Dispersion and Phase Exchange Process of Chemically Reactive Solute Through Circular Tube

Document Type : Regular Article

Authors

1 The Institute of Basic Science, Korea University, Seoul 02841, Republic of Korea

2 Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea

Abstract

This article explores how a chemically reactive solute will disperse across mobile to immobile phase when injected into the fluid flowing within a long circular tube. To model this process, we utilized mathematical modeling, including advection-diffusion equations for flow of fluid within the tube and first-order chemical reaction equations to account for reversible and irreversible reactions on the tubes’ wall. We proposed a numerical method based on an explicit finite difference scheme to solve the governing equations for the dispersion of a chemically reactive solute. We used an upwind method with a conservative representation in the diffusion component to discretize the advection-diffusion equation. To ensure the stability of our proposed numerical scheme, we computed the time step constraint condition so that the maximum principle for the discrete governing equation holds. We also verified the performance of our proposed scheme through computational results that were compared with previous studies. One of our key findings was that the depletion coefficient D0 achieved a quasi-steady state for larger absorption rates. We also observed that the advection coefficient  initially increased with an increasing absorption rate, but eventually declined due to phase exchange kinetics. The dispersion coefficient  also decreased with a rising absorption rate due to a low-velocity gradient in the middle region. Our study showed that rapid distributions are possible under certain conditions, such as a high Damköhler number (Da≥10 ) and a high absorption rate (Γ>5). Computational results show that the proposed scheme can be useful in developing an efficient pulmonary drug delivery system for periodic inhalation of drugs to determine the optimal frequency of injection.

