Deep learning model of fixed bed adsorption breakthrough curve hybrid-driven by data and physical information

doi:10.11949/0438-1157.20221609

Abstract

Abstract:

A deep learning model of fixed bed adsorption breakthrough curve hybrid-driven by the data and physical information (PINN_MOD) was proposed in this work. A combined method of external data constraint enhanced by penalty factors and residual-based adaptive refinement strategy was adopted to gradually optimize the neural network parameters to approximate the solution of the partial differential equation (PDE) of the dynamic binary gas adsorption process of fixed bed by minimizing the loss function. The physics-informed neural network (PINN) model was generally used to solve the forward and inverse problem of the one-dimensional single-component convection-diffusion and fixed-bed adsorption PDE models with high fidelity. However, there are convergence difficulties when it is used to solve the one-dimensional binary fixed-bed adsorption PDE model on long-time scale. In this paper, the traditional finite differential method (FDM) was first used to solve the PDE problem of one-dimensional binary fixed bed adsorption, and then the component concentration data in the spatiotemporal region obtained by FDM simulations were adopted as an external constraint of the PINN model to solve the PDE of one-dimensional binary fixed bed adsorption. Taking the CO₂/N₂ mixture (molar ratio 30∶70) adsorption models in fixed bed packed with different MOFs (CALF-20 and UTSA-16) as an example, the PINN_MOD model was used to calculate the outlet CO₂ breakthrough curve of fixed bed. The FDM calculation results can be well replicated, which proves that the model can effectively obtain high-fidelity PDE solutions only relying on a small amount of external data. It is confirmed that the PINN_MOD model could obtain the high fidelity solutions by relying on only a few external data. The proposed model is expected to play an important role in the development of novel metal-organic framework adsorbents for gas separation applications.

Key words: physics-informed neural network, finite differential method, fixed bed adsorption, breakthrough curve, convection-diffusion model, partial differential equation

摘要：

提出一种基于数据与物理信息混合驱动的固定床吸附穿透深度学习模型(PINN_MOD)，采用基于残差自适应网格加密策略联合惩罚因子增强外部数据约束方法，通过最小化损失函数逐步调整神经网络参数逼近固定床双组分气体动态吸附过程偏微分方程(PDE)的解。嵌入物理信息神经网络(PINN)模型可以高保真地求解一维单组分对流-扩散模型和一维单组分固定床吸附模型PDE的正向解和逆向解，但在求解长时间尺度一维双组分固定床吸附模型PDE时存在收敛困难。利用传统有限差分方法(FDM)首先计算一维双组分固定床吸附穿透PDE问题，然后将FDM模拟获得的时空区域内组分浓度数据作为外部约束，联合PINN模型一起求解一维双组分固定床吸附穿透PDE。以填充CALF-20和UTSA-16两种MOF材料的固定床吸附CO₂/N₂(摩尔比30∶70)混合物为例，采用PINN_MOD模型计算出组分CO₂固定床出口穿透曲线，能够较好地复制FDM计算结果，证实了该模型仅依赖于少量外部数据就能有效地获得PDE高保真解。PINN_MOD模型有望在开发面向气体分离应用的新型金属有机骨架(MOF)材料吸附剂方面发挥重要作用。

关键词: 内嵌物理信息神经网络, 有限差分方法, 固定床吸附, 穿透曲线, 对流-扩散模型, 偏微分方程

CLC Number:

TQ 028.8

Xuanjun WU, Chao WANG, Zijian CAO, Weiquan CAI. Deep learning model of fixed bed adsorption breakthrough curve hybrid-driven by data and physical information[J]. CIESC Journal, 2023, 74(3): 1145-1160.

吴选军, 王超, 曹子健, 蔡卫权. 数据与物理信息混合驱动的固定床吸附穿透深度学习模型[J]. 化工学报, 2023, 74(3): 1145-1160.

Figures/Tables 12

Fig.1 Scheme of 1-D gas adsorption modeling in fixed bed

Table 1 Dimensionless partial differential equations for gas adsorptions in fixed bed

方程	公式
质量平衡方程	$∂ c ¯ i ∂ τ + ∂ c ¯ i ∂ x ¯ + D i, g ∂ s ¯ i ∂ τ = 1 P e i ∂ 2 c ¯ i ∂ x ¯ 2$
LDF传质方程	$∂ s ¯ i ∂ τ = G i, f s ¯ i * - s ¯ i$
DSLangmuir方程	$s ¯ i * = 1 q s q s a t_a, i b 0_a, i e x p (- Δ H a, i / R T) c i, 0 c ¯ i 1 + ∑ j = 1 n b 0_a, j e x p (- Δ H a, j / R T) c j, 0 c ¯ j + q s a t_b, i b 0_b, i e x p (- Δ H b, i / R T) c i, 0 c ¯ i 1 + ∑ j = 1 n b 0_b, j e x p (- Δ H b, j / R T) c j, 0 c ¯ j$
初始条件与边界条件	$τ = 0 : c ¯ i = 0, s ¯ i = 0$
	$x ¯ = 0 : c ¯ i = c ¯ i, 0 (τ > 0)$ (Dirichlet)
	$x ¯ = 1 : ∂ c ¯ i ∂ x ¯ = 0 (τ > 0)$
	$x ¯ = 0 : ∂ c ¯ i ∂ x ¯ = 1 P e i (c ¯ i - c ¯ i, 0) (τ > 0)$ (Danckwerts)
无量纲中间变量	$c ¯ i = c i / c i, 0; s ¯ i = s i / s i, 0 ; x ¯ = x / L; D i, g = ρ s 1 - ε b ε b s i, 0 c i, 0; G i, f = k i, f L v; P e i = L v D i, a x$

