Global optimization of mixed gas multistage membrane separation system based on decomposition algorithm

doi:10.11949/0438-1157.20250261

Abstract

Abstract:

Mixed gas multistage membrane separation is an efficient separation technology, which achieves efficient separation of various components in mixed gas through the synergistic effect of multistage membrane system. However, optimizing its design requires establishing complex Mixed-integer nonlinear programming (MINLP) models, which are difficult to solve. This paper presents a global optimization method for multi-stage membrane separation systems for gas mixture. Using a decomposition algorithm, the complex MINLP problem is decomposed into a mixed-integer programming (MIP) problem and two nonlinear programming (NLP) problems. The separation sequences are enumerated via the MIP model, followed by the implementation of a multithreaded parallel computing approach where each thread sequentially optimizes the assigned sequences through two NLP models of distinct accuracy levels. The global optimal solution is determined by integrating all optimization results. Through a case study of natural gas sweetening, the total annual cost (tac) of the membrane separation system obtained is 7.35% lower than the optimal result in the literature. In addition, a case of CO₂ capture from blast furnace gas verifies that the method can be extended to the optimization of membrane separation system with variable pressure.

Key words: separation, membrane, optimization design, superstructure, parallel computing

摘要：

混合气体多级膜分离是一种高效的分离技术，通过多级膜系统的协同作用实现混合气体中各组分的高效分离。但对其进行优化设计时需要建立复杂的混合整数非线性规划（MINLP）模型，求解困难。本文提出了一种混合气体多级膜分离系统的全局优化方法，使用分解算法将复杂的MINLP问题分解为混合整数规划（MIP）问题和非线性规划（NLP）问题，通过MIP模型枚举分离序列，引入多线程并行计算方法，将各线程中的分离序列代入两个不同精度的NLP模型中序贯优化，综合所有优化结果后确定全局最优解。通过天然气脱硫案例得到的膜分离系统年总成本（tac）较文献最优结果降低7.35%，通过高炉煤气中捕集CO₂案例验证了该方法可拓展至可变压力的膜分离系统优化。

关键词: 分离, 膜, 优化设计, 超结构, 并行计算

CLC Number:

TQ 028.1

Jie WANG, Qucheng LIN, Xianming ZHANG. Global optimization of mixed gas multistage membrane separation system based on decomposition algorithm[J]. CIESC Journal, 2025, 76(9): 4670-4682.

王杰, 林渠成, 张先明. 基于分解算法的混合气体多级膜分离系统全局优化[J]. 化工学报, 2025, 76(9): 4670-4682.

Figures/Tables 13

Fig.1 Superstructure of multistage membrane separation systems

Fig.2 Algorithm diagram

Table 1 Related parameters of example 1

参数	数值
初始流量/(kmol/h)	36
进料压力/MPa	3.500
渗透压力/MPa	0.105
温度/K	313.15
CO₂纯度/%	2
CH₄回收率/%	90
进料组成/%（摩尔分数）	CO₂: 19, CH₄: 73,C₂₊: 7, H₂S: 1
渗透率/(kmol/(m²·MPa·h))	0.10656/0.005328/0.0021312/0.085248

Table 2 Objective function-related parameters of Example 1

参数	描述	数值	参数	描述	数值
$C c p$	压缩机成本/(USD/kW)	1000	$K s g$	原始天然气成本/(USD/1000 m³)	35
$f a c ̂$	固定成本的年化因子/%	27	$K w k$	营运资本比率/%	10
$K h v$	天然气总热值/(MJ/m³)	43	$t m$	膜使用寿命/a	3
$K m$	膜成本/(USD/m²)	200	$t o p$	年工作时间/(d/a)	300
$K m r$	膜更换费用/(USD/m²)	90	$η'$	压缩机效率/%	70
$K m t$	维修率/(%/a)	5

