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International Journal of Chemical Reactor Engineering

Ed. by de Lasa, Hugo / Xu, Charles Chunbao

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Volume 16, Issue 8


Volume 9 (2011)

Volume 8 (2010)

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Volume 1 (2002)

Transient Flow and Heat Transfer Characteristics of non-Newtonian Supercritical Third-Grade Fluid (CO2) past a Vertical Cylinder

G. Janardhana Reddy / Ashwini Hiremath / Hussain Basha / N.S. Venkata Narayanan
Published Online: 2018-06-22 | DOI: https://doi.org/10.1515/ijcre-2017-0232


The present study deals with the time-dependent natural convective supercritical third-grade fluid flow past a vertical cylinder. A new thermodynamic model for the supercritical carbon di-oxide (CO2) has been derived. In this model the thermal expansion coefficient is characterized as a function of pressure, temperature and compressibility factor. This model uses the Redlich-Kwong equation of state (RK-EOS). The numerically calculated thermal expansion coefficient values of CO2 are validated with available experimental results. The governing non-linear coupled partial differential equations are solved by using Crank-Nicolson method. The obtained numerical data is described in terms of velocity, temperature, skin-friction and Nusselt number through the graphs and tables for the different set of physical parameters. It is observed that the unsteady velocity is an increasing function of reduced pressure and reduced temperature; whereas it is a decreasing function with respect to third-grade fluid parameter. The temperature field is enhanced near the critical point for the increasing values of third-grade fluid parameter. In supercritical fluid region for the increasing values of reduced pressure and reduced temperature, the skin-friction values are magnified against time. Also, the average heat transfer rate decreases for increasing values of third-grade fluid parameter.

Keywords: supercritical third-grade fluid; carbon di-oxide; natural convection; vertical cylinder; reduced pressure; reduced temperature


  • Abbasbandy, S., and T. Hayat. 2011. “On Series Solution for Unsteady Boundary Layer Equations in a Special Third Grade Fluid.” Communication in Nonlinear Science and Numerical Simulation 16: 3140–3146.CrossrefGoogle Scholar

  • Arai, Y., T. Sako, and Y. Takebayashi. 2002. Supercritical Fluids,, . Springer-verlag Berlin Heidelberg.Google Scholar

  • Bae, Yoon-Yeong, Hwan Yeol Kim, et al. 2009. “Convective Heat Transfer to Co2 at Supercritical Pressure Flowing Vertically Upward in Tubes and an Annular Channel.” Experimental Thermal and Fluid Science 33: 329–339.CrossrefGoogle Scholar

  • Bejan, A. 2013. Convection Heat Transfer. John Wiley & Sons, Inc.Google Scholar

  • Bhavnani, S. H., and A. E. Bergles. 1990. “Effect of Surface Geometry and Orientation on Laminar Natural Convection Heat Transfer from a Vertical Flat Plate with Transverse Roughness Elements.” International Journal of Heat and Mass Transfer 33: 965–981.CrossrefGoogle Scholar

  • Blundell, S. J., and K. M. Blundell. 2006. Concepts in Thermal Physics. Oxford University Press.Google Scholar

  • Cao, Yuhui, and Xin-Rong Zhang. 2012. “Flow and Heat Transfer Characteristics of Supercritical CO2 in a Natural Circulation Loop.” International Journal of Thermal Sciences 58: 52–60.CrossrefGoogle Scholar

  • Domingo, C., and P. S. Peternault. 2016. Supercritical Fluid Nanotechnology. CRC press Taylor and Francis group.Google Scholar

  • Dunn, J. E., and K. R. Rajagopal. 1995. “Fluids of Differential Type: Critical Review and Thermodynamic Analysis”.” International Journal Engineering Science 33: 689–729.CrossrefGoogle Scholar

  • Ede, A. J. 1967. Advances in Free Convection. East Kilbride Scotland: National Engineering Laboratory.Google Scholar

  • Fosdick, R. L., and K. R. Rajagopal, Thermodynamics and Stability of Fluids of Third Grade, Proceedings of the Royal Society of London. Series A, 369(1980) 351–377.CrossrefGoogle Scholar

  • Hayat, T., M. Awais, S. Asghar, and S. Obaidat. 2012. “Unsteady Flow of Third Grade Fluid with Soret and Dufour Effects.” ASME Journal of Heat Transfer 134: 062001–1.CrossrefGoogle Scholar