Table 1 Dimensionless partial differential equations for gas adsorptions in fixed bed

方程	公式
质量平衡方程	$∂ c ¯ i ∂ τ + ∂ c ¯ i ∂ x ¯ + D i, g ∂ s ¯ i ∂ τ = 1 P e i ∂ 2 c ¯ i ∂ x ¯ 2$
LDF传质方程	$∂ s ¯ i ∂ τ = G i, f s ¯ i * - s ¯ i$
DSLangmuir方程	$s ¯ i * = 1 q s q s a t_a, i b 0_a, i e x p (- Δ H a, i / R T) c i, 0 c ¯ i 1 + ∑ j = 1 n b 0_a, j e x p (- Δ H a, j / R T) c j, 0 c ¯ j + q s a t_b, i b 0_b, i e x p (- Δ H b, i / R T) c i, 0 c ¯ i 1 + ∑ j = 1 n b 0_b, j e x p (- Δ H b, j / R T) c j, 0 c ¯ j$
初始条件与边界条件	$τ = 0 : c ¯ i = 0, s ¯ i = 0$
	$x ¯ = 0 : c ¯ i = c ¯ i, 0 (τ > 0)$ (Dirichlet)
	$x ¯ = 1 : ∂ c ¯ i ∂ x ¯ = 0 (τ > 0)$
	$x ¯ = 0 : ∂ c ¯ i ∂ x ¯ = 1 P e i (c ¯ i - c ¯ i, 0) (τ > 0)$ (Danckwerts)
无量纲中间变量	$c ¯ i = c i / c i, 0; s ¯ i = s i / s i, 0 ; x ¯ = x / L; D i, g = ρ s 1 - ε b ε b s i, 0 c i, 0; G i, f = k i, f L v; P e i = L v D i, a x$

Fig.2 Structural framework of physics-informed deep learning for gas adsorption in fixed bed

Fig.3 Breakthrough curves of convection-diffusion models in fixed bed

Fig.4 Contour of concentration errors in the time-space domain for convection-diffusion PINN models

Fig.5 Single-component breakthrough curves of dispersed adsorption models in fixed bed

Fig.6 Time-space domain for single-component gas adsorption models (black lines refer to the isohypse)

Table 2 The fitting parameters for DSLangmuir isotherms of single component gas adsorbed in various MOFs

MOFs	CO₂			N₂
MOFs	q_sat/(mol/kg)	b₀/Pa^-1	-ΔH(kJ/mol)	q_sat/(mol/kg)	b₀/Pa^-1	-ΔH/(kJ/mol)
UTSA-16	1.406	3.257×10^-11	54.38	3.419	1.972×10^-9	13.79
UTSA-16	4.077	1.891×10^-5	21.45^②	—^③	—^③	—^③
CALF-20	2.086	7.379×10^-8①	42.16	2.457	1.562×10^-8①	18.78
CALF-20	2.525	1.105×10^-8①	34.74^②	—^③	—^③	—^③

Fig.7 Isotherms of gases adsorbed in various MOF materials (solid points refer to the experimental data from the references[33, 50-51] and solid lines refer to dual-site Langmuir fitting curves)

Fig.8 3D structures of MOFs and breakthrough curves of CO2/N2 (30∶70) binary mixture in fixed bed

Table 3 The operation parameters for binary component adsorption fixed bed packed with various MOFs

MOF	参数	符号/单位	取值	MOF	参数	符号/单位	取值
UTSA-16	床高	L/m	1.0	CALF-20	床高	L/m	0.0786
	空隙率	ε_b	0.37		空隙率	ε_b	0.40
	内径	d_in/m	0.364		内径	d_in/m	0.0282
	密度	ρ_s/(kg/m³)	1659		密度	ρ_s/（kg/m³)	570
	黏度	μ/(kg/(m·s))	1.72×10^-5		黏度	μ/(kg/(m·s))	1.812×10^-5
	扩散系数	D_ax/(m/s²)	8.091×10^-4		扩散系数	D_ax/(m/s²)	1.496×10^-5
	传质系数	$k f, C O 2$ /s^-1	0.1631		传质系数	$k f, C O 2$ /s^-1	0.1598
		$k f, N 2$ /s^-1	0.2044		传质系数	$k f, N 2$ /s^-1	0.1973

Table 3 The operation parameters for binary component adsorption fixed bed packed with various MOFs

MOF	参数	符号/单位	取值	MOF	参数	符号/单位	取值
UTSA-16	床高	L/m	1.0	CALF-20	床高	L/m	0.0786
	空隙率	ε_b	0.37		空隙率	ε_b	0.40
	内径	d_in/m	0.364		内径	d_in/m	0.0282
	密度	ρ_s/(kg/m³)	1659		密度	ρ_s/（kg/m³)	570
	黏度	μ/(kg/(m·s))	1.72×10^-5		黏度	μ/(kg/(m·s))	1.812×10^-5
	扩散系数	D_ax/(m/s²)	8.091×10^-4		扩散系数	D_ax/(m/s²)	1.496×10^-5
	传质系数	$k f, C O 2$ /s^-1	0.1631		传质系数	$k f, C O 2$ /s^-1	0.1598
		$k f, N 2$ /s^-1	0.2044		传质系数	$k f, N 2$ /s^-1	0.1973

Fig.9 The comparison of breakthrough curves of the CO2/N2(30∶70) mixture adsorbed in fixed bed packed at 298 K and 97 kPa

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