Research Papers: Heat and Mass Transfer

On the Relaxation Time Scales of the Classical Thermodynamic Model for Heat Transfer in Quiescent Compressible Fluids

[+] Author and Article Information
Leonardo S. de B. Alves

Laboratório de Mecânica Teórica e Aplicada,
Departamento de Engenharia Mecânica,
Universidade Federal Fluminense,
Niterói, Rio de Janeiro 24210-240, Brazil
e-mail: leonardo.alves@mec.uff.br

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 27, 2016; final manuscript received April 16, 2016; published online June 7, 2016. Assoc. Editor: Antonio Barletta.

J. Heat Transfer 138(10), 102004 (Jun 07, 2016) (9 pages) Paper No: HT-16-1037; doi: 10.1115/1.4033462 History: Received January 27, 2016; Revised April 16, 2016

An approximate solution of the classical thermodynamic model for compressible heat transfer of a quiescent supercritical fluid under microgravity leads to the well-known piston effect relaxation time tPE=tD/(γ01)2, where tD is the thermal diffusion relaxation time and γ0 is the ratio between specific heats. This relaxation time represents an upper bound for the asymptotic bulk temperature behavior during very early times, which shows a strong algebraic relaxation due to the piston effect. This paper demonstrates that an additional relaxation time associated with the piston effect exists in this classical thermodynamic model, namely, tE=tD/γ0. Furthermore, it shows that tE represents the time required by the bulk temperature to reach steady-state. Comparisons with a numerical solution of the compressible Navier–Stokes equations as well as experimental data indicate the validity of this new analytical expression and its physical interpretation.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Dahl, D. , and Moldover, M. R. , 1972, “ Thermal Relaxation Near the Critical Point,” Phys. Rev. A, 6(5), pp. 1915–1920. [CrossRef]
Nitsche, K. , and Straub, J. , 1987, “ The Critical Hump of CV Under Microgravity, Results From D-Spacelab Experiment,” 6th European Symposium on Material Sciences Under Microgravity Conditions, ESA Paper No. SP-256.
Boukari, H. , Shaumeyer, J. N. , Briggs, M. E. , and Gammon, R. W. , 1990, “ Critical Speeding Up in Pure Fluids,” Phys. Rev. A, 41(4), pp. 2260–2263. [CrossRef] [PubMed]
Onuki, A. , Hao, H. , and Ferrell, R. A. , 1990, “ Fast Adiabatic Equilibration in a Single-Component Fluid Near the Liquid-Vapor Critical Point,” Phys. Rev. A, 41(4), pp. 2256–2259. [CrossRef] [PubMed]
Zappoli, B. , Bailly, D. , Garrabos, Y. , LeNeindre, B. , Guenoun, P. , and Beysens, D. , 1990, “ Anomalous Heat Transport by the Piston Effect in Supercritical Fluids Under Zero Gravity,” Phys. Rev. A, 41(4), pp. 2264–2267. [CrossRef] [PubMed]
Zappoli, B. , 2003, “ Near-Critical Fluid Hydrodynamics,” C. R. Mec., 331(10), pp. 713–726. [CrossRef]
Barmatz, M. , Hahn, I. , Lipa, J. A. , and Duncan, R. V. , 2007, “ Critical Phenomena in Microgravity: Past, Present and Future,” Rev. Mod. Phys., 79(1), pp. 1–52. [CrossRef]
Carlès, P. , 2010, “ A Brief Review of the Thermophysical Properties of Supercritical Fluids,” J. Supercrit. Fluids, 53(1–3), pp. 2–11. [CrossRef]
Shen, B. , and Zhang, P. , 2013, “ An Overview of Heat Transfer Near the Liquid-Gas Critical Point Under the Influence of the Piston Effect: Phenomena and Theory,” Int. J. Therm. Sci., 71, pp. 1–19. [CrossRef]
Onuki, A. , and Ferrell, R. A. , 1990, “ Adiabatic Heating Effect Near the Gas-Liquid Critical Point,” Physica A, 164(2), pp. 245–264. [CrossRef]
Ferrell, R. A. , and Hao, H. , 1993, “ Adiabatic Temperature Changes in a One-Component Fluid Near the Liquid–Vapor Critical Point,” Physica A, 197(1–2), pp. 23–46. [CrossRef]
Garrabos, Y. , Bonetti, M. , Beysens, D. , Perrot, F. , Fröhlich, T. , Carlès, P. , and Zappoli, B. , 1998, “ Relaxation of a Supercritical Fluid After a Heat Pulse in the Absence of Gravity Effects: Theory and Experiments,” Phys. Rev. E, 57(5), pp. 5665–5681. [CrossRef]
Carlès, P. , Zhong, F. , Weilert, M. , and Barmatz, M. , 2005, “ Temperature and Density Relaxation Close to the Liquid-Gas Critical Point: An Analytical Solution for Cylindrical Cells,” Phys. Rev. E, 71(4), p. 041201. [CrossRef]
