Selection of cubic equations of state for separation of CO2-C2H6 azeotrope

doi:10.11949/0438-1157.20181314

Abstract

Abstract:

Accurate selection of vapor-liquid equilibrium thermodynamic model is essential to the design and operation of CO₂-C₂H₆ azeotrope separation process. On the basis of collected experimental data, the reliabilities of four types of cubic equations of state including vdW, RK, SRK and PR combined with vdW, Margles and CVD mixing rules for predicting vapor-liquid equilibrium properties of CO₂ pure substance, CO₂-C₂H₆ azeotrope and n-C₅H₁₂-CO₂-C₂H₆ ternary system are evaluated according to the principle of vapor-liquid fugacity equilibrium. In addition, average absolute deviation is adopted to select cubic state equations. The results show that the SRK equation of state calculates the vapor-liquid equilibrium property of CO₂ pure material with the highest precision. The PR equation of state combined with Margles mixing rule can accurately calculate the vapor-liquid equilibrium characteristics of CO₂-C₂H₆ azeotrope system. However, for n-C₅H₁₂-CO₂-C₂H₆ ternary system, the calculating accuracy of SRK combined with Margles mixing rule is obviously better than that of vdW, RK and PR. Furthermore, binary interaction parameters have obvious effects on the calculating accuracy of cubic equations of state for the mixture system. In order to improve the calculation accuracy, the binary interaction parameter is calibrated for all of the studied EOSs regarding CO₂-C₂H₆ azeotropic system and n-C₅H₁₂-CO₂-C₂H₆ ternary system.

Key words: CO₂-C₂H₆ azeotrope, cubic equation of state, fugacity equilibrium, mixing rule, binary interaction parameters, average absolute deviation

摘要：

汽液平衡热力学模型的准确选取对CO₂-C₂H₆共沸物分离流程的设计和操作分析至关重要。在汽液平衡实验数据的基础上，依据逸度平衡原则，评估vdW、RK、SRK和PR立方型状态方程结合vdW、Margles和CVD混合规则预测CO₂纯物质、CO₂-C₂H₆共沸物和n-C₅H₁₂-CO₂-C₂H₆三元体系汽液平衡的可靠性，采用平均绝对误差的方法进行状态方程的选取。结果表明：SRK状态方程计算CO₂纯物质汽液平衡性质的精度最高；PR状态方程结合Margles混合规则可以准确计算CO₂-C₂H₆共沸体系汽液平衡特性；对于n-C₅H₁₂-CO₂-C₂H₆三元体系，SRK状态方程结合Margles混合规则计算精度明显优于vdW、RK和PR状态方程。通过试差迭代法优化CO₂-C₂H₆共沸体系和n-C₅H₁₂-CO₂-C₂H₆三元体系的二元交互作用参数，状态方程的计算精度得到明显提高。

关键词: CO₂-C₂H₆共沸物, 立方型状态方程, 逸度平衡, 混合规则, 二元交互作用参数, 平均绝对误差

CLC Number:

TE 645

Haiqin WANG, Minglong FAN, Zubin ZHANG. Selection of cubic equations of state for separation of CO₂-C₂H₆ azeotrope[J]. CIESC Journal, 2019, 70(9): 3228-3237.

王海琴, 范明龙, 张足斌. CO₂-C₂H₆共沸物分离的立方型状态方程选取[J]. 化工学报, 2019, 70(9): 3228-3237.

Figures/Tables 12

Table 1 Formula for cubic equations of state and fugacity coefficient

状态方程	公式形式	特征常数	纯物质逸度系数	混合物逸度系数
vdW	$p = R T V - b - a V 2$	$a = 2764 R 2 T c 2 p c$ $b = 18 R T c p c$	$l n φ = V V - b - 2 a R T V - l n p V - b R T - 1$	$l n φ i = l n V V - b m + 1 V - b m ? n b m ? n i - 1 R T V ? n 2 a m ? n i - l n Z$
RK	$p = R T V - b - a T 0.5 V V + b$	$a = 0.42748 R 2 T c 2.5 p c$ $b = 0.08664 R T c p c$	$l n φ = Z - 1 - l n p V - b R T - a b R T 1.5 l n 1 + b V$	$l n φ i = 1 b m ? n b m ? n i Z - 1 - l n p V - b m R T + a m b m R T 1.5 1 b m ? n b m ? n i - 1 a m ? n 2 a m ? n i l n 1 + b m V$
SRK	$p = R T V - b - a V V + b$	$a = 0.42748 R 2 T c 2 α T r, w p c$ $α T r, w 0.5 = 1 + m 1 - T r 0.5$ $m = 0.480 + 1.574 ω - 0.176 ω 2$ $T r = T T c$ ; $? b = 0.08664 R T c p c$	$l n φ = Z - 1 - l n p V - b R T - a b R T l n 1 + b V$	$l n φ i = 1 b m ? n b m ? n i Z - 1 - l n p V - b m R T + a m b m R T 1 b m ? n b m ? n i - 1 a m ? n 2 a m ? n i l n 1 + b m V$
PR	$p = R T V - b - a V V + b + b V - b$	$a = 0.457235 R 2 T c 2 α T r, w p c$ $α T r, w 0.5 = 1 + m 1 - T r 0.5$ $m = 0.37464 + 1.54226 ω - 0.26992 ω 2$ $T r = T T c$ ; $? b = 0.077796 R T c p c$	$l n φ = Z - 1 - l n p V - b R T - a 22 b R T × l n V + 2 + 1 b V - 2 - 1 b$	$l n φ i = 1 b m ? n b m ? n i Z - 1 - l n p V - b m R T + a m 22 b m R T × 1 b m ? n b m ? n i - 1 a m ? n 2 a m ? n i l n V + 2 + 1 b m V - 2 - 1 b m$

