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Wall-retardation effects on particles settling through non-Newtonian fluids in parallel plates

Abstract

Walls can exert a retardation effect on particles settling in bounded fluid media. In this work, the parallel plate retardation effect was studied for particles falling in non-Newtonian fluids along the centreline of parallel plate ducts. The eccentric effect was also investigated for those particles which approached the wall. For spheres settling in sodium carboxymethylcellulose (CMC) solutions, the variation in wall factors against the size ratio of the sphere’s diameter to the parallel plate wall spacing shows a non-linear trend; the particle settling velocity is independent at small size ratio, and then decreases quickly with increase in size ratio. A new correlation was presented covering a wider range of size ratios (0.02 < λ < 0.83) in the flow region of 0.0011 < Re < 9.75. When particles settle in polyacrylamide solutions, the fluid elasticity reduces the wall-retardation effect and it can be deduced that the drag reduction mechanism of some polyacrylamide solutions may weaken the wall retardation effect. As the spheres settling in the CMC solutions approach the wall, the neighbouring wall exerts no retardation effect at small size ratios (≤ 0.8). Then the settling velocity reduces sharply, while the effect is negligible for polyacrylamide solutions. In comparison with cylinders, the actuating range of the neighbouring wall is smaller for parallel plates.

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References

  1. Ambari, A., Gauthier-Manuel, B., & Guyon, E. (1985). Direct measurement of tube wall effect on the Stokes force. Physics of Fluids, 28, 1559–1561. DOI: 10.1063/1.864990.

    Article  Google Scholar 

  2. Arsenijevi´c, Z. Lj., Grbavči´c, ˇZ. B., Gari´c-Grulovi´c, R. V., & Boškovi´c-Vragolovi´c, N. M. (2010). Wall effects on the velocities of a single sphere settling in a stagnant and countercurrent fluid and rising in a co-current fluid. Powder Technology, 203, 237–242. DOI: 10.1016/j.powtec.2010.05.013.

    Article  Google Scholar 

  3. Ataíde, C. H., Pereira, F. A. R., & Barrozo, M. A. S. (1999). Wall effects on the terminal velocity of spherical particles in Newtonian and non-Newtonian fluids. Brazilian Journal of Chemical Engineering, 16, 387–394. DOI: 10.1590/s0104-66321999000400007.

    Article  Google Scholar 

  4. Balaramakrishna, P. V., & Chhabra, R. P. (1992). Sedimentation of a sphere along the axis of a long square duet filled with non-Newtonian liquids. The Canadian Journal of Chemical Engineering, 70, 803–807. DOI: 10.1002/cjce.5450700427.

    CAS  Article  Google Scholar 

  5. Bougas, A., & Stamatoudis, M. (1993). Wall factor for acceleration and terminal velocity of falling spheres at high Reynolds numbers. Chemical Engineering & Technology, 16, 314–317. DOI: 10.1002/ceat.270160506.

    CAS  Article  Google Scholar 

  6. Brenner, H. (1961). The slow motion of a sphere through a viscous fluid towards a plane surface. Chemical Engineering Science, 3–4, 242–251. DOI: 10.1016/0009-2509(61)80035-3.

    Article  Google Scholar 

  7. Chang, H. D. (1982). Correlation of turbulent drag reduction in dilute polymer solutions with rheological properties by an energy dissipation model. PhD. thesis, Texas A&M University, College Station, TX, USA.

    Google Scholar 

  8. Chhabra, R. P., Tiu, C., & Uhlherr, P. H. T. (1977). Wall effect for sphere motion in inelastic non-Newtonian fluids. In Proceedings of the 6th Australasian Hydraulics and Fluid Mechanics Conference, December 5–9, 1977. Adelaide, Australia.

    Google Scholar 

  9. Chhabra, R. P., & Uhlherr, P. H. T. (1980). Wall effect for high Reynolds number motion of spheres in shear thinning fluids. Chemical Engineering Communications, 5, 115–124. DOI: 10.1080/00986448008935958.

    CAS  Article  Google Scholar 

  10. Chhabra, R. P., Tiu, C., & Uhlherr, P. H. T. (1981). A study of wall effects on the motion of a sphere in viscoelastic fluids. The Canadian Journal of Chemical Engineering, 59, 771–775. DOI: 10.1002/cjce.5450590619.