Keywords

Main Subjects

References

Ani, E. C., Wallis, S., Kraslawski, A., & Agachi, P. S. (2009). Development, calibration and evaluation of two mathematical models for pollutant transport in a small river. Environmental Modelling & Software, 24(10), 1139–1152. https://doi.org/10.1016/j.envsoft.2009.03.008
Aris, R. (1956). On the dispersion of a solute in a fluid     flowing through a tube. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 235(1200), 67–77. https://doi.org/10.1016/S1874-5970(99)80009-5
Abidin, S. N. A. M. Z., Jaafar, N. A., & Ismail, Z. (2021). Herschel-Bulkley model of blood flow through a stenosed artery with the effect of chemical reaction on solute dispersion. Malaysian Journal of Fundamental and Applied Sciences, 17(4), 457–474. https://doi.org/10.11113/mjfas.v17n4.2144
Barati, P., & Saghafian, M. (2022). Optimum geometric bifurcation under pulsating flow assuming minimum energy consumption in cardiovascular system, an extension on murray’s law. Journal of Applied Fluid Mechanics, 15(3), 687–695. https://doi: 10.47176/jafm.15.03.33190
Bég, O. A., & Roy, A. K. (2022). Moment analysis of unsteady bi-component species (drug) transport with coupled chemical reaction in non-Newtonian blood flow. Chinese Journal of Physics, 77, 1810–1826. https://doi.org/10.1016/j.cjph.2022.04.003
Ben-Tal, A. (2006). Simplified models for gas exchange in the human lungs. Journal of theoretical biology, 238(2), 474–495. https://doi.10.1016/j.jtbi.2005.06.005
Bel Hadj Taher, A., Kanfoudi, H., & Zgolli, R. (2022). numerical prediction approach of cavitation erosion based on 3D simulation flow. Journal of Applied Fluid Mechanics, 15(4), 1165–1177. https://doi: 10.47176/jafm.15.04.1016
Chatwin, P. C. (1970). The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe. Journal of Fluid Mechanics, 43(2), 321–352. https://doi.10.1017/S0022112070002409
Das, P., Sarifuddin, S., Rana, J., & Mandal, P. K. (2021). Solute dispersion in transient Casson fluid flow through stenotic tube with exchange between phases. Physics of Fluids, 33(6). https://doi.org/10.1063/5.0052770
Das, P., Sarifuddin, Rana, J., & Kumar Mandal, P. (2022). Unsteady solute dispersion in the presence of reversible and irreversible reactions. Proceedings of the Royal Society A, 478(2264), 20220127. https://doi.10.1098/rspa.2022.0127
Debnath, S., Saha, A. K., Siddheshwar, P. G., & Roy, A. K. (2019). On dispersion of a reactive solute in a pulsatile flow of a two-fluid model. Journal of Applied Fluid Mechanics, 12(3), 987–1000. https://doi.org/10.29252/jafm.12.03.29101 
 Debnath, S., Saha, A. K., Mazumder, B. S., & Roy, A. K. (2020). On transport of reactive solute in a pulsatile Casson fluid flow through an annulus. International Journal of Computer Mathematics, 97(11), 2303-2319. https://doi.org/10.1080/00207160.2019.1695047
Davidson, M. R., & Schroter, R. C. (1983). A theoretical model of absorption of gases by the bronchial wall. Journal of Fluid Mechanics, 129, 313–335. https://doi.10.1017/S0022112083000786
Dhand, C., Prabhakaran, M. P., Beuerman, R. W., Lakshminarayanan, R., Dwivedi, N., & Ramakrishna, S. (2014). Role of size of drug delivery carriers for pulmonary and intravenous administration with emphasis on cancer therapeutics and lung-targeted drug delivery. RSC advances, 4(62), 32673–32689. https://doi.org/10.1039/C4RA02861A
Fu, X., Gao, R., & Wu, Z. (2016). Additional longitudinal displacement for contaminant dispersion in wetland flow. Journal of Hydrology, 532, 37–45. https://doi.org/10.1016/j.jhydrol.2015.10.064
Gill, W. N., & Sankarasubramanian, R. (1970). Exact analysis of unsteady convective diffusion. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 316(1526), 341–350. https://doi.org/10.1098/rspa.1970.0083
Giddings, J. C., & Eyring, H. (2002). A molecular dynamic theory of chromatography. The Journal of Physical Chemistry, 59(5), 416–421. https://doi.10.1021/J150527A009
Ibrahim, A., Meyrueix, R., Pouliquen, G., Chan, Y. P., & Cottet, H. (2013). Size and charge characterization of polymeric drug delivery systems by Taylor dispersion analysis and capillary electrophoresis. Analytical and bioanalytical chemistry, 405, 5369–5379. https://doi.10.1007/s00216-013-6972-4
Jiang, W., Zeng, L., Fu, X., & Wu, Z. (2022). Analytical solutions for reactive shear dispersion with boundary adsorption and desorption. Journal of Fluid Mechanics, 947, A37. https://doi.10.1017/jfm.2022.656.
Jiang, Y., & Grotberg, J. B. (1993). Bolus contaminant dispersion in oscillatory tube flow with conductive walls. https://doi.10.1115/1.2895507
Kori, J. (2020). Effect of first order chemical reactions on the dispersion coefficient associated with laminar flow through fibrosis affected lung. Journal of biomechanics, 99, 109494. https://doi.10.1016/j.jbiomech.2019.109494
Kori, J., & Pratibha. (2022). Effect of first order chemical reactions through tissue-blood interface on the partial pressure distribution of inhaled gas. Computer Methods in Biomechanics and Biomedical Engineering, 25(1), 84–96. https://doi.10.1080/10255842.2021.1932839
