Skip to main content
open-access modelling concepts to the mass continuity equation in hollow fibres

ZEOCAT-3D and innovative open-access modelling concepts to the mass continuity equation in hollow fibres

Curious about the modelling of the mass continuity equation in a hollow-fibre membrane? Read more for a historical perspective and related analytical and numerical solutions in open-source and commercial software in this article by the APTL/CERTH team

By Grigorios Pantoleontos; Ioanna Marina Anagnostara; Maria Syrigou; Athanasios Konstandopoulos (Aerosol & Particle Technology Laboratory, Chemical Process & Energy Resources Institute, Centre for Research & Technology Hellas, APTL/CPERI/CERTH)

In 2021, the ZEOCAT-3D team at the Aerosol & Particle Technology Laboratory from the Chemical Process & Energy Resources Institute of the Centre for Research & Technology Hellas (APTL/CPERI/CERTH) set up the modelling equations to predict the behaviour of membrane-based processes at play within the project. Specifically, we concentrated treating the feedstock natural gas and biogas used within the pilot reactor in the ZEOCAT-3D project.

Scientists aim at designing and creating models that allow for complete control over the results they hope to obtain. They do so while maximizing the versatility that concerns the mathematical definition and the equations involved in the process. Such development usually invokes the so-called Equation-based modelling. An example of equation-based modelling comprises a dimensionless set of partial differential equations (PDEs) with custom boundary conditions (BCs).

As the open-source software community grows steadily, any researcher may combine custom-made solvers and obtain an inexpensive and accurate representation of the variables of interest deriving from a set of equations. Thanks to such software, the researcher can perform dynamic or steady-state simulation runs and optimise the corresponding process at an acceptable computational cost.

In this respect, the open-access Carbon Capture Science and Technology journal of the Elsevier group recently published our study called: "Solutions of the mass continuity equation in hollow fibers for fully developed flow with some notes on the Lévêque correlation". The purpose of this study is to provide a generic computational framework using equation-based modelling in the open-source software (SageMath) and benchmarking commercial mathematical packages (Maple). Such a framework may apply to all relevant PDEs described in this article by merely altering the corresponding flux equation at the boundaries.

In this study, we solve the mass continuity equation for fully developed laminar flow in the lumen side of a hollow-fibre membrane contactor for the lumen-wall BC. While devoting much attention to the carbon-capture membrane-based gas absorption process, we extend the corresponding model, incorporating various linear or nonlinear BCs that account for different membrane-based processes. We also build upon our previously published work, namely the article: ”Modelling, simulation, and membrane wetting estimation in gas-liquid contacting processes”, written by G. Pantoleontos, T. Theodoridis, M. Mavroudi, E.S. Kikkinides, D. Koutsonikolas, S.P. Kaldis and A.E. Pagana, and published in 2017 by The Canedian Journal of Chemical Engineering; and the article: “Analytical and numerical solutions of the mass continuity equation in the lumen side of a hollow fiber membrane contactor with linear or nonlinear boundary conditions” by G. Pantoleontos, S.P. Kaldis, D. Koutsonikolas, G. Skodras, G.P. Sakellaropoulos, published in Chemical Engineering Communications in 2010.

A single hollow fibre flow pattern with gaseous mixture flowing in the lumen-side and counter-current mode of operation depicting the corresponding boundary conditions
Figure 1: A single hollow fibre flow pattern with gaseous mixture flowing in the lumen-side and counter-current mode of operation depicting the corresponding boundary conditions. Source: APTL/CPERI/CERTH

Such a model reveal as particularly useful in predicting the behaviour of the membrane-based biogas upgrading to biomethane. Said process is described and implemented in the

ZEOCAT-3D task regarding the pre-treatment of feedstock upstream to the Methane DehydroAromatisation (MDA) reactor – resulting in improved CO2 removal with the least possible loss of methane (CH4).

In addition, this work extensively discusses the Graetz problem and the Lévêque correlation resorting to the prerequisites, their limitations and applications. It shows that the constant wall concentration case (Dirichlet boundary condition) imposed by the Graetz-Lévêque postulations is a sub-case of the mixed Neumann-Dirichlet linear BC, overestimating the performance of membrane contactors.

When the mass transfer is determined by a linear lumen-wall BC, the existing analytical solution derived by the separation of variables method is very accurate and practical. This happens even in the region that is very close to the entrance of the computational domain, avoiding the need for numerical or entrance-region solutions.

We extended the analysis by solving the nonlinear lumen-wall BCs with the method-of-lines approach and by discretizing the radial domain using the Gauss-Jacobi-Lobatto orthogonal collocation and integrating the resulting initial-value ordinary differential equations system.

Contour plot of the dimensionless  concentration in the fibre for lumen-wall nonlinear BC
Figure 2: Contour plot of the dimensionless concentration in the fibre for lumen-wall nonlinear BC

Within the article, we illustrate all cases (the analytical solution of the linear BC case and the numerical analysis of the nonlinear BCs) in a step by step hands-on approach with the aid of current computational tools, such as the open-source SageMath and the commercial software Maple. We provide the scripts so that the reader may directly validate the code and compare it with other available approximations.

The overall treatment of the corresponding nonlinear PDEs in the Maple computing platform proves more effective than other approaches. Additionally, the open-source computational software SageMath is a promising alternative, providing an affordable and competent equation-based modelling platform with unlimited capabilities of development and content.

Following this article, together with the ZEOCAT-3D consortium, we will be focusing on the practical implementation of feedstock pre-treatment. If interested in knowing more, continue to follow our blog. If you have any questions or want to share any insight on our new article, do not hesitate to contact us directly.


We are professional and reliable provider since we offer customers the most powerful and beautiful themes. Besides, we always catch the latest technology and adapt to follow world’s new trends to deliver the best themes to the market.