Abstract
Lucas–Washburn equation is a fundamental expression which is used to describe capillary rise in porous materials according to average pore radius, liquid viscosity, surface tension, contact angle and time. However, a traditional equation is overestimating a real capillary rise height of liquid in the material, since it models pores as straight and circular capillaries, though in reality porous materials, such as biochar, have tortuous capillaries with different aperture forms. It is also known that cellulosic materials are characterised by their swelling capacity, which also can affect the process of capillary rise. Therefore, a modified model including a parameter describing the pores’ form and swelling parameters (volumetric swelling, energy loss coefficient and radius of swelled capillary) was developed. Experiments of water capillary rise in the biofilter tubes were conducted: the biochar made from different primary feedstocks, size of particles and modifications with steam of the biomedia. It was shown that the model is suitable for the prediction of short time (until 5 h) water capillary rise process in biochar due to low relative maximum error. Both experimental and modelling results showed that higher biochar porosity, average capillary radius, volumetric swelling and wettability govern higher velocity of capillary rise. Meanwhile, liquids with higher surface tension and dynamic viscosity lower the capillary rise speed in the biochar.
Similar content being viewed by others
Availability of Data and Material
Due to the nature of this research, participants of this study did not agree for their data to be shared publicly, so supporting data is not available.
References
Masoodi, R., & Pillai, K. M. (2010). Darcy’s law-based model for wicking in paper-like swelling porous media. AIChE Journal, 56(9), 2257–2267. https://doi.org/10.1002/aic.12163
Cai, J., Hu, X., Standnes, D. C., & You, L. (2012). An analytical model for spontaneous imbibition in fractal porous media including gravity. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 414, 228–233.
Washburn, E. W. (1921). The dynamics of capillary flow. Physics Review, 17(3), 273–283.
Cai, J., Yu, B., Zou, M., & Luo, L. (2010). Fractal characterization of spontaneous co-current imbibition in porous media. Energy & Fuels, 24, 1860–1867.
Cai, J., Perfect, E., Cheng, C. -L., & Hu, X. (2014). Generalized modeling of spontaneous imbibition based on Hagen-Poiseuille flow in tortuous capillaries with variably shaped apertures. Langmuir, 30(18), 5142–5151.
Siddique, J. I., & Kara, A. (2016). Capillary rise of magnetohydrodynamics liquid into deformable porous material. Journal of Applied Fluid Mechanics, 9(6), 2837–2843.
Masoodi, R., & Pillai, K. M. (2013). Wicking in porous materials (p. 2). Traditional and modern modeling approaches: CRC Press.
Ima, C. S., & Mann, D. D. (2011). Hygroscopic expansion of biofilter media consisting of woodchips. Australian Journal of Agricultural Engineering, 2(1), 5–7.
Dorado, A. D., Lafuente, F. J., Gabriel, D., & Gamisans, X. (2010). A comparative study based on physical charactersitics of suitable packing materials in biofiltration. Environmental Technology, 31(2), 193–204.
Mochado, D. R., Hasson, D., & Semiat, R. (1999). Effect of solvent properties on permeate flow through nanofiltration membranes. Part I: Investigation of parameters affecting solvent flux. Journal of Membrane Science, 163, 93–102.
Fu, Z., Guo, Z., Yuan, Z., & Wang, Z. (2007). Swelling and shrinkage behavior of raw and processed coals during pyrolysis. Fuel, 86, 418–425.
Xie, K. C. (2015). Coal swelling. In: Structure and Reactivity of Coal: A Survey of Selected Chinese Coals, 305–335.
Lucas, R. (1918). Ueber das Zeitgesetz des Kapillaren Aufstiegs von Flussigkeiten. Kolloid-Z, 23, 15–22.
Tsunawaza, Y., Yokoyama, T., & Nishiyama, N. (2016). An experimental study on the rate and mechanism of capillary rise in sandstone. Progress in Earth and Planetary Science, 3(8), 1–10.
Shi, S., & Gardner, D. J. (2000). A new model to determine contact angles on swelling polymer particles by the column wicking method. Journal of Adhesion Science and Technology, 14(2), 301–314.
Markl, D., Yassin, S., Wilson, D. I., Goodwin, D. J., Anderson, A., & Zeitler, J. A. (2017). Mathematical modelling of liquid transport in swelling pharmaceutical immediate release tablets. International Journal of Pharmaceutics, 526, 1–10.
Ha, J., Kim, J., Jung, Y., Yun, G., Kim, D. -N., Kim, H. -Y. (2018). Poro-elasto-capillary wicking of cellulose sponges. Science Advances, 4(3). https://doi.org/10.1016/B978-012369522-2/50012-9
Komkiene, J., & Baltrenaite, E. (2015). Biochar as adsorbent for removal of heavy metal ions [Cadmium(II), Copper(II), Lea(II), Zinc(II)] from aqueous phase. International Journal of Environmental Science and Technology, 13, 471–482. https://doi.org/10.1007/s13762-015-0873-3
Katyal, A., & Morisson, R. D. (2007). Forensic applications of contaminant transport models in the subsurface. In: Introduction to Environmental Forensics, Second Edition 513–575.
Peng, J., & Wan, A. (1998). Effect of ionic strength on Henry’s constants of volatile organic compound. Chemosphere, 36(13), 2731–2741.
