1
|
Peng X, Chen L, You L, Jin Y, Zhang C, Ren S, Kapteijn F, Wang X, Gu X. Improved Synthesis of Hollow Fiber SSZ-13 Zeolite Membranes for High-Pressure CO 2/CH 4 Separation. Angew Chem Int Ed Engl 2024; 63:e202405969. [PMID: 38760324 DOI: 10.1002/anie.202405969] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2024] [Revised: 04/30/2024] [Accepted: 05/15/2024] [Indexed: 05/19/2024]
Abstract
High-silica CHA zeolite membranes are highly desired for natural gas upgrading because of their separation performance in combination with superior mechanical and chemical stability. However, the narrow synthesis condition range significantly constrains scale-up preparation. Herein, we propose a facile interzeolite conversion approach using the FAU zeolite to prepare SSZ-13 zeolite seeds, featuring a shorter induction and a longer crystallization period of the membrane synthesis on hollow fiber substrates. The membrane thickness was constant at ~3 μm over a wide span of synthesis time (24-96 h), while the selectivity (separation efficiency) was easily improved by extending the synthesis time without compromising permeance (throughput). At 0.2 MPa feed pressure and 303 K, the membranes showed an average CO2 permeance of (5.2±0.5)×10-7 mol m-2 s-1 Pa-1 (1530 GPU), with an average CO2/CH4 mixture selectivity of 143±7. Minimal defects ensure a high selectivity of 126 with a CO2 permeation flux of 0.4 mol m-2 s-1 at 6.1 MPa feed pressure, far surpassing requirements for industrial applications. The feasibility for successful scale-up of our approach was further demonstrated by the batch synthesis of 40 cm-long hollow fiber SSZ-13 zeolite membranes exhibiting CO2/CH4 mixture selectivity up to 400 (0.2 MPa feed pressure and 303 K) without using sweep gas.
Collapse
Affiliation(s)
- Xingyu Peng
- State Key Laboratory of Materials-Oriented Chemical Engineering, College of Chemical Engineering, Nanjing Tech University, No. 30 Puzhu South Road, Nanjing, 211816, P. R. China
| | - Lingjie Chen
- State Key Laboratory of Materials-Oriented Chemical Engineering, College of Chemical Engineering, Nanjing Tech University, No. 30 Puzhu South Road, Nanjing, 211816, P. R. China
- Quzhou Membrane Material Innovation Institute, No. 99 Zheda Road, Quzhou, 324000, P. R. China
| | - Lekai You
- State Key Laboratory of Materials-Oriented Chemical Engineering, College of Chemical Engineering, Nanjing Tech University, No. 30 Puzhu South Road, Nanjing, 211816, P. R. China
| | - Yang Jin
- State Key Laboratory of Materials-Oriented Chemical Engineering, College of Chemical Engineering, Nanjing Tech University, No. 30 Puzhu South Road, Nanjing, 211816, P. R. China
| | - Chun Zhang
- State Key Laboratory of Materials-Oriented Chemical Engineering, College of Chemical Engineering, Nanjing Tech University, No. 30 Puzhu South Road, Nanjing, 211816, P. R. China
| | - Shengyuan Ren
- State Key Laboratory of Materials-Oriented Chemical Engineering, College of Chemical Engineering, Nanjing Tech University, No. 30 Puzhu South Road, Nanjing, 211816, P. R. China
| | - Freek Kapteijn
- Chemical Engineering Department, Delft University of Technology, Van der Maasweg 9, Delft, 2629 HZ, The, Netherlands
| | - Xuerui Wang
- State Key Laboratory of Materials-Oriented Chemical Engineering, College of Chemical Engineering, Nanjing Tech University, No. 30 Puzhu South Road, Nanjing, 211816, P. R. China
- Quzhou Membrane Material Innovation Institute, No. 99 Zheda Road, Quzhou, 324000, P. R. China
| | - Xuehong Gu
- State Key Laboratory of Materials-Oriented Chemical Engineering, College of Chemical Engineering, Nanjing Tech University, No. 30 Puzhu South Road, Nanjing, 211816, P. R. China
- Quzhou Membrane Material Innovation Institute, No. 99 Zheda Road, Quzhou, 324000, P. R. China
| |
Collapse
|
2
|
Wang L, Li Y, Pu L, Yang M, Lu H, Gu X, Wang X. Copolyimide membranes fabricated by nonsolvent-induced phase separation for helium extraction from natural gas. Sep Purif Technol 2023. [DOI: 10.1016/j.seppur.2023.123455] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/24/2023]
|
3
|
Liu L, Wu Q, Wang S, Lai W, Zheng P, Wang C, Wei X, Luo S. Highly Selective and Hydrocarbon-Resistant Polyimide Hollow Fiber Membranes for Helium Recovery from Natural Gas. Ind Eng Chem Res 2022. [DOI: 10.1021/acs.iecr.2c03907] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
