Mass-heat analogy and global optimization of mass exchange network based on generalized heat exchanger network

doi:10.16085/j.issn.1000-6613.2024-0020

Abstract

Abstract:

Mass exchange network is an important way to recover pollutants or impurities efficiently and economically in process systems, the small-scale characteristics of which have certain limitations on their solution domain and global optimization performance. Based on the theory of mass transfer and energy transfer comparison, this paper assumed the tray mass providing an effective mass transfer of per unit height tray, established the comparison relationship between tray column tower and heat exchanger. On this basis, the small-scale mass exchange network was analogized to the generalized heat exchanger network. Furthermore, the node-based non-structural model and the random walk algorithm with compulsive evolution were used to globally optimize the generalized heat exchanger network. Finally, the optimal generalized heat exchanger network was regressed to mass exchanger network to make it satisfy the mass transfer feasibilities. The example analysis showed that the method can effectively expand the search space of mass exchanger network, thus to improve the diversity of stream matching and global optimization performance. Meanwhile, flexibly adjusting the analogy scale and coordination coefficient can further enrich the optimization path of MEN, and improve the quality of the optimal solutions. The obtained optimal solutions for the R2S3 and R2S2 examples were both superior to the best solutions in the literature.

Key words: process system, mass exchange network, mass-heat analogy, generalized heat exchanger network, global optimization, random walk algorithm with compulsive evolution

摘要：

质量交换网络是过程系统高效经济回收污染物或杂质的重要途径，其中组分浓度的小尺度特征对于其求解域和全局优化性能存在一定限制。基于质量传递和能量传递比拟理论，本文假设了单位高度塔板提供有效传质的塔板质量，建立了非连续传质的板式塔和广义换热器的比拟关系；在此基础上，将小尺度质量交换网络比拟为广义换热网络，进而采用节点非结构模型和强制进化随机游走算法对广义换热网络进行全局优化；最后，将优化所得的广义换热网络回归为质量交换网络，使其满足传质可行性约束。算例分析表明，该方法可有效拓展质量交换网络搜索空间，提升流股匹配的多样性和全局优化性能。同时，灵活调整比拟尺度和协调系数能够进一步丰富优化路径，提升最优解的质量，获得了R2S3算例和R2S2算例优于文献最优的结构。

关键词: 过程系统, 质量交换网络, 质能比拟, 广义换热网络, 全局优化, 强制进化随机游走算法

CLC Number:

TQ021.8

XIAO Yuan, CHEN Yi, LIU Siqi, CUI Guomin. Mass-heat analogy and global optimization of mass exchange network based on generalized heat exchanger network[J]. Chemical Industry and Engineering Progress, 2025, 44(1): 121-134.

肖媛, 陈怡, 刘思琪, 崔国民. 基于广义换热网络的质量交换网络质能比拟及全局优化[J]. 化工进展, 2025, 44(1): 121-134.

Figures/Tables 25

比拟量	质量（板式塔）	能量（换热器）
强度量	组分浓度C/kg·kg^–1	温度T/K
传递性能	对流传质系数h_m/m·s^–1	表面传热系数h/kW·m^-2·K^-1
热质交换设备
传递驱动力	浓度差(LMCD、AMCD)/kg·kg^–1	温度差(LMTD、AMTD)/K
传递性能	总传质系数K_m/m·s^–1	总传热系数K/kW·m^-2·K^-1
传递量	传质量M_A/kg·s^–1	换热量Q_A/kW
设备尺寸	提供有效传质的塔板质量 $μ_{0} \times N$ /kg·m^–1	换热面积A/m²

比拟量	质量（板式塔）	能量（换热器）
强度量	组分浓度C/kg·kg^–1	温度T/K
传递性能	对流传质系数h_m/m·s^–1	表面传热系数h/kW·m^-2·K^-1
热质交换设备
传递驱动力	浓度差(LMCD、AMCD)/kg·kg^–1	温度差(LMTD、AMTD)/K
传递性能	总传质系数K_m/m·s^–1	总传热系数K/kW·m^-2·K^-1
传递量	传质量M_A/kg·s^–1	换热量Q_A/kW
设备尺寸	提供有效传质的塔板质量 $μ_{0} \times N$ /kg·m^–1	换热面积A/m²

变量	释义	下角标
QH_i,im,ib	各热节点上换热单元的换热量，kW	i∈N_R, im∈N_sgH,ib∈N_bH, j∈N_s, jm∈N_sgC,jb∈N_bC, ip∈3, jp∈3。 ip=1: 流股编号； ip=2: 主节点编号； ip=3: 分支编号
QC_j,jm,jb	各冷节点上换热单元的换热量，kW
SPH_i,im,ib	热流股分支的分流比
SPC_j,jm,jb	冷流股分支的分流比
$F c_{p_{j}}$	冷流股的热容流率，kW/K
Z_i,im,ib	每个热节点上有无换热单元的 0-1变量
NH_C_i,im,ib,ip	热节点连的冷节点
NC_H_j,jm,jb,jp	冷节点连的热节点

