The main physical properties of liquid metals which differ from other fluids are:
a high thermal conductivity
a high melting point (except mercury and sodium-potassium)
a high boiling point at atmospheric pressure
a low electrical resistivity.
These characteristics are used for particular heat transfer applications, especially at high temperature. Sodium and some alloys are used to cool fast neutron nuclear reactors, which operate at low pressure and high temperature. Lithium or specific alloys are used to cool the fusion reactors blankets; magnetohydrodynamic phenomena are used in metallurgical processes. Chemically, some liquid metals are highly reactive with oxygen. When used, special precautions should be taken to avoid contact with air, water, etc.
Basic information on heat transfer of liquid metals was compiled by Kutateladze (1959) and detailed data on sodium and sodium-potassium alloy were given by Foust (1976) including chemistry, physical properties, heat transfer correlations and industrial applications. Two other liquid metals handbooks were previously edited by Lyon (1952) and Jackson (1955).
For forced and natural convection the governing equations are the same as for any fluid, i.e., conservation of mass, momentum and energy.
Forced convection is governed by two dimensionless parameters: the Reynolds Number ( Re) and the Peclet Number (Pe). The specificity of liquid metals is in the low value of Prandtl number, so Pe << Re. The influence of Prandtl number on temperature distribution in a heated tube is shown in Figure 1.
Liquid metals heat transfer coefficients are high, a fact which could be advantageous but could also be a drawback. The forced convection heat transfer correlations for liquid metals depend on the Peclet number, while for classical fluids (water, air) they depend on the Reynolds number.
The heat transfer correlations for liquid metals in a heated tube have been reported by Foust (1976):
For a uniform heat flux,
(1)For a uniform wall temperature,
(2)
Alternative expressions are given in the article on Liquid Metal Heat Transfer. Heat transfer correlations for liquid metals flow in annuli and between parallel plates have also been reported by Foust (1976).
Longitudinal flow and crossflow through rod or tube bundles have been extensively studied for nuclear reactor applications. The heat transfer correlations are reported by Foust (1976), showing the dependency on the pitch/diameter and on the geometrical arrangement of the bundle.
Natural convection is governed by two dimensionless parameters: the Grashof Number ( Gr) and the Boussinesq number ( Bo). With the following definition
Bo = (Gr)(Pr)^{2} = (Ra)(Pr), liquid metals are characterized by Bo << Ra << Gr. Natural convection heat transfer correlations of liquid metals depend on the Boussinesq number while for classical fluids, they depend on Rayleigh number.
There is a large variety of natural convection configurations, but the basic one is natural convection along a heated plate. Sheriff (1979) has presented a review of the following cases:
for the vertical plate with a uniform wall temperature, the local Nusselt number is:
and for the vertical plate with a uniform heat flux,
where Gr_{ x} = βgx^{3}ΔT/ν^{2} and Gr = βgx^{4}/ν^{2} and x is the vertical distance along the plate.
The configurations of downward-facing plate, upward-facing plate and the influence of inclination have also been reported by Sheriff (1979).
Boiling of liquid metals has been extensively studied in the frame of nuclear reactors safety analyses [Kottowski ( 1994)]. General features related to liquid metal boiling have been described by Dwyer (1976). Foust (1976) has reported various analytical calculations and experimental data, which tend to show a high level of superheat at boiling inception for sodium and potassium. However, this seems to be highly dependent on the purity of the liquid metals, the wetting of the surface, the heat-up kinetics (dT/dt) and the absence of microbubbles in the liquid. Most of the work performed since 20 years ago in subassembly configurations present a low level of superheat, which is clearly more representative of industrial applications.
Apart from superheat, effect of thermal conductivity and high saturation temperature, the boiling of liquid metals at normal pressure presents many analogies with boiling of water at normal pressure: high liquid-to-vapor density ratio, similar flow configurations (bubbly, slug, etc.). Friction pressure drop correlations developed for water are generally used for liquid metals. The heat transfer coefficient under nucleate boiling is very high. A survey of pool boiling of liquid metals is proposed by Shah (1992).
One important feature for liquid metals boiling is the prediction of Critical Heat Flux (CHF). For water CHF depends strongly on the flow conditions. Several correlations have been established, mainly for sodium flow in nuclear reactor subassemblies. Kottowski (1982) proposes the following correlation for CHF:
for a hydraulic diameter d in the range: 0.002 to 0.010 m and for a heating length in the range: 0.3 to 1.5 m.
Further information on boiling of liquid metals is given in the section on Liquid Metal Heat Transfer.