Affiliation(s)
- Lu Liu
- School of Rare Earths, University of Science and Technology of China, Hefei230026, China
- Ganjiang Innovation Academy, Chinese Academy of Sciences, Ganzhou341119, China
- CAS Key Laboratory of Green Process and Engineering, State Key Laboratory of Multiphase Complex Systems, Beijing Key Laboratory of Ionic Liquids Clean Process, Institute of Process Engineering, Chinese Academy of Science (CAS), Beijing100190, China
| | - Qi Wu
- School of Rare Earths, University of Science and Technology of China, Hefei230026, China
- Ganjiang Innovation Academy, Chinese Academy of Sciences, Ganzhou341119, China
- CAS Key Laboratory of Green Process and Engineering, State Key Laboratory of Multiphase Complex Systems, Beijing Key Laboratory of Ionic Liquids Clean Process, Institute of Process Engineering, Chinese Academy of Science (CAS), Beijing100190, China
| | - Shunli Wang
- CAS Key Laboratory of Green Process and Engineering, State Key Laboratory of Multiphase Complex Systems, Beijing Key Laboratory of Ionic Liquids Clean Process, Institute of Process Engineering, Chinese Academy of Science (CAS), Beijing100190, China
| | - Wei Lai
- Ganjiang Innovation Academy, Chinese Academy of Sciences, Ganzhou341119, China
- CAS Key Laboratory of Green Process and Engineering, State Key Laboratory of Multiphase Complex Systems, Beijing Key Laboratory of Ionic Liquids Clean Process, Institute of Process Engineering, Chinese Academy of Science (CAS), Beijing100190, China
| | - Peijun Zheng
- CAS Key Laboratory of Green Process and Engineering, State Key Laboratory of Multiphase Complex Systems, Beijing Key Laboratory of Ionic Liquids Clean Process, Institute of Process Engineering, Chinese Academy of Science (CAS), Beijing100190, China
| | - Can Wang
- School of Rare Earths, University of Science and Technology of China, Hefei230026, China
- Ganjiang Innovation Academy, Chinese Academy of Sciences, Ganzhou341119, China
- CAS Key Laboratory of Green Process and Engineering, State Key Laboratory of Multiphase Complex Systems, Beijing Key Laboratory of Ionic Liquids Clean Process, Institute of Process Engineering, Chinese Academy of Science (CAS), Beijing100190, China
| | - Xin Wei
- SINOPEC (Beijing) Research Institute of Chemical Industry Co., Ltd, Beijing100013, China
| | - Shuangjiang Luo
- CAS Key Laboratory of Green Process and Engineering, State Key Laboratory of Multiphase Complex Systems, Beijing Key Laboratory of Ionic Liquids Clean Process, Institute of Process Engineering, Chinese Academy of Science (CAS), Beijing100190, China
| |
Collapse
|
4
|
Wang L, Li Y, Zhang P, Chen X, Nian P, Wei Y, Lu H, Gu X, Wang X. Thermally rearranged poly(benzoxazole-co-imide) composite membranes on α-Al2O3 support for helium extraction from natural gas. J Memb Sci 2022. [DOI: 10.1016/j.memsci.2022.120614] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
|
5
|
Influence of Draw Ratio and Take-Up Velocity on Properties of Ultrafiltration Hollow Fiber Membranes from Polyethersulfone. FIBERS 2022. [DOI: 10.3390/fib10030029] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
This study aimed to reveal the influence of the draw ratio and take-up speed on the pore size distribution and morphology of the hollow fiber ultrafiltration membrane selective layer. To this end, spinnerets with ring ducts of 1.8 and 1.3 mm were employed, whereas the external diameter of the obtained fiber was kept equal. Atomic force microscopy and scanning electron microscopy were employed to study the morphology of the selective layer. Liquid–liquid displacement porosimetry was used to determine the limiting pore size distribution. The produced polyethersulfone ultrafiltration membranes had a robust, sponge-like porous structure, permeance 1000 L/(m2·h·bar), smooth selective layer, and mean pore size 25 nm. It was found that limiting pore sizes are affected more by the change in the take-up speed, whereas the surface pore sizes, roughness, and morphology are controlled by the draw ratio. It was shown that excessive draw causes the selective layer stretching and crop-up of the porous sublayer. Consequently, the diameters of the spinneret ring duct and the bore needle should match the hollow fiber outer and lumen diameters, respectively.