变量	释义	下角标
QH_i,im,ib	各热节点上换热单元的换热量，kW	i∈N_R, im∈N_sgH,ib∈N_bH, j∈N_s, jm∈N_sgC,jb∈N_bC, ip∈3, jp∈3。 ip=1: 流股编号； ip=2: 主节点编号； ip=3: 分支编号
QC_j,jm,jb	各冷节点上换热单元的换热量，kW
SPH_i,im,ib	热流股分支的分流比
SPC_j,jm,jb	冷流股分支的分流比
$F c_{p_{j}}$	冷流股的热容流率，kW/K
Z_i,im,ib	每个热节点上有无换热单元的 0-1变量
NH_C_i,im,ib,ip	热节点连的冷节点
NC_H_j,jm,jb,jp	冷节点连的热节点

GHEN-NNM	MEN-NNM	释义
QH_i,im,ib	MR_i,im,ib $= Q H_{i, i m, i b} \times C_{m a x} / (T_{m a x} \times c_{p_{j}})$	富节点上换热单元的传质量，kg/s
QC_j,jm,jb	MS_j,jm,jb $= Q C_{j, j m, j b} \times C_{m a x} / (T_{m a x} \times c_{p_{j}})$	贫节点上换热单元的换热量，kg/s
SPH_i,im,ib	SPR_i,im,ib = SPH_i,im,ib	富流股分支的分流比
SPC_j,jm,jb	SPS_j,jm,jb = SPC_j,jm,jb	贫流股分支的分流比
$F C_{p_{j}}$	L_j = $F c_{p_{j}} / c_{p}$	贫流股的热容流率，kg/s
Z_i,im,ib	ZZ_i,im,ib = Z_i,im,ib	每个富节点有无换热单元的0-1变量
NH_C_i,im,ib,ip	NR_S_i,im,ib,ip = NH_C_i,im,ib,ip	富节点连的贫节点
NC_H_j,jm,jb,jp	NS_R_j,jm,jb,jp = NC_H_j,jm,jb,jp	贫节点连的富节点
i∈N_R, im∈N_sgH, ib∈N_bH; j∈N_s, jm∈N_sgC, jb∈N_bC, ip∈3, jp∈3。 ip=1, 流股编号; ip=2, 主节点编号; ip=3, 分支编号

GHEN-NNM	MEN-NNM	释义
QH_i,im,ib	MR_i,im,ib $= Q H_{i, i m, i b} \times C_{m a x} / (T_{m a x} \times c_{p_{j}})$	富节点上换热单元的传质量，kg/s
QC_j,jm,jb	MS_j,jm,jb $= Q C_{j, j m, j b} \times C_{m a x} / (T_{m a x} \times c_{p_{j}})$	贫节点上换热单元的换热量，kg/s
SPH_i,im,ib	SPR_i,im,ib = SPH_i,im,ib	富流股分支的分流比
SPC_j,jm,jb	SPS_j,jm,jb = SPC_j,jm,jb	贫流股分支的分流比
$F C_{p_{j}}$	L_j = $F c_{p_{j}} / c_{p}$	贫流股的热容流率，kg/s
Z_i,im,ib	ZZ_i,im,ib = Z_i,im,ib	每个富节点有无换热单元的0-1变量
NH_C_i,im,ib,ip	NR_S_i,im,ib,ip = NH_C_i,im,ib,ip	富节点连的贫节点
NC_H_j,jm,jb,jp	NS_R_j,jm,jb,jp = NC_H_j,jm,jb,jp	贫节点连的富节点
i∈N_R, im∈N_sgH, ib∈N_bH; j∈N_s, jm∈N_sgC, jb∈N_bC, ip∈3, jp∈3。 ip=1, 流股编号; ip=2, 主节点编号; ip=3, 分支编号