Liquid metals are good electrical conductors, and they interact with electromagnetic fields so that various MHD applications are possible. Analysis of MHD phenomena needs to solve simultaneously the Navier-Stokes and the Maxwell equations. Berton (1991) presents the basic dimensionless parameters deduced from the system of equations:
the Hartmann Number: Ha = BL
the hydrodynamic Reynolds Number, Re, and the magnetic Reynolds number R_{ m} = μ_{0}σuL
the Stuart number: N = σB^{2}L/ρu = Ha^{2}/Re
the Prandtl Number, Pr, and the Batchelor number (also known as the magnetic Prandtl number); Ba = νμ_{0}σ,
where B is the magnetic flux in Teslas, μ_{0} the permeability of the space (4 π × 10^{−7} henry m^{−1}) and σ is the electrical conductance in Siemens.
Various applications of MHD in liquid metals have been developed in industrial processes: metallurgy, liquid metal cooled nuclear reactors, prototypes for fusion reactors, ship propulsion, power conversion. An overview of these applications has been included in the Proceedings of "Energy Transfer in MHD flows" Conference in Cadarache (1991).
Table 1.
Quantity | Symbol | SI Unit |
Magnetic field | B | T |
Critrcal pressure | p_{c} | Pa |
Vacuum magnetic permeability | μ_{0} | H m^{−1} |
Fluid magnetic permeablity | μ | H m^{−1} |
Electrical conductivity | σ | H m^{−1} |
REFERENCES
Berton, R. (1991) Magnétohydrodynamique. Ed. Masson.
Chen, J. C. and Bishop, A. A. (1970) Liquid-metal heat transfer and fluid dynamics. ASME Winter Annual Meeting. New York, USA.
Dwyer, O. E. (1976) Boiling Liquid Metal Heat Transfer. ANS. 244 East Ogden Avenue. Winsdale, ILL.
Foust, O. J. (1976) Sodium-NaK Engineering Handbook, ch. 2. Gordon and Breach. Science Publishers, Inc.
Jackson, C. B. (1955) Liquid Metals Handbook, ch. II: 3 edn.
Kottowski, H. M. Liquid metal thermal hydraulics. 1NFORUM Verlags GmbH.
Kottowski, H., Saraterri, C. (1982) Conveetive heat transfer and critical heat flux at liquid metal boiling. I0th Liquid Metal Boiling Working Group. Karlsruhe, Oct. 27, 29. Available from JRC Ispra.
Lyon, R. N. (1952) Liquid Metals Handbook, ch. 5. 2nd edn.
Shah, M. M. (1992) A survey of experimental heat transfer data for nucleate pool boiling of liquid metals and a new correlation. Int. J. Heat and Fluid Flow. 13. DOI: 10.1016/0142-727X(92)90007-V
Sheriff, N. and Davies, N. W. (1979) Liquid metal natural convection from plane surfaces: a review including recent sodium measurements. Int. J. Heat and Fluid Flow. 1. DOI: 10.1016/0142-727X(79)90002-X
Les références
- Berton, R. (1991) MagnÃ©tohydrodynamique. Ed. Masson.
- Chen, J. C. and Bishop, A. A. (1970) Liquid-metal heat transfer and fluid dynamics. ASME Winter Annual Meeting. New York, USA.
- Dwyer, O. E. (1976) Boiling Liquid Metal Heat Transfer. ANS. 244 East Ogden Avenue. Winsdale, ILL.
- Foust, O. J. (1976) Sodium-NaK Engineering Handbook, ch. 2. Gordon and Breach. Science Publishers, Inc.
- Jackson, C. B. (1955) Liquid Metals Handbook, ch. II: 3 edn.
- Kottowski, H. M. Liquid metal thermal hydraulics. 1NFORUM Verlags GmbH.
- Kottowski, H., Saraterri, C. (1982) Conveetive heat transfer and critical heat flux at liquid metal boiling. I0th Liquid Metal Boiling Working Group. Karlsruhe, Oct. 27, 29. Available from JRC Ispra.
- Lyon, R. N. (1952) Liquid Metals Handbook, ch. 5. 2nd edn.
- Shah, M. M. (1992) A survey of experimental heat transfer data for nucleate pool boiling of liquid metals and a new correlation. Int. J. Heat and Fluid Flow. 13. DOI: 10.1016/0142-727X(92)90007-V
- Sheriff, N. and Davies, N. W. (1979) Liquid metal natural convection from plane surfaces: a review including recent sodium measurements. Int. J. Heat and Fluid Flow. 1. DOI: 10.1016/0142-727X(79)90002-X