Table 2 Objective function-related parameters of Example 1

参数	描述	数值	参数	描述	数值
$C c p$	压缩机成本/(USD/kW)	1000	$K s g$	原始天然气成本/(USD/1000 m³)	35
$f a c ̂$	固定成本的年化因子/%	27	$K w k$	营运资本比率/%	10
$K h v$	天然气总热值/(MJ/m³)	43	$t m$	膜使用寿命/a	3
$K m$	膜成本/(USD/m²)	200	$t o p$	年工作时间/(d/a)	300
$K m r$	膜更换费用/(USD/m²)	90	$η'$	压缩机效率/%	70
$K m t$	维修率/(%/a)	5

Fig.3 Optimal membrane separation system of Example 1[24]

Table 3 Optimization results and literature comparison of Example 1

项目	Qi和Henson^[17]	Ramírez-Santos等^[22]	Taifan和Maravelias^[24]	本研究
第一级膜面积/m²	182.75	127.65	140.06	174.75
第二级膜面积/m²	197.92	181.08	151.40	111.58
第三级膜面积/m²	13.33	14.00	10.63	6.00
总膜面积/m²	394.00	322.73	302.09	292.33
压缩机功率/kW	12.61	12.87	10.66	7.31
CH₄纯度/%	88.71	89.09	88.84	88.75
年总成本/(USD/1000 m³)	10.971	9.095	8.501	7.876
求解时间/s	—	—	1185	698

Table 4 Comparison of optimization results of Example 1 with simulation results of PRO Ⅱ

项目	$F 1$	$F 1 R T O T - F 2$	$F 1 P T O T$	$F 2 R T O T$	$F 2 P T O T - F 3$	$F 3 R T O T$	$F 3 P T O T$
本研究	38.24	29.53	8.71	26.65	2.88	2.24	0.64
PRO Ⅱ	38.24	29.46	8.78	26.59	2.87	2.24	0.63
误差/%	0	0.24	-0.80	0.23	0.35	0	1.59

Table 4 Comparison of optimization results of Example 1 with simulation results of PRO Ⅱ

项目	$F 1$	$F 1 R T O T - F 2$	$F 1 P T O T$	$F 2 R T O T$	$F 2 P T O T - F 3$	$F 3 R T O T$	$F 3 P T O T$
本研究	38.24	29.53	8.71	26.65	2.88	2.24	0.64
PRO Ⅱ	38.24	29.46	8.78	26.59	2.87	2.24	0.63
误差/%	0	0.24	-0.80	0.23	0.35	0	1.59

Table 5 Related parameters of Example 2

参数	数值
初始流量/(mol/s)	1240.0
初始进料流股压力/bar	1.00
温度/K	308.15
N₂纯度/%	1
CO₂回收率/%	90
进料组成/%（摩尔分数）	CO₂: 23.2, CO: 22.6,N₂: 50.3, H₂: 3.9
渗透率/(mol/(m²·bar·s))	1000/20/15/85

Table 6 Objective function-related parameters of Example 2

参数	描述	数值	参数	描述	数值
$f a c ̂$	固定成本的年化因子/%	8.54	$M F c p$	压缩机材料因子	1.4
$C c p$	压缩机成本/EUR	10⁶	$I C F$	间接成本因素	1.8
$C v p$	真空泵成本因子/(EUR/kW)	1500	$t o p$	膜每年使用时间/(h/year)	8322
$C O E ̂$	电力成本/(EUR/(kW·h))	0.044	$U F 2000$	更新因子	1.42
$K m$	膜成本/(EUR/m²)	40	$γ$	绝热比	1.36
$K m f$	基础框架成本/EUR	286×10³	$η$	压缩机等熵效率/%	85
$K m r$	膜更换费用/(EUR/m²)	25	$λ$	真空泵等熵效率/%	85
$M C O 2$	CO₂摩尔质量/(g/mol)	44.01	$ν$	膜年更换率/%	20
$M D F c p$	压缩机模块因子	2.72	$φ$	机械效率/%	95

