Skip to main content
SpringerLink
Log in

Analytical solution of the problem of supercritical fluid instability in a heated channel

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

The paper represents an investigation into thermohydraulic instability in flow of a supercritical fluid with respect to a “density wave”. An analytical solution was obtained for the stability boundary separating stable and unstable modes of the fluid flow. Effects of the thermophysical properties and wall thickness on the flow stability were studied. It was shown that an increase in the thermal conductivity and the thickness of the wall leads to the increase in the flow stability. The theoretically obtained stability boundary was compared with experimental data obtained for the cooling system of superconducting magnets. Taking into account the thermal conjugation “wall-coolant” lifts the problem to the new higher level: an additional parameter is involved into the mathematical description, which causes qualitative changes in the character of the solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Abbreviations

A u :

Amplitude of the velocity oscillations, Eq. (16)

A v :

Amplitude of the oscillations of the specific volume, Eq. (16)

B :

Pressure parameter, Eq. (25)

F :

Parameter of the thermal conjugation of the wall, Eq. (13)

H :

Enthalpy (J/kg)

h :

Heat transfer coefficient (W/m2K)

j :

Mass velocity of the flow (kg/m2s)

l :

Length of the channel (m)

∆p :

Pressure drop at a throttle (Pa)

q :

Heat flux (W/m2)

q v :

Volumetric heat source (W/m3)

S :

Expansion parameter, Eq. (11)

S * :

Generalized expansion parameter, Eq. (30)

u :

Longitudinal velocity of the flow (m/s)

V:

Specific volume of the coolant (m3/kg)

x :

Longitudinal coordinate (m)

X :

Dimensionless longitudinal coordinate, Eq. (19)

\( \left\langle {} \right\rangle \) :

Averaging over the period of oscillations

β :

Eigenfrequency of the oscillation

β * :

Generalized eigenfrequency of the oscillation, Eq. (30)

δ :

Wall thickness (m)

κ :

Parameter, Eq. (15)

τ :

Time (s)

ξ :

Surface friction coefficient

Ω:

Complex oscillation frequency (1/s)

Ω0 :

Scale of the oscillation frequency (1/s), Eq. (7)

ω = Ω/Ω0 :

Nondimensional oscillation frequency

γ :

Increment of perturbations

\( \gamma_{ * } \) :

Generalized increment of perturbations, Eq. (30)

′:

Instantaneous oscillation value

0:

Stability boundary at B = 0

1:

Inlet cross-section of the channel

2:

Outlet cross-section of the channel

w :

Wall

References

  1. Narlikar AV (2005) Frontiers in superconductive materials. Springer, Berlin

    Book  Google Scholar 

  2. Daney DE (1979) An experimental study of thermal-induced flow oscillations in supercritical helium. J Heat Transf Trans ASME 101:9–14

    Article  Google Scholar 

  3. Wang Q, Kim K, Park H et al (2004) Heating surge and temperature oscillation in KSTAR PF and TF coils for plasma disruption under continuous plasma discharging conditions. IEEE Trans Appl Supercond 14:1451–1454

    Article  Google Scholar 

  4. Pioro IL, Duffey RB (2007) Heat transfer and hydraulic resistance at supercritical pressures in power engineering applications. ASME Press, New York

    Book  Google Scholar 

  5. Varaprasad SK, Tummalapalli B (2007) Supercritical flow experimental facility. University of Manitoba, Winnipeg

    Google Scholar 

  6. Labuntsov DA, Mirzoyan PA (1983) Analysis of boundaries of stability of motion of helium at supercritical parameters in heated channel. Therm Eng 30(3):121–123

    Google Scholar 

  7. Drazin PG (2002) Introduction to hydrodynamic stability. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  8. Zudin YB (1998) Calculation of the thermal effect of the wall on the thermohydraulic stability of a flow of liquid of supercritical parameters. High Temp 36:239–243

    Google Scholar 

  9. Zudin YB (1998) The stability of a flow of liquid of supercritical parameters with respect to density-waves. High Temp 36:975–978

    Google Scholar 

  10. Zudin YB (2000) The stability of a flow of liquid of supercritical parameters with respect to density-waves. High Temp 38:156–157

    Article  Google Scholar 

  11. Zudin YB (2011) Theory of periodic conjugate heat transfer, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  12. Baehr HD, Stephan K (2011) Heat and mass transfer, 3rd edn. Springer, Berlin

    Book  Google Scholar 

  13. Bell WW (2004) Special functions for scientists and engineers. Dover books on mathematics. Dover Pubn Inc, Mineola

    Google Scholar 

  14. Ambrosini W (2011) Assessment of flow stability boundaries in a heated channel with different fluids at supercritical pressure. Ann Nuclear Energy 38 (2, 3) 615–627

    Google Scholar 

  15. Weigand B (2004) Analytical methods for heat transfer and fluid flow problems. Springer, Berlin

    Book  MATH  Google Scholar 

  16. Shevchuk IV (2009) Convective heat and mass transfer in rotating disk systems. Springer, Berlin

    Book  MATH  Google Scholar 

  17. Shevchuk IV (2006) Unsteady conjugate laminar heat transfer of a rotating non-uniformly heated disk: application to the transient experimental technique.Internat. J Heat Mass Transf 49(19–20):3530–3537

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to the Prof. Alexander A. Avdeev (Head of Russian Nuclear-Power Machine Building Research Institute), Prof. Bernhard Weigand (Head of Institute of Aerospace Thermodynamics University Stuttgart) and Dr. Igor V. Shevchuk (MBtech Group GmbH & Co. KGaA) for their very useful comments and numerous discussions.

Author information

Authors and Affiliations

  1. Russian State Atomic Energy Corporation, Russian Nuclear-Power Machine Building Research Institute, u. Kosmonavta Volkova, 6a, 125171, Moscow, Russia

    Yuri B. Zudin

Authors
  1. Yuri B. Zudin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuri B. Zudin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zudin, Y.B. Analytical solution of the problem of supercritical fluid instability in a heated channel. Heat Mass Transfer 49, 585–593 (2013). https://doi.org/10.1007/s00231-012-1105-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-012-1105-8

Keywords

Search

Navigation