Collapse
|
6
|
Medi B, Vesali-Naseh M, Hamedani MH. A generalized numerical framework for solving cocurrent and counter-current membrane Models for gas separation. Heliyon 2022; 8:e09053. [PMID: 35309396 PMCID: PMC8924320 DOI: 10.1016/j.heliyon.2022.e09053] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2021] [Revised: 01/09/2022] [Accepted: 03/01/2022] [Indexed: 11/26/2022] Open
Abstract
Membrane separation has become a panacea for various scientific and engineering problems, including water treatment, gas separation, purification, hemodialysis, and drug delivery. Modeling and simulation of such systems are necessary for the design, analysis, and optimization of membrane separation processes. Despite numerous studies, an efficient numerical solution of such systems is an open problem, especially when speed and reliability matter. In this study, a generalized numerical framework for solving cocurrent and counter-current membrane models is proposed, which hinges on a straightforward and reliable Gauss-Seidel method with successive over-relaxation. The results confirm the speed and reliability of the proposed algorithm, while it is validated by the experimental data for the separation of a mixture of CH4 and CO2, as well as a mixture of He, CO2, N2, and CH4. The permeate outlet pressure estimation error can be reduced to any value as low as ∼10−14%, while the computational time on a personal laptop is not more than 4.5 s. This algorithm can be readily implemented in various programming languages and commercial software applications.
Collapse
|
7
|
De Pascale M, Benedetti FM, Lasseuguette E, Ferrari MC, Papchenko K, Degli Esposti M, Fabbri P, De Angelis MG. Mixed Matrix Membranes Based on Torlon ® and ZIF-8 for High-Temperature, Size-Selective Gas Separations. MEMBRANES 2021; 11:membranes11120982. [PMID: 34940483 PMCID: PMC8703552 DOI: 10.3390/membranes11120982] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/11/2021] [Revised: 12/05/2021] [Accepted: 12/07/2021] [Indexed: 11/16/2022]
Abstract
Torlon® is a thermally and plasticization-resistant polyamide imide characterized by low gas permeability at room temperature. In this work, we aimed at improving the polymer performance in the thermally-enhanced He/CO2 and H2/CO2 separations, by compounding Torlon® with a highly permeable filler, ZIF-8, to fabricate Mixed Matrix Membranes (MMMs). The effect of filler loading, gas size, and temperature on the MMMs permeability, diffusivity, and selectivity was investigated. The He permeability increased by a factor of 3, while the He/CO2 selectivity decreased by a factor of 2, when adding 25 wt % of ZIF-8 at 65 °C to Torlon®; similar trends were observed for the case of H2. The MMMs permeability and size-selectivity were both enhanced by temperature. The behavior of MMMs is intermediate between the pure polymer and pure filler ones, and can be described with models for composites, indicating that such materials have a good polymer/filler adhesion and their performance could be tailored by acting on the formulation. The behavior observed is in line with previous investigations on MMMs based on glassy polymers and ZIF-8, in similar conditions, and indicates that ZIF-8 can be used as a polymer additive when the permeability is a controlling aspect, with a proper choice of loading and operative temperature.
Collapse
Affiliation(s)
- Matilde De Pascale
- Department of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, 40131 Bologna, Italy; (M.D.P.); (F.M.B.); (K.P.); (M.D.E.); (P.F.)
- GVS S.p.A via Guido Rossa 30, 40069 Zola Predosa, Italy
| | - Francesco Maria Benedetti
- Department of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, 40131 Bologna, Italy; (M.D.P.); (F.M.B.); (K.P.); (M.D.E.); (P.F.)