Lau, M. W., & Ng, C. O. (2009). On the early development of dispersion in flow through a tube with wall reactions. In New Trends in Fluid Mechanics Research: Proceedings of the Fifth International Conference on Fluid Mechanics (Shanghai, 2007), Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-75995-9_224
Le, A. D., & Tran, H. T. (2022). Improvement of mass transfer rate modeling for prediction of cavitating flow. Journal of Applied Fluid Mechanics, 15(2), 551-561. https://doi: 10.47176/jafm.15.02.33231
Mazumder, B. S., & Paul, S. (2012). Dispersion of reactive species with reversible and irreversible wall reactions. Heat and Mass Transfer, 48, 933–944. https://doi.org/10.1007/s00231-011-0920-7
Mazumder, B. S., & Das, S. K. (1992). Effect of boundary reaction on solute dispersion in pulsatile flow through a tube. Journal of Fluid Mechanics, 239, 523-549.
https://doi.org/10.1017/S002211209200452X
Mohseni, M., & Domfeh, M. K. (2023). Numerical analysis of transient vortex formation at the outlet of a tank containing gas-liquid phases. Journal of Applied Fluid Mechanics, 16(11), 2235–2248. https://doi: 10.47176/jafm.16.11.1942
Ng, C. O., & Rudraiah, N. (2008). Convective diffusion in steady flow through a tube with a retentive and absorptive wall. Physics of Fluids, 20(7). https://doi.10.1063/1.2958322
Paramanantham, S. S. S., Nagulapati, V. M., & Lim, H. (2022). Numerical investigation of the influence of microchannel geometry on the droplet generation process. Journal of Applied Fluid Mechanics, 15(5), 1291–1305. https://doi: 10.47176/jafm.15.05.1126
Rana, J., & Murthy, P. V. S. N. (2016). Solute dispersion in pulsatile Casson fluid flow in a tube with wall absorption. Journal of Fluid Mechanics, 793, 877–914. https://doi.10.1017/jfm.2016.155
Rana, J., & Murthy, P. V. S. N. (2017). Unsteady solute dispersion in small blood vessels using a two-phase Casson model. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2204), 20170427. https://doi.10.1098/rspa.2017.0427
Roy, A. K., & Bég, O. A. (2021). Mathematical modelling of unsteady solute dispersion in two-fluid (micropolar-Newtonian) blood flow with bulk reaction. International Communications in Heat and Mass Transfer, 122, 105169. https://doi.org/10.1016/j.icheatmasstransfer.2021.105169
Shaw, S., Ganguly, S., Sibanda, P., & Chakraborty, S. (2014). Dispersion characteristics of blood during nanoparticle assisted drug delivery process through a permeable microvessel. Microvascular Research, 92, 25–33. https://doi.10.1016/j.mvr.2013.12.007
Saini, A., Katiyar, V. K., & Pratibha. (2014). Effects of first-order chemical reactions on the dispersion coefficient associated with laminar flow through the lungs. International Journal of Biomathematics, 7(02), 1450021. https://doi.10.1142/S1793524514500211
Sankarasubramanian, R., & Gill, W. N. (1973). Unsteady convective diffusion with interphase mass transfer. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 333(1592), 115–132. https://doi.10.1098/rspa.1973.0051
Shankar, A., & Lenhoff, A. M. (1991). Dispersion and partitioning in short coated tubes. Industrial & engineering chemistry research, 30(5), 828–835. https://doi.10.1016/j.jtbi.2005.06.005.
Taylor, G. I. (1953). Dispersion of soluble matter in solvent flowing slowly through a tube. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 219(1137), 186–203. https://doi.org/10.1098/rspa.1953.0139
Venditti, C., Giona, M., & Adrover, A. (2022). Exact moment analysis of transient/asymptotic dispersion properties in periodic media with adsorbing/desorbing walls. Physics of Fluids, 34(12). https://doi.10.1063/5.0130648
Wang, Y. F., & Huai, W. X. (2019). Random walk particle tracking simulation on scalar diffusion with irreversible first-order absorption boundaries. Environmental Science and Pollution Research, 26, 33621–33630. https://doi.org/10.1007/s11356-019-06422-1
Wu, Z., Zeng, L., Chen, G. Q., Li, Z., Shao, L., Wang, P., & Jiang, Z. (2012). Environmental dispersion in a tidal flow through a depth-dominated wetland. Communications in Nonlinear Science and Numerical Simulation, 17(12), 5007–5025. https://doi.org/10.1016/j.cnsns.2012.04.006
Wu, Z., & Chen, G. Q. (2014). Approach to transverse uniformity of concentration distribution of a solute in a solvent flowing along a straight pipe. Journal of Fluid Mechanics, 740, 196–213. https://doi.10.1017/jfm.2013.648
Wu, Z., Zhou, D., Li, S., Yang, J., Chen, G., & Li, X. (2022). Numerical Analysis of the Effect of Streamlined Nose Length on Slipstream of High-Speed Train Passing through a Tunnel. Journal of Applied Fluid Mechanics, 15(6), 1933–1945. https://doi.org/10.47176/jafm.15.06.1189
Yang, X., Hu, Y., Gong, Z., Jian, J., & Liu, Z. (2021). Numerical study of combined drag reduction bases on vortex generators and riblets for the ahmed body using IDDES methodology. Journal of Applied Fluid Mechanics, 15(1), 193–207. https://doi: 10.47176/jafm.15.01.32832
Zhang, D. X., Lu, Z. M., Liu, Y. L., & Chiu-On, N. G. (2009). Numerical simulation of the dispersion in oscillating flows with reversible and irreversible wall reactions. Journal of Hydrodynamics, Ser. B, 21(4), 482-490. https://doi.org/10.1016/S1001-6058(08)60174-2 

Files

History

  • Received: 11 September 2023
  • Revised: 31 October 2023
  • Accepted: 25 November 2023
  • Available online: 30 January 2024

Share

How to cite

Statistics