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