- Osmoses Inc., 444 Somerville Ave, Somerville, MA 02143, USA
| | - Elsa Lasseuguette
- School of Engineering, University of Edinburgh, Sanderson Building, Robert Stevenson Road, Edinburgh EH9 3FB, Scotland, UK; (E.L.); (M.-C.F.)
| | - Maria-Chiara Ferrari
- School of Engineering, University of Edinburgh, Sanderson Building, Robert Stevenson Road, Edinburgh EH9 3FB, Scotland, UK; (E.L.); (M.-C.F.)
| | - Kseniya Papchenko
- Department of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, 40131 Bologna, Italy; (M.D.P.); (F.M.B.); (K.P.); (M.D.E.); (P.F.)
- School of Engineering, University of Edinburgh, Sanderson Building, Robert Stevenson Road, Edinburgh EH9 3FB, Scotland, UK; (E.L.); (M.-C.F.)
| | - Micaela Degli Esposti
- Department of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, 40131 Bologna, Italy; (M.D.P.); (F.M.B.); (K.P.); (M.D.E.); (P.F.)
- Italian Consortium for Science and Technology of Materials (INSTM), 50121 Firenze, Italy
| | - Paola Fabbri
- Department of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, 40131 Bologna, Italy; (M.D.P.); (F.M.B.); (K.P.); (M.D.E.); (P.F.)
- Italian Consortium for Science and Technology of Materials (INSTM), 50121 Firenze, Italy
| | - Maria Grazia De Angelis
- Department of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, 40131 Bologna, Italy; (M.D.P.); (F.M.B.); (K.P.); (M.D.E.); (P.F.)
- School of Engineering, University of Edinburgh, Sanderson Building, Robert Stevenson Road, Edinburgh EH9 3FB, Scotland, UK; (E.L.); (M.-C.F.)
- Italian Consortium for Science and Technology of Materials (INSTM), 50121 Firenze, Italy
- Correspondence:
| |
Collapse
|
8
|
Dai Z, Deng J, He X, Scholes CA, Jiang X, Wang B, Guo H, Ma Y, Deng L. Helium separation using membrane technology: Recent advances and perspectives. Sep Purif Technol 2021. [DOI: 10.1016/j.seppur.2021.119044] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
|
9
|
Physical and Mechanical Properties of Hollow Fiber Membranes and Technological Parameters of the Gas Separation Process. MEMBRANES 2021; 11:membranes11080583. [PMID: 34436345 PMCID: PMC8398972 DOI: 10.3390/membranes11080583] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/20/2021] [Revised: 07/16/2021] [Accepted: 07/19/2021] [Indexed: 11/16/2022]
Abstract
The porous layer of composite and asymmetric hollow fiber membranes acts as a support and is exposed to strong mechanical stresses. The effect of external pressure on the polymer structure and, as a consequence, the separation characteristics of the membrane remains unsolved. Based on the solution of the Lamé approach to the calculation of the stress state of a hollow cylinder, a method of calculation was proposed for hollow fiber membranes. Calculations were based on the approximation of the isotropic nature of the physical and mechanical characteristics of the selective layer and substrate. Permissible deformation of the membrane’s selective layer was determined from the linear sector of strain-on-stress dependence, where Hooke’s law was performed. For these calculations, commercial polyethersulfone membranes were chosen with an inner and/or outer selective layer and with the following values of Young’s modulus of 2650 and 72 MPa for the selective and porous layers, respectively. The results obtained indicate that the dependence of the maximum allowable operating pressure on the substrate thickness asymptotically trends to a certain maximum value for a given membrane. Presented data showed that membranes with outer selective layer can be operated at higher working pressure. Optimal parameters for hollow fiber gas separation membrane systems should be realized, solving the optimization problem and taking into account the influence of operating, physicochemical and physicomechanical parameters on each other.
