A-to-Z Guide to Thermodynamics,
Heat & Mass Transfer, and Fluids Engineering

Liquid Metals are a specific class of coolants. Their basic advantage is a high molecular thermal conductivity which, for identical flow parameters, enhances heat transfer coefficients. Another distinguishing feature of liquid metals is the low pressure of their vapors, which allows their use in power engineering equipment at high temperatures and low pressure, thus alleviating solution of mechanical strength problems.

The most widespread liquid metals used in engineering are alkali metals. Among them sodium is first and foremost, used as a coolant of fast reactors and a working fluid of high-temperature heat pipes. Potassium is a promising working medium for space power plants. In some cases eutectic Na−K and Pb−Bi alloys and Hg, Li, and Ga are also used.

The high thermal conductivity and, hence, low Prandtl numbers of liquid metals imply that heat transfer by molecular thermal conduction is significant not only in the near-wall layer, but also in the flow core even in a fully developed turbulent flow. The thickness of the thermal boundary layer proves to be substantially larger than the thickness of the hydrodynamic boundary layer (see Single-phase Forced Convection Heat Transfer).

The hydrodynamic characteristics of liquid metal flows (friction factor and the coefficient of local resistance) are calculated by conventional formulas.

Fully developed heat transfer to liquid metals in tubes may be calculated using the following generalized relations:

For the case Tw = const (curve 1 in Figure 1)

(1) where Nu = αd/λ and Pe = ud/κ where α is the heat transfer coefficient, d the tube diameter, λ the thermal conductivity and κ the thermal diffusivity. For the case = const (curve 2 in Figure 1)

(2) At high values of Pe number Eqs. (1) and (2) approach each other (Figure 1). Figure 1. Single phase heat transfer relationships for liquid metals.

An approximate calculation of mean heat transfer in the entrance region in the case of turbulent flow can be performed with Eqs. (1) and (2) by introducing a correction factor for this region

(3) where 1 is the tube length. The length of the thermal entrance region is 10 to 15d.

For tube bundles in longitudinal flow, the following relationships may be used:

(4) and

(5) where Pe is calculated from the free-stream velocity and the outside tube diameter. These relationships are valid for the range of pitch-to-diameter ratio s/d = 1.2−1.75 and may also be used for staggered and in-line tube bundles in crossflow.

The thermal properties of liquid metal depend only slightly on temperature. Taking into account the small transverse temperature difference in liquid metal flow due to the high thermal conductivity, the effect of nonisothermal conditions in heat transfer is not significant and, as a rule, is not considered.

Bundles of fuel rods in triangular or square arrays are often used in reactors with a liquid metal coolant. In this case, due to the nonuniform flow past the central, peripheral and angular fuel rods, the heat transfer to them is different. There are also variations in heat transfer around the perimeter of the fuel rod. Normally, heat transfer is determined above all by the Pe number. However, the pitch to diameter ratio of the rods in the bundle, the arrangement of the rods and the presence of plugs mounted for equalizing coolant flow rate over the bundle section have proved to be important. A uniform temperature distribution around the fuel rod perimeter, except for the above factors, depends on the ratio of the coefficient of thermal conductivity of the liquid metal to that of the rod enclosure. These factors assume particular importance in tightly-packed bundles.

Heat transfer for the cases indicated above is calculated using unwieldy empirical relations that are valid, as a rule, within a narrow range of parameters. Details of these are presented in handbooks.

Heat transfer by natural convection from a horizontal cylinder is described by the formula

(6) where Nu = αd/λ, Pr = cpη/λ and Gr = gd3ρ2βΔT/η2 where cp is the specific heat capacity, η the viscosity, g the acceleration due to gravity, ΔT the temperature difference between the surface and the fluid and β the coefficient of volumetric thermal expansion. C = 0.67, n = 1/4 for Gr = 102−108, and C = 0.35, n = 1/3 for Gr > 108 For a vertical cylinder of height H and radius r, the equation:

(7) may be used for where NuH = αH/λ ; RaH = GrHPr = gβΔTH3/νκ.

The relation

(8) where C(φ) reduces from 0.069 to 0.049, with φ varying from 0 to 90°, is used to calculate heat transfer in a plane gap between two surfaces arranged at an angle φ to the horizontal in the 1.5 × 105 ≤ Ra ≤ 2.5 × 108 range.

It has been established that heat transfer in liquid metals depends, to a high degree, on fouling resistances at the wall-liquid interface. These resistances appear due to chemical or electrochemical interaction between the wall material and a coolant to produce a surface layer of intermetallics, carbides and other compounds or solid solutions with reduced thermal conductivity. Mass transfer and deposition of corrosion products are also possible on the heat exchange surface. In all cases, much importance is attached not only to the chemical compatibility of the liquid metal coolant and the wall material, but also to the degree of metal purity. This is given a special attention with special in-line systems for metal purification used in many systems.