    CAS  Article  Google Scholar 

  11. Chhabra, R. P., & Uhlherr, P. H. T. (1988). The influence of fluid elasticity on wall effects for creeping sphere motion in cylindrical tubes. The Canadian Journal of Chemical Engineering, 66, 154–157. DOI: 10.1002/cjce.5450660124.

    CAS  Article  Google Scholar 

  12. Chhabra, R. P. (1995). Wall effects on free-settling velocity of non-spherical particles in viscous media in cylindrical tubes. Powder Technology, 85, 83–90. DOI: 10.1016/0032-5910(95)03012-x.

    CAS  Article  Google Scholar 

  13. Chhabra, R. P. (1996). Wall effects on terminal velocity of non-spherical particles in non-Newtonian polymer solutions. Powder Technology, 88, 39–44. DOI: 10.1016/0032-5910(96)03100-2.

    CAS  Article  Google Scholar 

  14. Chhabra, R. P., Agarwal, S., & Chaudhary, K. (2003). A note on wall effect on the terminal falling velocity of a sphere in quiescent Newtonian media in cylindrical tubes. Powder Technology, 129, 53–58. DOI: 10.1016/s0032-5910(2)00164-x.

    CAS  Article  Google Scholar 

  15. D’Avino, G., Hulsen, M. A., Snijkers, F., Vermant, J., Greco, F., & Maffettone, P. L. (2008). Rotation of a sphere in a viscoelastic liquid subjected to shear flow. Part I: Simulation results. Journal of Rheology, 52, 1331–1346. DOI: 10.1122/1.2998219.

    Article  Google Scholar 

  16. Delidis, P., & Stamatoudis, M. (2009). Comparison of the velocities and the wall effect between spheres and cubes in the accelerating region. Chemical Engineering Communications, 196, 841–853. DOI: 10.1080/00986440802668182.

    CAS  Article  Google Scholar 

  17. Di Felice, R. (1996). A relationship for the wall effect on the settling velocity of a sphere at any flow regime. International Journal of Multiphase Flow, 22, 527–533. DOI: 10.1016/0301-9322(96)00004-3.

    Article  Google Scholar 

  18. Ely, J.W., Fowler, S. L., Tiner, R. L., Aro, D. J., Sicard, G. R., Jr., & Sigman, T. A. (2014). ”Slick water fracturing and small proppant” The future of stimulation or a slippery slope? In Proceedings of SPE Annual Technical Conference and Exhibition, October 27–29, 2014. Amsterdam, The Netherlands: Society of Petroleum Engineers. DOI: 10.2118/170784-ms.

    Google Scholar 

  19. Falade, A., & Brenner, H. (1985). Stokes wall effects for particles moving near cylindrical boundaries. Journal of Fluid Mechanics, 154, 145–162. DOI: 10.1017/s002211208500146x.

    CAS  Article  Google Scholar 

  20. Faxén, H. (1922). Der Widerstand gegen die Bewegung einer starren Kugel in einer zähen Fl¨ussigkeit, die zwischen zwei parallelen ebenen Wänden eingeschlossen ist. Annalen der Physik, 68, 89–119. DOI: 10.1002/andp.19223731003. (in German)

    Article  Google Scholar 

  21. Fidleris, V., & Whitmore, R. L. (1961). Experimental determination of the wall effect for spheres falling axially in cylindrical vessels. British Journal of Applied Physics, 12, 490–494. DOI: 10.1088/0508-3443/12/9/311.

    Article  Google Scholar 

  22. Gallego, F., & Shah, S. N. (2009). Friction pressure correlations for turbulent flow of drag reducing polymer solutions in straight and coiled tubing. Journal of Petroleum Science and Engineering, 65, 147–161. DOI: 10.1016/j.petrol.2008.12.013.

    CAS  Article  Google Scholar 

  23. Giesekus, H. (1982). A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. Journal of Non-Newtonian Fluid Mechanics, 11, 69–109. DOI: 10.1016/0377-0257(82)85016-7.

    CAS  Article  Google Scholar 

  24. Giesekus, H. (1983). Stressing behaviour in simple shear flow as predicted by a new constitutive model for polymer fluids. Journal of Non-Newtonian Fluid Mechanics, 12, 367–374. DOI: 10.1016/0377-0257(83)85009-5.