Collapse
|
10
|
Grushevenko EA, Borisov IL, Bakhtin DS, Volkov VV, Volkov AV. The Effect of the Type and Concentration of the Crosslinking Diene on the Gas Transport Properties of Membranes Based on Polyoctylmethylsiloxane. MEMBRANES AND MEMBRANE TECHNOLOGIES 2020. [DOI: 10.1134/s2517751620060037] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
|
11
|
Influence of Sodium Hypochlorite Treatment on Pore Size Distribution of Polysulfone/Polyvinylpyrrolidone Membranes. MEMBRANES 2020; 10:membranes10110356. [PMID: 33228077 PMCID: PMC7699321 DOI: 10.3390/membranes10110356] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/29/2020] [Revised: 11/16/2020] [Accepted: 11/18/2020] [Indexed: 11/17/2022]
Abstract
This work was focused on the study of hypochlorite treatment on the pore size distribution of membranes. To this end, ultrafiltration membranes from a polysulfone/polyvinylpyrrolidone blend with a sponge-like structure were fabricated and exposed to hypochlorite solutions with different active chlorine concentrations for 4 h at ambient temperature. Liquid–liquid displacement and scanning electron microscopy were employed to study the limiting and surface pores, respectively. After treatment with 50 ppm hypochlorite solution at pH = 7.2, a five-fold increase in water permeance up to 1400 L/(m2·h·bar) was observed, accompanied by a 40% increase in the limiting pore sizes and almost a three-fold increase in the porosity. After 5000 ppm treatment at pH = 11.5, a 40% rise in the maximum limiting pore size and almost a two-fold increase in the porosity and permeance was observed, whereas the mean pore size was constant. Apparently, changes in the membrane structure at pH = 11.5 were connected with polyvinylpyrrolidone (PVP) degradation and wash-out, whereas at lower pH and despite lower active chlorine concentration, this process was coupled with polysulfone (PSf) destruction and removal.
Collapse
|
12
|
Kagramanov GG, Gurkin VN, Farnosova EN. Effect of Gas Solubility on Efficiency of Membrane Processes with Separation of He/CH4 and CO2/CH4 Mixtures as an Example. MEMBRANES AND MEMBRANE TECHNOLOGIES 2020. [DOI: 10.1134/s251775162004006x] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
|
13
|
Kirsch V, Bazhenov S. Numerical simulation of solute removal from a cross-flow past a row of parallel hollow-fiber membranes. Sep Purif Technol 2020. [DOI: 10.1016/j.seppur.2020.116834] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
|
14
|
Grushevenko E, Borisov I, Knyazeva A, Volkov V, Volkov A. Polyalkylmethylsiloxanes composite membranes for hydrocarbon/methane separation: Eight component mixed-gas permeation properties. Sep Purif Technol 2020. [DOI: 10.1016/j.seppur.2020.116696] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
|
15
|
Yampolskii Y, Belov N, Alentiev A. Perfluorinated polymers as materials of membranes for gas and vapor separation. J Memb Sci 2020. [DOI: 10.1016/j.memsci.2019.117779] [Citation(s) in RCA: 44] [Impact Index Per Article: 11.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023]
|
16
|
Elaboration of High Permeable Macrovoid Free Polysulfone Hollow Fiber Membranes for Air Separation. FIBERS 2019. [DOI: 10.3390/fib7050043] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
In this work, polysulfone hollow fibers with oxygen permeance 70 L (STP)/(m2·h·bar) and selectivity α(O2/N2) = 6 were obtained. A decrease in the dope solution temperature allowed to diminish macrovoids due to the increase of the dope viscosity from 15.5 Pa·s at 62 °C to 35 Pa·s at 25 °C. To reduce the fiber diameter, thereby increasing the packing density, they were spun at high linear velocities. A hollow fiber membrane element was produced with effective membrane area 2.75 m2 and packing density 53%. Its air separation performance was evaluated to bridge laboratory studies and practical application.
Collapse
|
17
|
Simulation of Convection–Diffusion Transport in a Laminar Flow past a Row of Parallel Absorbing Fibers. FIBERS 2018. [DOI: 10.3390/fib6040090] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
A numerical simulation of the laminar flow field and convection–diffusion mass transfer in a regular system of parallel fully absorbing fibers for the range of Reynolds numbers up to Re = 300 is performed. An isolated row of equidistant circular fibers arranged normally to the external flow is considered as the simplest model for a hollow-fiber membrane contactor. The drag forces acting on the fibers with dependence on Re and on the ratio of the fiber diameter to the distance between the fiber axes, as well as the fiber Sherwood number versus Re and the Schmidt number, Sc, are calculated. A nonlinear regression formula is proposed for calculating the fiber drag force versus Re in a wide range of the interfiber distances. It is shown that the Natanson formula for the fiber Sherwood number as a function of the fiber drag force, Re, and Sc, which was originally derived in the limit of high Peclet numbers, is applicable for small and intermediate Reynolds numbers; intermediate and large Peclet numbers, where Pe = Re × Sc; and for sparse and moderately dense rows of fibers.
Collapse
|