Baltrėnas, P., Baltrėnaitė, E., & Spudulis, E. (2015). Biochar from pine and birch morphology and pore structure change by treatment in biofilter. Water, Air and Soil Pollution, 226(3), 1–14.
Liu, Q., Yasufuku, N., Miao, J., & Ren, J. (2014). An approach for quick estimation of maximum height of capillary rise. Soils and Foundations, 54(6), 1241–1245.
Shang, J., Flury, M., Harsh, J. B., & Zollars, R. L. (2008). Comparison of different methods to measure contact angles of soil colloids. Journal of Colloid and Interface Science, 328(2), 299–307.
Jeffery, S., Meinders, M. B. J., Stoof, C. R., Bezemer, T. M., van de Voorde, T. F. J., Mommer, L., & van Groenigen, J. W. (2015). Biochar application does not improve the soil hydrological function of a sandy soil. Geoderma, 251–252, 47–54.
Giffin, S., Littke, R., Klaver, J., & Urai, J. L. (2013). Application of BIB-SEM technology to characterize macropore morphology in coal. International Journal of Coal Geology, 114, 85–95.
Sienkiewicz, A., Krasucka, P., Charmas, B., Stefaniak, W., & Goworek, J. (2017). Swelling effects in cross-linked polymers by thermogravimetry. Journal of Thermal Analysis and Calorimetry, 130, 85–93.
Okechukwu, I. K. (2019). Determination of contact angle measurement of sub-bituminous and bituminous coal particles through capillary rise method using washburn equation for surface free energy and interfacial energy. International Journal of Advances Engineering and Technology, 3(2), 58–62.
Vinš, V., Hruby, J., Hykl, J., Blaha, J., & Šmid, B. (2013). Design of an experimental apparatus for measurement of the surface tension of metastable fluids. EPJ Web of Conferences, 45, 5.
A & D 2005. GX-13 Instruction Manual. Retrieved from https://www.aandd.jp/products/manual/manual_balances.html
Wangler, J., & Kohlus, R. (2017). Dynamics of capillary wetting of biopolymer powders. Chemical Engineering Technology, 40(9), 1552–1560.
Nagy, E., & Deak, A. J. (2013). Investigation of water capillary rise in soil columns made from clay mineral mixtures pretreated with cationic surfactants. Communications in Soil Science and Plant Analysis, 44(1–4), 749–757.
Yang, W., Shang, J., Baoguo, L., & Flury, M. (2019). Surface and colloid properties of biochar and implications for transoprt in porous media. Critical Reviews in Environmental Science and Technology, 50(23), 2484–2522. https://doi.org/10.1080/10643389.2019.1699381
Lou, K., Rajapaksha, A. U., Ok, Y. S., & Chang, S. X. (2016). Pyrolysis temperature and steam activation effects on sorption of phosfate on pine sawdust biochars in aqueous solutions. Chemical Speciation and Bioavailability, 28(1–4), 42–50. https://doi.org/10.1080/09542299.2016.1165080
Li, K. W., & Horne, R. N. (2004). An analytical scaling method for spontaneous imbibition in gas-water-rock systems. SPE Journal, 9, 322–329.
Huber, P., Gruner, S., Schafer, C., Knorr, K., & Kityk, A. V. (2007). Rheology of liquids in nanopores: A study on the capillary rise of water, n-Hexadecane and n-Tetracosane in mesoporous silica. The European Physical Journal Special Topics, 141, 101–105.
Pezron, I., Bourgain, G., & Quere, D. (1995). Imbibition of a fabric. Journal of Colloid and Interface Science, 173, 319–327.
Fries, N., & Dreyer, M. (2008). An analytical solution of capillary rise restrained by gravity. Colloid and Interface Science, 320, 259–263.
Barry, D. A., Parlange, J. Y., Sander, M., & Sivaplan, M. (1993). A class of exact-solutions for Richards equation. Journal of Hydrology, 142, 29–46.
Zeng, J., Lin, L., Tang, Y., Sun, Y., & Yuan, W. (2017). Fabrication and capillary characterization of micro-grooved wicks with reetrant cavity array. International Journal of Heat and Mass Transfer, 104, 918–929.
Li, Y., Zhang, C., Chen, C., Chen, H. (2018). Calculation of capillary rise of soils by SWCC model. Advances in Civil Engineering, 10. https://doi.org/10.1155/2018/5190354
Jaine, J. E., & Mucalo, M. R. (2015). Measurements of the wettability of catalyst support materials using the Washburn capillary rise technique. Powder Technology, 276, 123–128.
Brewer, C. E., Chuang, V. J., Masiello, C. A., Gonnermann, H., Gao, X., Dugan, B., Driver, L. E., Panzacchi, P., Zygourakis, K., & Davies, C. A. (2014). New approaches to measuring biochar density and porosity. Biomass and Bioenergy, 66, 176–185.
Author information
Authors and Affiliations
Contributions
Both authors have contributed equally to the work.
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Usevičiūtė, L., Baltrėnaitė-Gedienė, E. Modelling of a Capillary Rise Height of Biochar by Modified Lucas–Washburn Equation. Environ Model Assess 27, 29–43 (2022). https://doi.org/10.1007/s10666-021-09782-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10666-021-09782-6