Nakano, A. , and Shiraishi, M. , 2004, “ Numerical Simulation for the Piston Effect and Thermal Diffusion Observed in Supercritical Nitrogen,” Cryogenics, 44(12), pp. 867–873. [CrossRef]
Nakano, A. , and Shiraishi, M. , 2005, “ Piston Effect in Supercritical Nitrogen Around the Pseudo-Critical Line,” Int. Commun. Heat Mass Transfer, 32(9), pp. 1152–1164. [CrossRef]
Zhang, P. , and Shen, B. , 2009, “ Thermoacoustic Wave Propagation and Reflection Near the Liquid–Gas Critical Point,” Phys. Rev. E, 79(6), p. 060103. [CrossRef]
Miura, Y. , Yoshihara, S. , Ohnishi, M. , Honda, K. , Matsumoto, M. , Kawai, J. , Ishikawa, M. , Kobayashi, H. , and Onuki, A. , 2006, “ High-Speed Observation of the Piston Effect Near the Gas–Liquid Critical Point,” Phys. Rev. E, 74(1), p. 010101(R). [CrossRef]
Shen, B. , and Zhang, P. , 2010, “ On the Transition From Thermoacoustic Convection to Diffusion in a Near-Critical Fluid,” Int. J. Heat Mass Transfer, 53(21–22), pp. 4832–4843. [CrossRef]
Shen, B. , and Zhang, P. , 2011, “ Thermoacoustic Waves Along the Critical Isochore,” Phys. Rev. E, 83(1), p. 011115. [CrossRef]
Paolucci, S. , 1982, “ On the Filtering of Sound From the Navier–Stokes Equations,” Sandia National Laboratories, Technical Report No. SAND 82-8257.
Klainerman, S. , and Majda, A. , 1982, “ Compressible and Incompressible Fluids,” Commun. Pure Appl. Math., 35(5), pp. 629–651. [CrossRef]
Jounet, A. , Zappoli, B. , and Mojtabi, A. , 2000, “ Rapid Thermal Relaxation in Near-Critical Fluids and Critical Speeding Up: Discrepancies Caused by Boundary Effects,” Phys. Rev. Lett., 84(15), pp. 3224–3227. [CrossRef] [PubMed]
Nikolayev, V. S. , Dejoan, A. , Garrabos, Y. , and Beysens, D. , 2003, “ Fast Heat Transfer Calculations in Supercritical Fluids Versus Hydrodynamic Approach,” Phys. Rev. E, 67(061202), p. 061202. [CrossRef]
Teixeira, P. C. , and de B. Alves, L. S. , 2015, “ Modeling Supercritical Heat Transfer in Compressible Fluids,” Int. J. Therm. Sci., 88, pp. 267–278. [CrossRef]
Cotta, R. M. , 1993, Integral Transforms in Computational Heat and Fluid Flow, CRC Press, Boca Raton, FL.
Turkel, E. , 1993, “ Review of Preconditioning Methods for Fluid Dynamics,” Appl. Numer. Math., 12(1–3), pp. 257–284. [CrossRef]
Turkel, E. , 1999, “ Preconditioning Techniques in Computational Fluid Dynamics,” Annu. Rev. Fluid Mech., 31(1), pp. 385–416. [CrossRef]
Moler, C. , and Loan, C. V. , 2003, “ Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later,” SIAM Rev., 45(1), pp. 1–46. [CrossRef]
Wolfram, S. , 2003, The Mathematica Book, 5th ed., Wolfram Media/Cambridge University Press, London.
Delache, A. , Ouarzazi, M. N. , and Combarnous, M. , 2007, “ Spatio-Temporal Stability Analysis of Mixed Convection Flows in Porous Media Heated From Below: Comparison With Experiments,” Int. J. Heat Mass Transfer, 50(7–8), pp. 1485–1499. [CrossRef]
Hirata, S. C. , Goyeau, B. , Gobin, D. , Chandesris, M. , and Jamet, D. , 2009, “ Stability of Natural Convection in Superposed Fluid and Porous Layers: Equivalence of the One-and Two-Domain Approaches,” Int. J. Heat Mass Transfer, 52(1–2), pp. 533–536. [CrossRef]
Barletta, A. , and Rees, D. A. S. , 2012, “ Linear Instability of the Darcy–Hadley Flow in an Inclined Porous Layer,” Phys. Fluids, 24, p. 074104. [CrossRef]
Sphaier, L. A. , Barletta, A. , and Celli, M. , 2015, “ Unstable Mixed Convection in a Heated Inclined Porous Channel,” J. Fluid Mech., 778, pp. 428–450. [CrossRef]
Masuda, Y. , Aizawa, T. , Kanakubo, M. , Saito, N. , and Ikushima, Y. , 2002, “ One Dimensional Heat Transfer on the Thermal Diffusion and Piston Effect of Supercritical Water,” Int. J. Heat Mass Transfer, 45(17), pp. 3673–3677. [CrossRef]
de B. Alves, L. S. , 2013, “ An Integral Transform Solution for Unsteady Compressible Heat Transfer in Fluids Near Their Thermodynamics Critical Point,” Therm. Sci., 17(3), pp. 673–686. [CrossRef]
Özisik, M. N. , 1993, Heat Conduction, 2nd ed., Wiley Interscience, New York.
Schlichting, H. , 1986, Boundary-Layer Theory, 7th ed., McGraw-Hill, New York.
Guenoun, P. , Khalil, B. , Beysens, D. , Garrabos, Y. , Kammoun, F. , LeNeindre, B. , and Zappoli, B. , 1993, “ Thermal Cycle Around the Critical Point of Carbon Dioxide Under Reduced Gravity,” Phys. Rev. E, 47(3), pp. 1531–1545. [CrossRef]