Table 1 Formula for cubic equations of state and fugacity coefficient

状态方程	公式形式	特征常数	纯物质逸度系数	混合物逸度系数
vdW	$p = R T V - b - a V 2$	$a = 2764 R 2 T c 2 p c$ $b = 18 R T c p c$	$l n φ = V V - b - 2 a R T V - l n p V - b R T - 1$	$l n φ i = l n V V - b m + 1 V - b m ? n b m ? n i - 1 R T V ? n 2 a m ? n i - l n Z$
RK	$p = R T V - b - a T 0.5 V V + b$	$a = 0.42748 R 2 T c 2.5 p c$ $b = 0.08664 R T c p c$	$l n φ = Z - 1 - l n p V - b R T - a b R T 1.5 l n 1 + b V$	$l n φ i = 1 b m ? n b m ? n i Z - 1 - l n p V - b m R T + a m b m R T 1.5 1 b m ? n b m ? n i - 1 a m ? n 2 a m ? n i l n 1 + b m V$
SRK	$p = R T V - b - a V V + b$	$a = 0.42748 R 2 T c 2 α T r, w p c$ $α T r, w 0.5 = 1 + m 1 - T r 0.5$ $m = 0.480 + 1.574 ω - 0.176 ω 2$ $T r = T T c$ ; $? b = 0.08664 R T c p c$	$l n φ = Z - 1 - l n p V - b R T - a b R T l n 1 + b V$	$l n φ i = 1 b m ? n b m ? n i Z - 1 - l n p V - b m R T + a m b m R T 1 b m ? n b m ? n i - 1 a m ? n 2 a m ? n i l n 1 + b m V$
PR	$p = R T V - b - a V V + b + b V - b$	$a = 0.457235 R 2 T c 2 α T r, w p c$ $α T r, w 0.5 = 1 + m 1 - T r 0.5$ $m = 0.37464 + 1.54226 ω - 0.26992 ω 2$ $T r = T T c$ ; $? b = 0.077796 R T c p c$	$l n φ = Z - 1 - l n p V - b R T - a 22 b R T × l n V + 2 + 1 b V - 2 - 1 b$	$l n φ i = 1 b m ? n b m ? n i Z - 1 - l n p V - b m R T + a m 22 b m R T × 1 b m ? n b m ? n i - 1 a m ? n 2 a m ? n i l n V + 2 + 1 b m V - 2 - 1 b m$

Table 2 Different mixed rules for cubic equations of state

混合规则	混合物特征常数
混合规则	a _m	b _m
vdW	$a m = ∑ i = 1 N ∑ j = 1 N y i y j a i j$ ; $?$ $a i j = a i a j 1 - k i j$ ; $? ? (n 2 a m) ? n i = 2 ∑ j = 1 N y j a i j$	$b m = ∑ i = 1 N ∑ j = 1 N y i y j b i j$ ; $? b i j = b i + b j 2;$ $? (n b m) ? n i = 2 ∑ j = 1 N y j b i j$
Margles	$a m = ∑ i = 1 N ∑ j = 1 N y i y j a i j$ ; $? a i j = a i a j 1 - y i k i j - y j k j i$ ; $?$ $? (n 2 a m) ? n i = 2 ∑ j = 1 N y j a i j$
CVD	$a m = ∑ i = 1 N ∑ j = 1 N y i y j a i j b b i j k i j$ ; $? a i j = a i a j$ ; $?$ $b i j = b i b j$ $? (n 2 a m) ? n i = 2 ∑ j = 1 N y i a i j b m b i j k i j + b i b m - 1 ∑ i = 1 N ∑ j = 1 N y i y j a i j k i j b b i j k i j$