  • Hayat, T., M. Mustafa, and S. Asghar. 2010. “Unsteady Flow with Heat and Mass Transfer of a Third Grade Fluid over a Stretching Surface in the Presence of Chemical Reaction.” Nonlinear Analysis: Real World Applications 11: 3186–3197.CrossrefGoogle Scholar

  • Hayat, T., H. Nazar, M. Imtiaz, A. Alsaedi, and M. Ayub. 2017. “Axisymmetric Squeezing Flow of Third Grade Fluid in Presence of Convective Conditions.” Chinese Journal of Physics 55: 738–754.CrossrefGoogle Scholar

  • Hayat, T., A. Shafiq, and A. Alsaedi. 2015. “MHD Axisymmetric Flow of Third Grade Fluid by a Stretching Cylinder.” Alexandria Engineering Journal 54: 205–212.CrossrefGoogle Scholar

  • Hayat, T., A. Shafiq, A. Alsaedi, and M. Awais. 2013. “MHD Axisymmetric Flow of Third Grade Fluid between Stretching Sheets with Heat Transfer.” Computers & Fluids 86: 103–108.CrossrefGoogle Scholar

  • He, S., W. S. Kim, and J. D. Jackson. 2008. “A Computational Study of Convective Heat Transfer to Carbon Dioxide at a Pressure Just above the Critical Value.” Applied Thermal Engineering 28: 1662–1675.CrossrefGoogle Scholar

  • Huilgol, R. R. 1975. Continuum Mechanics of Viscoelastic Liquids. Delhi: Hindustan Publishing Corporation.Google Scholar

  • Imtiaz, M., A. Alsaedi, A. Shafiq, and T. Hayat. 2017. “Impact of Chemical Reaction on Third Grade Fluid Flow with Cattaneo-Christov Heat Flux.” Journal of Molecular Liquids 229: 501–507.CrossrefGoogle Scholar

  • Jaluria, Y. 1980. Natural Convection Heat and Mass Transfer. Oxford: pergamon press.Google Scholar

  • Jiang, Pei-Xue, Yu Zhang, and Run-Fu Shi. 2008a. “Experimental and Numerical Investigation of Convection Heat Transfer of Co2 at Supercritical Pressures in a Vertical Mini-Tube.” International Journal of Heat and Mass Transfer 51: 3052–3056.CrossrefGoogle Scholar

  • Jiang, Pei-Xue, Run-Fu Shi, Chen-Ru Zhao, and Yi-Jun Xu. 2008b. “Experimental and Numerical Study of Convection Heat Transfer of Co2 at Supercritical Pressures in Vertical Porous Tubes.” International Journal of Heat and Mass Transfer 51: 6283–6293.CrossrefGoogle Scholar

  • Jiang, Pei-Xue, Yi-Jun Xu, Jing Lv, Run-Fu Shi, S. He, and J. D. Jackson. 2004. “Experimental Investigation of Convection Heat Transfer of CO2 at Super-Critical Pressures in Vertical Mini-Tubes and in Porous Media.” Applied Thermal Engineering 24: 1255–1270.CrossrefGoogle Scholar

  • Kawai, S., Direct Numerical Simulation of Trans-Critical Turbulent Boundary Layers at Supercritical Pressures with Strong Real Fluid Effects, in 54th AIAA Aerospace Sciences Meeting, AIAA SciTech Forum (American Institute of Aeronautics and Astronautics (2016).Google Scholar

  • Kawai, S., H. Terashima, and H. Negishi. 2015. “A Robust and Accurate Numerical Method for Trans-Critical Turbulent Flows at Supercritical Pressure with an Arbitrary Equation of State.” Journal of Computational Physics 300: 116–135.CrossrefGoogle Scholar

  • Khonakdar, D. R., and M. R Raveshi. 2016. “Mixed Convection on a Vertical Plate in Supercritical Fluids by Selecting the Best Equation of State.” The Journal of Supercritical Fluids 107: 549–559.CrossrefGoogle Scholar

  • Kiran, E., P. G. Debenedetti, and C. J. Peters. 2000. Supercritical Fluids: Fundamental and Applications. NATO science series, Springer.Google Scholar

  • Lee, H. R., T. S. Chen, and B. F. Armaly. 1988. “Natural Convection along Slender Vertical Cylinders with Variable Surface Temperature.” Journal of Heat Transfer 110: 103–108.CrossrefGoogle Scholar