    CAS  Article  Google Scholar 

  25. Happel, J., & Brenner, H. (1983). Low Reynolds number hydrodynamics. The Hague, The Netherlands: Martinus Nijhoff Publishers. DOI: 10.1007/978-94-009-8352-6.

    Google Scholar 

  26. Higdon, J. J. L., & Muldowney, G. P. (1995). Resistance functions for spherical particles, droplets and bubbles in cylindrical tubes. Journal of Fluid Mechanics, 298, 193–210. DOI: 10.1017/s0022112095003272.

    CAS  Article  Google Scholar 

  27. Holz, T., Fischer, P., & Rehage, H. (1999). Shear relaxation in the nonlinear-viscoelastic regime of a Giesekus fluid. Journal of Non-Newtonian Fluid Mechanics, 88, 133–148. DOI: 10.1016/s0377-0257(99)00016-6.

    CAS  Article  Google Scholar 

  28. Ilic, V., Tullock, D., Phan-Thien, N., & Graham, A. L. (1992). Translation and rotation of spheres settling in square and circular conduits: Experiments and numerical predictions. International Journal of Multiphase Flow, 18, 1061–1075. DOI: 10.1016/0301-9322(92)90075-r.

    CAS  Article  Google Scholar 

  29. Kaiser, A. E., Graham, A. L., & Mondy, L. A. (2004). Non-Newtonian wall effects in concentrated suspensions. Journal of Non-Newtonian Fluid Mechanics, 116, 479–488. DOI: 10.1016/j.jnnfm.2003.11.004.

    CAS  Article  Google Scholar 

  30. Lali, A. M., Khare, A. S., Joshi, J. B., & Nigam, K. D. P. (1989). Behaviour of solid particles in viscous non-Newtonian solutions: Settling velocity, wall effects and bed expansion in solid-liquid fluidized beds. Powder Technology, 57, 39–50. DOI: 10.1016/0032-5910(89)80102-0.

    CAS  Article  Google Scholar 

  31. Li, C. L., Lafollette, R., Sookprasong, A., & Wang, S. (2013). Characterization of hydraulic fracture geometry in shale gas reservoirs using early production data. In Proceedings of the International Petroleum Technology Conference, March 26–28, 2013. Beijing, China: International Petroleum Technology Conference. DOI: 10.2523/16896-ms.

    Google Scholar 

  32. Liu, Y., & Sharma, M. M. (2005). Effect of fracture width and fluid rheology on proppant settling and retardation: An experimental study. SPE Annual Technical Conference and Exhibition, October 9–12, Dallas, TX, USA: Society of Petroleum Engineers. DOI: 10.2118/96208-ms.

    Google Scholar 

  33. Lorentz, H. A. (1907). Abhandlungen ¨uber theoretische Physik (chapter II, pp. 23–42). Leipzig, Germany: B. G. Teubner. (in German)

    Google Scholar 

  34. Machač, I., & Lecjaks, Z. (1995). Wall effect for a sphere falling through a non-Newtonian fluid in a rectangular duct. Chemical Engineering Science, 50, 143–148. DOI: 10.1016/0009-2509(94)00211-9.

    Article  Google Scholar 

  35. Madhav, G. V., & Chhabra, R. P. (1994). Settling velocities of non-spherical particles in non-Newtonian polymer solutions. Powder Technology, 78, 77–83. DOI: 10.1016/0032-5910(93)02761-x.

    CAS  Article  Google Scholar 

  36. Malhotra, S., & Sharma, M. M. (2012). Settling of spherical particles in unbounded and confined surfactant-based shear thinning viscoelastic fluids: An experimental study. Chemical Engineering Science, 84, 646–655. DOI: 10.1016/j.ces.2012.09. 010.

    CAS  Article  Google Scholar 

  37. Missirlis, K. A., Assimacopoulos, D., Mitsoulis, E., & Chhabra, R. P. (2001). Wall effects for motion of spheres in powerlaw fluids. Journal of Non-Newtonian Fluid Mechanics, 96, 459–471. DOI: 10.1016/s0377-0257(0)00189-0.

