0.164423 ω 2 + 0.016666 ω 3$ as $ω > 0.491$
Gasem (2001)	[9]	$α (T r) = e x p A + B T r 1 - T r m$ $A = 2, ?? B = 0.836, ?$ $m = 0.134 + 0.508 ω - 0.0467 ω 2$
Coquelet (2004)	[16]	$a T r = 1 + c 1 1 - T r + c 2 1 - T r 2 ? + c 3 1 - T r 3 2$ $c 1 = 0.1316 ω 2 + 1.4031 ω + 0.3906$ $c 2 = - 1.3127 ω 2 + 0.3015 ω - 0.1213$ $c 3 = 0.7661 ω + 0.3041$
汪萍(2004)	[17]	$a T r = 1 + k 1 - T r 2$ $k = k 0 + k 1 1 + T r 0.7 - T r ?$ as $T r ≤ 0.7$ $k 0 = 0.3784 + 1.5182 ω - 0.2663 ω 2 + 0.0923 ω 3$ $k 1 = - 0.0035 + 0.0028 ω + 0.8369 ω 2 - 1.7359 ω 3 + 0.8325 ω 4$ $k = k 0$ as $T r > 0.7$
Joshipura (2009)	[22]	$α (T r) = e x p m 1 - T r$ $m = 1.2294 ω + 0.4887$
Li-Yang (2011)	[18]	$α T r = e x p ? k 1 1 - T r + l l n ? 1 + ? k 2 1 - T r 2$ l= $0.81769$ $k 1 = 0.13280 - 0.05052 ω + 0.25948 ω 2$ $k 2 = 0.31355 + 1.86745 ω - 0.52604 ω 2$ $k 3 = 0.38856 + 1.40137 ω - 0.05387 ω 2 - 0.01824 ω 3$
Haghtalab (2011)	[23]	$α T r = e x p 1 - n l n T r$ $n = 1.7309 + 1.6571 ω + 0.1554 ω 2$
Saffari-Zahedi (2013)	[24]	$α T r = e x p ? k 1 T r + k 2 l n T r + k 3 1 - T r$ $k 1 = 0.003091 + 0.013145 ω$ $k 2 = - 0.006487 + 0.482173 ω$ $k 3 = 0.721306 + 3.586161 ω$
Hou (2015)	[25]	$α T r = e x p ? k 1 T r + k 2 l n T r + k 3 1 - T r$ $k 1 = 0.0033 + 0.0113 ω$ $k 2 = - 0.0085 + 0.4106 ω$ $k 3 = 0.7532 + 3.541 ω$
Valiollahi (2016)	[21]	$α T r = 1 + k (1 - T r) 2 1 + F (T) × F (P)$ $k = 0.3746 + 1.54226 ω - 0.26992 ω 2$ $F T = T r - 1 0.5$ $F P = f 1 f 2 + f 3 P r + f 4 P r 2$ $f 1 = 1.03 × e x p - e x p 9.3890 - 0.8524 ρ c$ $f 2 = 3744.556 × ρ c - 4.2525$ $f 3 = - 568313.250 × ρ c - 5.9531$ $f 4 = 1166215.400 × ρ c - 6.2409$
Le Guennec (2016)	[10]	$α (T r) = T r 2 M - 1 e x p L 1 - T r 2 M$ $L = 0.0877 + 0.6039 ω + 0.1290 ω 2$ $M = 0.8884 - 0.2600 ω + 0.1760 ω 2$
Forero (2016)	[19]	$α (T r) = e x p ? m 1 - T r n$ $n = γ β - 1 + β$ , $m = β - 1 n$ $β = 1.3469 + 1.4923 ω + 1.4252 χ - 0.1204 ω 2 + 2.8166 χ 2 + 0.3127 ω χ$ $γ = - 0.3759 - 1.4419 ω + 6.2044 χ - 0.6747 ω 2 + 15.586 χ 2 - 0.7058 ω χ$ $χ = l g P r T r = 0.6 + 1.7 ω + 1.552$
Twu-Mahmoodi (2017)	[11]	$α T r = α 0 + ω α 1 - α 0$ $α 0 = T r 2.8848 (0.92291 - 1) e x p 0.06683 (1 - T r 2.8848 × 0.92291)$ $α 0 = T r 1.4636 (0.87704 - 1) e x p 1.1744 (1 - T r 1.4636 × 0.87704)$
Coquelet-Mahmoodi (2017)	[11]	$α (T r) = e x p ? m 1 1 - T r 1 + m 2 1 - T r 2 + m 3 1 - T r 3 2$ $m 1 = 0.40464 + 1.3361 ω - 0.07987 ω 2$ $m 2 = - 0.08139 + 0.096493 ω - 0.85454 ω 2$ $m 3 = 0.31953 + 0.001007 ω - 0.88858 ω 2$
PR-PM (2017)	[11]	$α T r = e x p 2 c 1 1 - T r - c 2 1 - T r 2 + 23 c 3 1 - T r 3$ ， $0 ≤ c 3 ≤ 1.25 c 1$ $c 1 = 0.36818 + 1.4801 ω - 0.14407 ω 2$ $c 2 = 0.19422 + 1.9061 ω - 0.46577 ω 2$ $c 3 = 1.514 e x p - ω - 1.6645 1.5474 4$
PR-PM2 (2017)	[11]	$α T r = e x p 2 c 1 1 - T r - c 2 1 - T r 2 + 23 c 1 1 - T r 3$ $c 1 = 0.3464 + 1.4886 ω - 0.13347 ω 2$ $c 2 = 0.18339 + 1.8435 ω - 0.26929 ω 2$

Table 2 Basic information of 11 kinds of compounds

物质类别	中文名称	数据点数	蒸汽压数据参考文献
正构烷烃	C₁~C₂₀、C₂₂、C₂₄	1434	[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]
正构醇	C₁~C₁₀、C₁₂、C₁₄、C₁₆、C₁₈	826	[29,46,47,48,49,50,51,52,53,54,55,56,57]
芳烃	苯、甲苯、乙苯、二甲苯(邻、间、对)、萘、1-甲基萘	559	[34,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72]
卤代烃	1,1-二氯-1-氟乙烷(R-141b)	75	[73,74]
气体	氮气、氩、氪、氙、一氧化碳、氨	404	[28,75,76,77,78,79,80,81,82,83,84]
酸	乙酸、丙酸	66	[85,86,87]
醚	二甲醚、二丙醚、乙丙醚、甲基正丁基醚	142	[88,89,90,91]
酮	丙酮	47	[92,93,94]
酯	甲酸甲酯、乙酸甲酯、丙酸甲酯、丁酸甲酯、异丁酸甲酯、甲酸乙酯、乙酸乙酯、丙酸乙酯、甲酸丙酯、乙酸丙酯	645	[87,95,96,97,98]
杂环	1,4-二烷	66	[99,100]
水	水	189	[101]

Fig.1 ARDs of vapor pressures for different kinds of compounds with generalized alpha functions

Fig.2 ARDs of vapor pressures for normal alkanes with generalized alpha functions

Fig.3 RDs of generalized alpha functions on prediction of vapor pressures ofn-butane with increasing reduced temperature

Fig.4 ARDs of vapor pressures for normal alcohols with generalized alpha functions

Fig.5 RDs of generalized alpha functions on prediction of vapor pressures ofn-butanol with increasing reduced temperature

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