  • Liao, S. M., and T. S. Zhao. 2002a. “An Experimental Investigation of Convection Heat Transfer to Supercritical Carbon Dioxide in Miniature Tubes.” International Journal of Heat and Mass Transfer 45: 5025–5034.CrossrefGoogle Scholar

  • Liao, S. M., and T. S. Zhao. 2002b. “Measurements of Heat Transfer Coefficients from Supercritical Carbon Dioxide Flowing in Horizontal Mini/Micro Channels.” ASME Journal of Heat Transfer 124: 413–420.CrossrefGoogle Scholar

  • Long, Z. Q., P. Zhang, and B. Shen. 2015. “Natural Convection Heat Transfer of Supercritical Binary Fluid in a Long Closed Vertical Cylinder.” International Journal of Heat and Mass Transfer 80: 551–561.CrossrefGoogle Scholar

  • Massoudi, M., and I. Christie. 1995. “Effects of Variable Viscosity and Viscous Dissipation on the Flow of Third Grade Fluid in Pipe.” International Journal of Non-Linear Mechanics 30: 687–699.CrossrefGoogle Scholar

  • McHugh, M. A., and V. J. Krukonis. 1994. Supercritical Fluid Extraction, Principles and Practice, Second. Boston: Butterworth-Heinemann.Google Scholar

  • Muller, E. A., and L. A. Estevez. 1990. “Mixing Expansivities and Grashof Number in Supercritical Fluids Using Cubic Equations of State.” The Journal of Supercritical Fluids 3: 136–142.CrossrefGoogle Scholar

  • Nadeem, S., and S. Saleem. 2015. “Analytical Study of Third Grade Fluid over a Rotating Vertical Cone in Presence of Nanoparticles.” International Journal of Heat and Mass Transfer 85: 1041–1048.CrossrefGoogle Scholar

  • NIST (National Institute of Standards and Technology). https://www.nist.gov/

  • Onbasioglu, S. U., and H. Onbastoglu. 2004. “On Enhancement of Heat Transfer with Ribs.” Applied Thermal Engineering 24: 43–57.CrossrefGoogle Scholar

  • Pakdemirli, M. 1992. “The Boundary Layer Equations of Third-Grade Fluids.” International Journal of Non-Linear Mechanics 27: 785–793.CrossrefGoogle Scholar

  • Pandey, S., X. Chu, and E. Laurien. 2017. “Investigation of In-Tube Cooling of Carbon Dioxide at Supercritical Pressure by Means of Direct Numerical Simulation.” International Journal of Heat and Mass Transfer 114: 944–957.CrossrefGoogle Scholar

  • Pandey, S., E. Laurien, and Xu Chu. 2017. “A Modified Convective Heat Transfer Model for Heated Pipe Flow of Supercritical Carbon Dioxide.” International Journal of Thermal Sciences 117: 227–238.CrossrefGoogle Scholar

  • Poling, B. E., J. M. Prausnitz, and J. P. O’Connell. 2001. The Properties of Gases and Liquids, fifth. New York: McGraw-Hill.Google Scholar

  • Rajagopal, K. R., and A. S. Gupta. 1981. “On a Class of Exact Solutions to the Equations of Motion of a Second Grade Fluid.” International Journal of Engineering Science 19: 1009–1014.CrossrefGoogle Scholar

  • Rajagopal, K. R., and A. S. Gupta. 1984. “An Exact Solution for the Flow of a non-Newtonian Fluid past an Infinite Porous Plate.” Meccanica 19: 158–160.CrossrefGoogle Scholar

  • Rajagopal, K. R., and P. N. Kaloni. 1989. “Some Remarks on the Boundary Conditions for Fluids of Differential Type.” In Continuum Mechanics and Its Applications, edited by G.A.C. Graham and S.K. Malik, 935–942. New York: Hemisphere.Google Scholar

  • Rajagopal, K. R., A. Z. Szeri, and W. Troy. 1986. “An Existence Theorem for the Flow of non-Newtonian Fluid past an Infinite Porous Plate.” International Journal of Non-Linear Mechanics 21: 279–289.CrossrefGoogle Scholar

  • Rani, H. P., G. J. Reddy, and C. N Kim. 2013. “Transient Analysis of Diffusive Chemical Reactive Species for Couple Stress Fluid Flow over Vertical Cylinder.” Applied Mathematics and Mechanics (English Edition) 34 (8): 985–1000.CrossrefGoogle Scholar