    CAS  Article  Google Scholar 

  38. Miyamura, A., Iwasaki, S., & Ishii, T. (1981). Experimental wall correction factors of single solid spheres in triangular and square cylinders, and parallel plates. International Journal of Multiphase Flow, 7, 41–46. DOI: 10.1016/0301-9322(81)90013-6.

    Article  Google Scholar 

  39. Okuda, K. (1975). Pipe wall effects on suspension velocities of single freely-suspended spheres and on terminal velocities of single spheres in a pipe. Bulletin of JSME, 18, 1142–1150. DOI: 10.1299/jsme1958.18.1142.

    Article  Google Scholar 

  40. Snijkers, F., D’Avino, G., Maffettone, P. L., Greco, F., Hulsen, M. A., & Vermant, J. (2011). Effect of viscoelasticity on the rotation of a sphere in shear flow. Journal of Non-Newtonian Fluid Mechanics, 166, 363–372. DOI: 10.1016/j.jnnfm.2011. 01.004.

    CAS  Article  Google Scholar 

  41. Song, D. Y., Gupta, R. K., & Chhabra, R. P. (2009). Wall effects on a sphere falling in quiescent power law fluids in cylindrical tubes. Industrial & Engineering Chemistry Research, 48, 5845–5856. DOI: 10.1021/ie900176y.

    CAS  Article  Google Scholar 

  42. Tomac, I., & Gutierrez, M. (2015). Micromechanics of proppant agglomeration during settling in hydraulic fractures. Journal of Petroleum Exploration and Production Technology, 5, 417–434. DOI: 10.1007/s13202-014-0151-9.

    CAS  Article  Google Scholar 

  43. Uhlherr, P. H. T., & Chhabra, R. P. (1995). Wall effect for the fall of spheres in cylindrical tubes at high Reynolds number. The Canadian Journal of Chemical Engineering, 73, 918–923. DOI: 10.1002/cjce.5450730615.

    CAS  Article  Google Scholar 

  44. Unnikrishnan, A., & Chhabra, R. P. (1990). Slow parallel motion of cylinders in non-Newtonian media: wall effects and drag coefficient. Chemical Engineering and Processing: Process Intensification, 28, 121–125. DOI: 10.1016/0255-2701(90)80008-s.

    CAS  Article  Google Scholar 

  45. Verhulst, K., Moldenaers, P., & Minale, M. (2007). Drop shape dynamics of a Newtonian drop in a non-Newtonian matrix during transient and steady shear flow. Journal of Rheology, 51, 261–273. DOI: 10.1122/1.2426973.

    CAS  Article  Google Scholar 

  46. Wang, Y. T., & Miskimins, J. L. (2010). Experimental investigations of hydraulic fracture growth complexity in slickwater fracturing treatments. In Proceedings of the Tight Gas Completions Conference, November 2–3, 2010. San Antonio, TX, USA: Society of Petroleum Engineers. DOI: 10.2118/137515-ms.

    Book  Google Scholar 

  47. Yin, J. C., Xie, J., Datta-Gupta, A., & Hill, A. D. (2011). Improved characterization and performance assessment of shale gas wells by integrating stimulated reservoir volume and production data. In Proceedings of the SPE Eastern Regional Meeting, August 17–19, 2011. Columbus, OH, USA: Society of Petroleum Engineers. DOI: 10.2118/148969-ms.

    Book  Google Scholar 

  48. Zhang, G. D., Li, M. Z., Li, J. B., Yu, M., Qi, M. H., & Bai, Z. F. (2015). Wall effects on spheres settling through non-Newtonian fluid media in cylindrical tubes. Journal of Dispersion Science and Technology, 36, 1199–1207. DOI: 10.1080/01932691.2014.966309.

    CAS  Article  Google Scholar 

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Correspondence to Guo-Dong Zhang or Ming-Zhong Li.

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Zhang, GD., Li, MZ., Xue, JQ. et al. Wall-retardation effects on particles settling through non-Newtonian fluids in parallel plates. Chem. Pap. 70, 1389–1398 (2016). https://doi.org/10.1515/chempap-2016-0082

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Keywords

  • wall factor
  • settling velocity
  • viscoelastic
  • Giesekus model
  • eccentric effect
  • parallel plate