Table 2 Different mixed rules for cubic equations of state

混合规则	混合物特征常数
混合规则	a _m	b _m
vdW	$a m = ∑ i = 1 N ∑ j = 1 N y i y j a i j$ ; $?$ $a i j = a i a j 1 - k i j$ ; $? ? (n 2 a m) ? n i = 2 ∑ j = 1 N y j a i j$	$b m = ∑ i = 1 N ∑ j = 1 N y i y j b i j$ ; $? b i j = b i + b j 2;$ $? (n b m) ? n i = 2 ∑ j = 1 N y j b i j$
Margles	$a m = ∑ i = 1 N ∑ j = 1 N y i y j a i j$ ; $? a i j = a i a j 1 - y i k i j - y j k j i$ ; $?$ $? (n 2 a m) ? n i = 2 ∑ j = 1 N y j a i j$
CVD	$a m = ∑ i = 1 N ∑ j = 1 N y i y j a i j b b i j k i j$ ; $? a i j = a i a j$ ; $?$ $b i j = b i b j$ $? (n 2 a m) ? n i = 2 ∑ j = 1 N y i a i j b m b i j k i j + b i b m - 1 ∑ i = 1 N ∑ j = 1 N y i y j a i j k i j b b i j k i j$

Fig.1 p-T diagram of vapor-liquid equilibrium for pure CO2

Fig.2 AAD of carbon dioxide vapor-liquid fugacity coefficient

Fig.3 p-xy diagram of vapor-liquid equilibrium data for CO2-C2H6 system

Fig.4 Sum of AAD of equilibrium constant of carbon dioxide and ethane at different temperatures

Table 3 AAD of EOSs with new binary interaction parameters kij of CO2-C2H6 azeotropic system

温度、压力范围	混合规则	状态方程	$k C O 2, C 2 H 6$	$k C 2 H 6, C O 2$	$A A D C O 2 + A A D C 2 H 6$
温度、压力范围	混合规则	状态方程	$k C O 2, C 2 H 6$	$k C 2 H 6, C O 2$	优化后	优化前
244.6～288 K, 1.43～5.63 MPa	vdW	vdW	0.1589	0.1424	0.2243	0.4435
		RK	0.2137	0.2937	0.2889	0.3212
		SRK	0.3257	0.2077	0.2406	0.2691
		PR	0.2850	0.1901	0.2160	0.2541
	Margles	vdW	0.1981	0.0948	0.2212	0.4460
		RK	0.2384	0.1600	0.2224	0.3122
		SRK	0.2754	0.1618	0.1889	0.2500
		PR	0.2668	0.1445	0.1841	0.2765
	CVD	vdW	0.1707	0.1994	0.3943	0.3952
		RK	0.1002	0.1661	0.3872	0.3894
		SRK	0.0099	0.2212	0.3854	0.3914
		PR	0.1779	0.1485	0.3859	0.3876

Table 3 AAD of EOSs with new binary interaction parameters kij of CO2-C2H6 azeotropic system

温度、压力范围	混合规则	状态方程	$k C O 2, C 2 H 6$	$k C 2 H 6, C O 2$	$A A D C O 2 + A A D C 2 H 6$
温度、压力范围	混合规则	状态方程	$k C O 2, C 2 H 6$	$k C 2 H 6, C O 2$	优化后	优化前
244.6～288 K, 1.43～5.63 MPa	vdW	vdW	0.1589	0.1424	0.2243	0.4435
		RK	0.2137	0.2937	0.2889	0.3212
		SRK	0.3257	0.2077	0.2406	0.2691
		PR	0.2850	0.1901	0.2160	0.2541
	Margles	vdW	0.1981	0.0948	0.2212	0.4460
		RK	0.2384	0.1600	0.2224	0.3122
		SRK	0.2754	0.1618	0.1889	0.2500
		PR	0.2668	0.1445	0.1841	0.2765
	CVD	vdW	0.1707	0.1994	0.3943	0.3952
		RK	0.1002	0.1661	0.3872	0.3894
		SRK	0.0099	0.2212	0.3854	0.3914
		PR	0.1779	0.1485	0.3859	0.3876

Fig.5 Ternary diagram of vapor-liquid equilibrium data for

Fig.6 Sum of AAD of equilibrium constant of carbon dioxide, ethane and n-pentane at different temperatures

Table 4 AAD of EOSs with new binary interaction parameters kij of CO2-C2H6-n-C5H12 ternary system