  • Reddy, G. J., A. Hiremath, and M. Kumar. 2018. “Computational Modeling of Unsteady Third-Grade Fluid Flow over a Vertical Cylinder: A Study of Heat Transfer Visualization.” Results in Physics 8: 671–682.CrossrefGoogle Scholar

  • Rivlin, R. S., and J. L. Ericksen. 1955. “Stress Deformation Relations for Isotropic Materials.” Journal Rational Mechanics and Analysis 4: 323–425.Google Scholar

  • Rolando, A., Natural Convection Heat Transfer in Supercritical Fluids, M.Sc. Thesis, Mechanical Engineering, University of Puerto Rico, 2004.Google Scholar

  • Sajid, M., T. Hayat, and S. Asghar. 2007. “Non-Similar Solution for the Axisymmetric Flow of a Third-Grade Fluid over a Radially Stretching Sheet.” Acta Mechanica 189: 193–205.CrossrefGoogle Scholar

  • Sarkar, M. K. S., and D. N. Basu. 2017. “Numerical Appraisal on the Suitability of Supercritical Condition in Natural Circulation Loop with Isothermal Boundary Conditions.” International Journal of Thermal Sciences 111: 30–40.CrossrefGoogle Scholar

  • Schowalter, W. R. 1978. Mechanics of non-Newtonian Fluids. Oxford: Pergamon Press.Google Scholar

  • Shariati, A., and C. J. Peters. 2003. “Recent Developments in Particle Design Using Supercritical Fluids.” Current Opinion in Solid State and Materials Science 7: 371–383.CrossrefGoogle Scholar

  • Shen, B., and P. Zhang. 2013. “An Overview of Heat Transfer near the Liquid-Gas Critical Point under the Influence of the Piston Effect: Phenomena and Theory.” International Journal of Thermal Sciences 71: 1–19.CrossrefGoogle Scholar

  • Shen, B., and P. Zhang. 2015. “Three-Dimensional Thermoconvection from a Non-Uniformly Heated Plate near the Liquid-Vapor Critical Point.” International Journal of Thermal Sciences 89: 136–153.CrossrefGoogle Scholar

  • Sinclai, L. K., J. W. Tester, J. F. H. Thompson, and R.V. Fox. 2018. “The Role of Water in Extraction and Separation of Rare Earth Elements in Supercritical Carbon Dioxide.” Journal of Supercritical Fluids. Article In Press, Accepted Manuscript.Google Scholar

  • Soave, G. 1993. “20 Years of Redlich-Kwong Equation of State.” Fluid Phase Equilibria 82: 345–359.CrossrefGoogle Scholar

  • Span, R., and W. Wangar. 1996. “A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple Point Temperature to 1100 K at Pressures up to 800 MPa.” Journal of Physical and Chemical Reference Data 25 (6): 1509–1596.CrossrefGoogle Scholar

  • Sparrow, E. M., and J. L. Gregg. 1956. “Laminar Free Convection Heat Transfer from the Outer Surface of a Vertical Circular Cylinder.” Transactions of the ASME 78: 18–23.Google Scholar

  • Teymourtash, A. R., D. R. Khonakdar, and M. R. Raveshi. 2013. “Natural Convection on a Vertical Plate with Variable Heat Flux in Supercritical Fluids.” The Journal of Supercritical Fluids 74: 115–127.CrossrefGoogle Scholar

  • Truesdell, C., and W. Noll. 1965. “,” in The Non-Linear Field Theories of Mechanics,. Vol. III/3. New York: Springer.Google Scholar

  • Zhang, Lina, Minshan Liu, Qiwu Dong, and Songwei Zhao. 2011. “Numerical Research of Heat Transfer of Supercritical Co2 in Channels.” Energy and Power Engineering 3: 167–173.CrossrefGoogle Scholar

  • Zhao, Z., D. Cheb, Y. Zhangb, S. Yaoa, K. Zhanga, and Y. Lina. 2016. “Numerical Investigation on Conjugate Heat Transfer to Supercritical CO2 in Membrane Helical Coiled Tube Heat Exchangers.” Numerical Heat Transfer, Part A 69 (9): 977–995.CrossrefGoogle Scholar

About the article

Received: 2017-12-02

Accepted: 2018-04-29

Revised: 2018-03-09

Published Online: 2018-06-22

Citation Information: International Journal of Chemical Reactor Engineering, Volume 16, Issue 8, 20170232, ISSN (Online) 1542-6580, DOI: https://doi.org/10.1515/ijcre-2017-0232.

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