Table 6 Objective function-related parameters of Example 2

参数	描述	数值	参数	描述	数值
$f a c ̂$	固定成本的年化因子/%	8.54	$M F c p$	压缩机材料因子	1.4
$C c p$	压缩机成本/EUR	10⁶	$I C F$	间接成本因素	1.8
$C v p$	真空泵成本因子/(EUR/kW)	1500	$t o p$	膜每年使用时间/(h/year)	8322
$C O E ̂$	电力成本/(EUR/(kW·h))	0.044	$U F 2000$	更新因子	1.42
$K m$	膜成本/(EUR/m²)	40	$γ$	绝热比	1.36
$K m f$	基础框架成本/EUR	286×10³	$η$	压缩机等熵效率/%	85
$K m r$	膜更换费用/(EUR/m²)	25	$λ$	真空泵等熵效率/%	85
$M C O 2$	CO₂摩尔质量/(g/mol)	44.01	$ν$	膜年更换率/%	20
$M D F c p$	压缩机模块因子	2.72	$φ$	机械效率/%	95

Fig.4 Optimal membrane separation system of Example 2[22]

Table 7 Otimization results and literature comparison of Example 2

项目	Ramírez-Santos等^[22]	恒压	变压
第一级膜面积/ m²	105192	92072	106672
第二级膜面积/ m²	3058	7333	7476
第三级膜面积/ m²	7229	3774	4490
总膜面积/ m²	115479	103179	118638
进料压力/bar	2.29	2.29	2.14
第一级渗透压力/bar	0.20	0.20	0.20
第二级渗透压力/bar	0.71	0.71	0.70
第三级渗透压力/bar	0.43	0.43	0.41
总功率/MW	—	7.67	7.53
年总成本/(EUR/t)	28.9	27.0	26.9
求解时间/s	—	16129	30172

Table 8 Comparison of optimization results of Example 2 under constant pressure with simulation results of PRO Ⅱ

项目	$F 1$	$F 1 R T O T$	$F 1 P T O T - F 2$	$F 2 R T O T - F 3$	$F 2 P T O T$	$F 3 R T O T$	$F 3 P T O T$
本研究	1398	971	427	235	192	158	77.1
PRO Ⅱ	1404	975	429	240	189	164	75.6
误差/%	-0.4	-0.4	-0.5	-2.1	1.6	-3.7	2.0

Table 8 Comparison of optimization results of Example 2 under constant pressure with simulation results of PRO Ⅱ

项目	$F 1$	$F 1 R T O T$	$F 1 P T O T - F 2$	$F 2 R T O T - F 3$	$F 2 P T O T$	$F 3 R T O T$	$F 3 P T O T$
本研究	1398	971	427	235	192	158	77.1
PRO Ⅱ	1404	975	429	240	189	164	75.6
误差/%	-0.4	-0.4	-0.5	-2.1	1.6	-3.7	2.0

Table 9 Comparison of optimization results of Example 2 under variable pressure with simulation results of PRO Ⅱ

项目	$F 1$	$F 1 R T O T$	$F 1 P T O T - F 2$	$F 2 R T O T - F 3$	$F 2 P T O T$	$F 3 R T O T$	$F 3 P T O T$
本研究	1417	971	446	269	177	177	92.1
PRO Ⅱ	1424	975	449	275	174	184	90.8
误差/%	-0.5	-0.4	-0.7	-2.2	1.7	-3.8	1.4

Table 9 Comparison of optimization results of Example 2 under variable pressure with simulation results of PRO Ⅱ

项目	$F 1$	$F 1 R T O T$	$F 1 P T O T - F 2$	$F 2 R T O T - F 3$	$F 2 P T O T$	$F 3 R T O T$	$F 3 P T O T$
本研究	1417	971	446	269	177	177	92.1
PRO Ⅱ	1424	975	449	275	174	184	90.8
误差/%	-0.5	-0.4	-0.7	-2.2	1.7	-3.8	1.4

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