MEN(R2S3)参数^[26]
MEN	G(L)/kg·s^–1	$C y_{i}^{i n} (C x_{j}^{i n})$ /kg·kg^–1	$C y_{i}^{o u t} (C x_{j}^{o u t})$ /kg·kg^–1		$C y_{j}^{* i n}$ /kg·kg^–1	$C y_{j}^{* o u t}$ /kg·kg^–1
R1	2	0.05	0.01
R2	1	0.03	0.006
S1	5	0.005	0.015		0.01	0.03
S2	3	0.01	0.03		0.0153	0.0459
S3	∞	0.0013	0.015		0.001923	0.01165
S1、S2免费，S3运行费用系数： $f_{o, 3} /$ USD·kg^-1·s=309600；传质设备费用系数： $f_{C, N}$ /USD·a^-1=4552
比拟参数
C_max/kg·kg^–1	T_max/℃	(T_max/C_max)/℃·kg^–1·kg	ρ/kg·m^-3	c_p /kJ·kg^-1·K^-1	μ₀/kg·m^-1	C₀
0.005	100	10000	1000	10	10000	1.0
GHEN(H2C3)参数
GHEN	Fc_p =Gc_p /kJ·K^-1·s^-1		Tⁱⁿ/℃			T^out/℃
H1	20		500			100
H2	10		300			60
C1	50		100			300
C2	30		153			459
C3	10		19.23			116.5
C1、C2免费，C3运行费用系数： $f_{o, 3} /$ USD·kg^-1·s=309600；换热器面积费用系数： $f_{C, N}$ /USD·a^-1=4552

MEN(R2S3)参数^[26]
MEN	G(L)/kg·s^–1	$C y_{i}^{i n} (C x_{j}^{i n})$ /kg·kg^–1	$C y_{i}^{o u t} (C x_{j}^{o u t})$ /kg·kg^–1		$C y_{j}^{* i n}$ /kg·kg^–1	$C y_{j}^{* o u t}$ /kg·kg^–1
R1	2	0.05	0.01
R2	1	0.03	0.006
S1	5	0.005	0.015		0.01	0.03
S2	3	0.01	0.03		0.0153	0.0459
S3	∞	0.0013	0.015		0.001923	0.01165
S1、S2免费，S3运行费用系数： $f_{o, 3} /$ USD·kg^-1·s=309600；传质设备费用系数： $f_{C, N}$ /USD·a^-1=4552
比拟参数
C_max/kg·kg^–1	T_max/℃	(T_max/C_max)/℃·kg^–1·kg	ρ/kg·m^-3	c_p /kJ·kg^-1·K^-1	μ₀/kg·m^-1	C₀
0.005	100	10000	1000	10	10000	1.0
GHEN(H2C3)参数
GHEN	Fc_p =Gc_p /kJ·K^-1·s^-1		Tⁱⁿ/℃			T^out/℃
H1	20		500			100
H2	10		300			60
C1	50		100			300
C2	30		153			459
C3	10		19.23			116.5
C1、C2免费，C3运行费用系数： $f_{o, 3} /$ USD·kg^-1·s=309600；换热器面积费用系数： $f_{C, N}$ /USD·a^-1=4552

网络	X	$Δ L_{X}$	X_min	φ_ev,X	X_g	φ_g,x	τ
GHEN	QH	100	5	0.95×0.3	100	0.001	0.01
	SP	0.03	0.01	0.05×0.3	1
	$F c_{p_{N S}}$	0.01	0.01	0.0001	1
MEN	MR	5×10^-5	5×10^-5	0.7×0.3	0.002	0.005	0.001
	SP	0.05	0.01	0.3×0.3	1
	L_NS	0.03	0.02	0.01	0.5

网络	X	$Δ L_{X}$	X_min	φ_ev,X	X_g	φ_g,x	τ
GHEN	QH	100	5	0.95×0.3	100	0.001	0.01
	SP	0.03	0.01	0.05×0.3	1
	$F c_{p_{N S}}$	0.01	0.01	0.0001	1
MEN	MR	5×10^-5	5×10^-5	0.7×0.3	0.002	0.005	0.001
	SP	0.05	0.01	0.3×0.3	1
	L_NS	0.03	0.02	0.01	0.5

文献	方法	单元数	塔板总数	AOC/USD·a^-1	TAC/USD·a^-1
Hallale and Fraser ^[26]	PDM	7	—	—	345416
Isafiade and Fraser ^[27]	IBMS & DICOPT	6	26	219816	338168
Comeaus ^[28]	—	7	28	205800	333300
Xiao等^[23]	NV-NSM & RWCE	6	26	217938	336290
Xiao等^[23]	NV-NSM & RWCE	7	21	233911	329503
本文（图9）	质能比拟	8	21	227607	323199
本文（图12）	质能比拟	6	24	212552	321800

GHEN	10⁴℃·kg^–1·kg		2×10⁴℃·kg^–1·kg		10⁵℃·kg^–1·kg
GHEN	Tⁱⁿ/℃	T^out/℃	Tⁱⁿ/℃	T^out/℃	Tⁱⁿ/℃	T^out/℃
H1	250	50	500	100	2500	500
H2	150	30	300	60	1500	300
C1	50	150	100	300	500	1500
C2	76.5	229.5	153	459	765	2295
C3	9.615	58.25	19.23	116.5	96.15	582.5