Grahic Jump Location
Fig. 1

Eigenvalue λN convergence with respect to the number of terms N in the summation series solution, showing that λN→1/γ0 in the limit as N→∞

Grahic Jump Location
Fig. 2

Dimensionless temperature absolute error δΘ versus dimensionless position ξ at dimensionless time γ0 τ=0.001 with γ0=16.2652 [18]

Grahic Jump Location
Fig. 3

Dimensionless temperature Θ versus dimensionless position ξ near the left wall for different values of γ0 τ with γ0=50. Lines are analytical solutions and points are numerical solutions.

Grahic Jump Location
Fig. 4

Dimensionless bulk temperatures Θb versus dimensionless time γ0 τ=t/tE from the approximate and generalized integral transform solutions with γ0=50 and the pure thermal diffusion solution

Grahic Jump Location
Fig. 5

Least-mean squared error δτE∗ of equilibration time interpolation (58) versus its number of terms I for different γ0 values

Grahic Jump Location
Fig. 6

Equilibration time τE∗ normalized by γ0 versus number of terms N in summation series solution for different γ0 values. Points represent the datasets and lines represent their respective interpolation functions based on Eq. (58).

Grahic Jump Location
Fig. 7

Relaxation time estimate τE∞ normalized by γ0 versus specific heat ratio γ0. Dashed line represents theoretical value based on expression (50).



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In