Developing a fast and robust numerical method for the simulation of cocurrent hollow fiber gas separation membranes for process flowsheet synthesis

Abstract

Membrane separation technology has become an important part of process flowsheet synthesis in which it can be combined with other unit operations for enhanced separation of gasses and liquids. Since process flowsheet synthesis requires computationally efficient numerical algorithms, in this work, an accurate, fast, and robust computer model for the simulation of cocurrent hollow fiber membranes is proposed. The algorithm is based on a modified shooting method with quadratic interpolation, which requires no derivative computations versus classical optimization methods. Moreover, for solving the set of differential equations, three different integration methods were employed and compared over wide range of operating conditions. Based on both speed and robustness, the best method was found to be the variable order numerical differentiation formula as it succeeded in solving the problem fast and accurately under all tested points. The maximum computational time was 3.4 s, while the maximum absolute permeate outlet pressure estimation error was 0.37%, which is remarkable. Also, the method was validated with both experimental and simulation studies in the literature. Hence the proposed algorithm is suitable for process flowsheet synthesis, process design, and optimization studies.

A generalized numerical framework for solving cocurrent and counter-current membrane models for gas separation

“…In the following, the proposed numerical method is assessed based on the computational time for wide ranges of membrane length (0.1–2.5 m) and feed pressure (4–70 bar). As discussed in our previous work [ 29 ], these ranges of operating conditions could give rise to some stiff problems, which might not be solved by the conventional variable step Runge-Kutta methods (specifically Runge-Kutta (4,5) formula, the Dormand-Prince pair [ 37 ]). Hence, the proposed numerical method is assessed on the entire ranges as given above.…”

“…The maximum obtained computational time was below 4.5 s for the given ranges of membrane length and feed pressure. For the cocurrent mode, the proposed method is overall slower than the shooting method ( Figure 6 b) described elsewhere [ 29 ].

Figure 6 Computational time of (a) the GS method (cocurrent mode), (b) the shooting method of Medi and Nomvar [ 29 ] (cocurrent mode), and (c) the GS method (counter-current mode).

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“…For the cocurrent mode, the proposed method is overall slower than the shooting method ( Figure 6 b) described elsewhere [ 29 ].

Figure 6 Computational time of (a) the GS method (cocurrent mode), (b) the shooting method of Medi and Nomvar [ 29 ] (cocurrent mode), and (c) the GS method (counter-current mode). The values in legends are feed pressure (bar).

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“…However, since the boundary conditions do not necessarily fall in a favorable position, the problem appears as a boundary value problem, which cannot be solved by the conventional numerical integration methods such as the Runge-Kutta methods. Instead, other methods such as the shooting method [ 29 ] or the GS iterations are required [ 33 ].…”

“…Despite this rich research background, finding an efficient numerical method in terms of accuracy and speed is still an open challenge. In our previous study, a cocurrent HFM module was simulated for multicomponent gas separation by a mathematical model based on the modified shooting method [ 29 ]. Using the variable order numerical differentiation formula (NDF), the permeate outlet pressure estimation error of ∼0.37% and computational time of 3.4 s were obtained.…”