MEN(R2S2)参数^[26]
MEN	G(L)/kg·s^–1	$C y_{i}^{i n} (C x_{j}^{i n})$ /kg·kg^–1	$C y_{i}^{o u t} (C x_{j}^{o u t})$ /kg·kg^–1	$C y_{j}^{* i n}$ /kg·kg^–1	$C y_{j}^{* o u t}$ /kg·kg^–1
R1	0.9	0.07	0.0003	—	—
R2	0.1	0.051	0.0001	—	—
S1	2.3	0.0006	0.031	0.00087	0.04495
S2	∞	0.0002	0.0035	0.000052	0.00091
S1运行费用系数： $f_{o, 1} /$ USD·kg^-1·s=117360；S2运行费用系数： $f_{o, 2} /$ USD·kg^-1·s=176040；传质设备费用系数： $f_{C, N}$ /USD·a^-1=4552
比拟参数
ρ/kg·m^-3	C_p /kJ·kg^-1·K^-1		μ₀/ kg·m^-1	C₀
1000	10		10000	1.0
GHEN(H2C2)参数
GHEN		H1	H2	C1	C2
Fc_p =Gc_p /kJ·K^-1·s^-1		9	1	23	10
C1运行费用系数： $f_{o, 1} /$ USD·kg^-1·s=117360；C2运行费用系数： $f_{o, 2} /$ USD·kg^-1·s=176040；换热器面积费用系数： $f_{C, N}$ /USD·a^-1=4552

MEN(R2S2)参数^[26]
MEN	G(L)/kg·s^–1	$C y_{i}^{i n} (C x_{j}^{i n})$ /kg·kg^–1	$C y_{i}^{o u t} (C x_{j}^{o u t})$ /kg·kg^–1	$C y_{j}^{* i n}$ /kg·kg^–1	$C y_{j}^{* o u t}$ /kg·kg^–1
R1	0.9	0.07	0.0003	—	—
R2	0.1	0.051	0.0001	—	—
S1	2.3	0.0006	0.031	0.00087	0.04495
S2	∞	0.0002	0.0035	0.000052	0.00091
S1运行费用系数： $f_{o, 1} /$ USD·kg^-1·s=117360；S2运行费用系数： $f_{o, 2} /$ USD·kg^-1·s=176040；传质设备费用系数： $f_{C, N}$ /USD·a^-1=4552
比拟参数
ρ/kg·m^-3	C_p /kJ·kg^-1·K^-1		μ₀/ kg·m^-1	C₀
1000	10		10000	1.0
GHEN(H2C2)参数
GHEN		H1	H2	C1	C2
Fc_p =Gc_p /kJ·K^-1·s^-1		9	1	23	10
C1运行费用系数： $f_{o, 1} /$ USD·kg^-1·s=117360；C2运行费用系数： $f_{o, 2} /$ USD·kg^-1·s=176040；换热器面积费用系数： $f_{C, N}$ /USD·a^-1=4552

GHEN	10⁴℃·kg^–1·kg		2×10⁴℃·kg^–1·kg		10⁵℃·kg^–1·kg		2×10⁵℃·kg^–1·kg
GHEN	Tⁱⁿ/℃	T^out/℃	Tⁱⁿ/℃	T^out/℃	Tⁱⁿ/℃	T^out/℃	Tⁱⁿ/℃	T^out/℃
H1	700	3	1400	6	7000	30	14000	60
H2	510	1	1020	2	5100	10	10200	20
C1	8.7	449.5	17.4	899	87	4.495	174	8990
C2	0.52	9.1	1.04	18.2	5.2	91	10.4	182

算例	（T_max/C_max）/℃·kg^–1·kg	GHEN/USD·a^-1	MEN/USD·a^-1	单元数
R2S3	10⁴	319617	340020	8
	2×10⁴	321779	323199	8
	10⁵	340130	342077	8
R2S2	10⁵	418365	411043	5
	2×10⁴	444661	431570	5
	10⁴	509784	437389	4

协调系数	GHEN/USD·a^-1	MEN/USD·a^-1	总塔板数	单元个数
0.8	323983	337797	21	6
1.0	321779	323199	21	8
1.1	321792	321800	24	6
1.2	320281	338702	23	7
1.3	320915	334183	26	8
1.4	321098	333534	22	5
2.0	319995	334603	22	5

文献	方法	单元数	塔板总数	AOC/USD·a^-1	TAC/USD·a^-1
Papalexandri等^[29]	Hyper MINLP	3	—	—	918000
Azeez等^[30]	SBS	5	37	300433	469968
Hallale and Fraser ^[26]	PDM	5	25	313293	427093
都健等^[12]	SWS & GA-SA	4	22	321500	421644
侯创等^[7]	SWS & 取整函数	4	20	320126	411166
Zhou等^[9]	NNM & RWCE	4	21	310332	410476
本文（图10）	质能比拟	5	17	324555	411043
本文（图13）	质能比拟	4	20	318124	409164

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