温度、压力范围	混合规则	EOS	$k C O 2, C 2 H 6$	$k C 2 H 6, C O 2$	$k C O 2, n - C 5 H 12$	$k n - C 5 H 12, C O 2$	$k C 2 H 6, n - C 5 H 12$	$k n - C 5 H 12, C 2 H 6$	$A A D C O 2 + A A D C 2 H 6 + A A D n - C 5 H 12$
温度、压力范围	混合规则	EOS	$k C O 2, C 2 H 6$	$k C 2 H 6, C O 2$	$k C O 2, n - C 5 H 12$	$k n - C 5 H 12, C O 2$	$k C 2 H 6, n - C 5 H 12$	$k n - C 5 H 12, C 2 H 6$	优化后	优化前
253～283 K, 0.6896～5.46 MPa	vdW	vdW	-0.6837	0.0875	0.9587	-0.3942	0.3495	0.3443	1.8261	2.9008
		RK	-0.6438	0.0921	0.9977	0.8328	0.2997	0.6939	1.8108	2.4772
		SRK	-0.4869	0.2656	1.0962	-0.4959	0.3834	3.4815	2.3790	2.5333
		PR	-0.3722	0.1694	1.0288	-0.4821	0.3716	3.3173	2.1855	2.5360
	Margles	vdW	1.7717	-0.4295	1.0937	0.0148	0.4679	-0.0955	1.6319	2.2090
		RK	1.9416	-0.5333	1.2083	0.3128	0.3289	-0.0739	1.0382	2.2478
		SRK	0.8621	-0.0888	1.3591	0.3986	0.3603	0.3035	0.9192	2.3850
		PR	0.9872	-0.1547	1.3165	0.3438	0.3594	0.2123	0.9584	2.3725
	CVD	vdW	-0.2764	-0.1711	-0.4405	-0.3684	0.3663	0.3177	2.9648	2.9725
		RK	-2.8909	-2.8909	-0.1320	-0.1320	-0.2929	-0.2929	2.1863	2.2646
		SRK	0.2875	0.3152	0.8591	0.8564	0.3805	0.3737	2.3533	2.3741
		PR	0.3293	0.3422	0.8850	0.8874	0.3924	0.3877	2.3303	2.3547

Table 4 AAD of EOSs with new binary interaction parameters kij of CO2-C2H6-n-C5H12 ternary system

温度、压力范围	混合规则	EOS	$k C O 2, C 2 H 6$	$k C 2 H 6, C O 2$	$k C O 2, n - C 5 H 12$	$k n - C 5 H 12, C O 2$	$k C 2 H 6, n - C 5 H 12$	$k n - C 5 H 12, C 2 H 6$	$A A D C O 2 + A A D C 2 H 6 + A A D n - C 5 H 12$
温度、压力范围	混合规则	EOS	$k C O 2, C 2 H 6$	$k C 2 H 6, C O 2$	$k C O 2, n - C 5 H 12$	$k n - C 5 H 12, C O 2$	$k C 2 H 6, n - C 5 H 12$	$k n - C 5 H 12, C 2 H 6$	优化后	优化前
253～283 K, 0.6896～5.46 MPa	vdW	vdW	-0.6837	0.0875	0.9587	-0.3942	0.3495	0.3443	1.8261	2.9008
		RK	-0.6438	0.0921	0.9977	0.8328	0.2997	0.6939	1.8108	2.4772
		SRK	-0.4869	0.2656	1.0962	-0.4959	0.3834	3.4815	2.3790	2.5333
		PR	-0.3722	0.1694	1.0288	-0.4821	0.3716	3.3173	2.1855	2.5360
	Margles	vdW	1.7717	-0.4295	1.0937	0.0148	0.4679	-0.0955	1.6319	2.2090
		RK	1.9416	-0.5333	1.2083	0.3128	0.3289	-0.0739	1.0382	2.2478
		SRK	0.8621	-0.0888	1.3591	0.3986	0.3603	0.3035	0.9192	2.3850
		PR	0.9872	-0.1547	1.3165	0.3438	0.3594	0.2123	0.9584	2.3725
	CVD	vdW	-0.2764	-0.1711	-0.4405	-0.3684	0.3663	0.3177	2.9648	2.9725
		RK	-2.8909	-2.8909	-0.1320	-0.1320	-0.2929	-0.2929	2.1863	2.2646
		SRK	0.2875	0.3152	0.8591	0.8564	0.3805	0.3737	2.3533	2.3741
		PR	0.3293	0.3422	0.8850	0.8874	0.3924	0.3877	2.3303	2.3547

Fig.7 Extractive distillation process designed

Table 5 Mole fraction and recovery rate of carbon dioxide and ethane on column top distillates

状态方程	产品摩尔分数/%				产品回收率/%
	CO₂			C₂H₆	CO₂	C₂H₆
	计算值	实验值^[38]	相对误差	C₂H₆	CO₂	C₂H₆
vdW	83.75	95.5	12.3	81.83	93.82	94.96
RK	94.63		0.91	95.46	98.08	96.56
SRK	95.19		0.32	96.93	98.68	98.09
PR	95.01		0.51	96.56	98.53	97.53

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Selection of cubic equations of state for separation of CO₂-C₂H₆ azeotrope

CO₂-C₂H₆共沸物分离的立方型状态方程选取

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