"Metal-metal" heat exchangers with a bilateral flow of single-phase coolant past the wall are calculated by conventional relations. However, the design of such apparatus — primarily shell-and-tube heat exchangers — has a specific feature. Due to the relatively low specific heat of liquid metals, heating of the coolant is comparable to, and in some cases appreciably exceeds, the value of the governing temperature difference. Thus, a correct allowance for the bypass leakages of the coolant along the shell and the contribution of zones of deteriorated flow along the heat exchange surface are of crucial significance. In this case a "zone-by-zone" approach in designing heat exchangers has proven to be efficient.

One of the principal specific features of boiling of most of liquid metals — alkali metals above all — is that the superheat for the incipience of boiling may amount to tens, and in some cases, hundreds of degree. This is due to the good wetting of the solid metal surfaces by alkali metals, growing solubility of inert gases with temperature, and a small slope in the vapor pressure curve dp/dTs. Thus, measures should be taken, whenever necessary, for reducing the incipient boiling superheat and ensuring the reproducibility of its value. Note that mercury, which poorly wets most technical surfaces, has extremely low incipient superheat.

Pool boiling of alkali metals at moderate heat loads is characterized by unstable boiling, i.e., spontaneous switching from a natural convection regime (curve 1 in Figure 2) to developed boiling (curve 2 in Figure 2) and vice versa (circles in Figure 2). This transition under the condition = const is accompanied by wall temperature fluctuations; these are greatest at low pressure and low heat flux. Figure 2. Pool boiling of liquid metals: spontaneous switching between natural connection (I) and developed boiling (2).

The fraction of the overall heat flux accounted for by bubble evaporation during growth on the heating surface is between 20−60°. The remainder of the heat is transferred by convection to the liquid bulk, or is removed with the superheated liquid surrounding the bubbles rising to the free surface. These effects are linked with the high thermal conductivity of liquid metals.

The heat transfer coefficient αpb, for nucleate pool boiling of alkali metals is described by an empirical formula that is typical for boiling of most of liquids under similar conditions

(9) where αpb is expressed in W/m2K, in W/m2, p in MPa. For sodium, A = 22.4, m = 0.67, n = 0.4 in the 5 to 30 kPa range and A = 7.55, m = 0.67, n = 0.1 in the 30 to 150 kPa range. For potassium, A = 6.35, m = 0.67, n = 0.1 in the 10 to 200 kPa range.

Critical heat fluxes which bring about the transition from nucleate to film boiling, are described for alkali metals by the empirical formula

(10) where λ is the coefficient of thermal conductivity and Pcr, the critical pressure

Flow regimes of two-phase alkali metal flows are the same as those of ordinary liquids (see Forced Convection Boiling). However, owing to the high incipient boiling superheat and a high ratio of specific volumes of vapor and liquid phase (low operating pressures), the regions of bubble and slug flow regimes may correspond to an extremely narrow range of vapor quality or be missing altogether. The annular-dispersed flow regime (see Annular Flow) is predominant. Due to the high thermal conductivity, the temperature difference across the liquid film in annular flow (ΔTlf) is small. Even at high evaporation rates, the temperature difference across the liquid-vapor interface ΔTev is small also. Thus, the overall difference between the wall and the saturation temperature ΔT w = T w −Ts = ΔTlf + ΔTev is insufficient for incipience of vapor bubbles on the heating surface. This means that classic boiling with vapor bubbles growing on the wall is usually absent. Phase transition occurs by evaporation from the interface, to which heat is supplied by thermal conduction through the liquid film.

The heat transfer coefficient for forced convection boiling is determined as:

(11) where ΔTev can be estimated by the Hertz-Knudsen correlation; ΔTlf, by conventional relations for thermal conduction; and film thickness can be determined sufficiently accurately from the Martinelli-Lockart relation (see Forced Convection Boiling) for a given vapor quality. Typical values of heat transfer coefficients lie between 5 × 104 and 105 W/m2K. It should be noted that αfcb is independent of .

Burnout for a forced, two-phase liquid metal flow is commonly linked with the dryout of the near-wall liquid film. The boundary quality which gives rise to the dryout of the film depends on pressure and mass flow rate. It is fairly high and varies from 0.8 to 0.9 for pressures typical of alkali metals. A further increase in can be attained applying porous coatings, coiled tubes or other means facilitating retention on the wall of moisture deposited from the flow core. The length of the zone of the complete film dryout is Δx ≈ 0.05−0.1.

Condensation of liquid metals readily wetting the surface obeys the same laws as condensation of conventional liquids does (see Condensation). As a rule, this is a film-type condensation. The high thermal conductivity of liquid metals leads to a drastic reduction of the contribution of liquid film thermal resistance to overall heat transfer resistance during condensation (for nonmetallic liquids it is the basic contribution). Simultaneously, the contribution of resistance at the vapor-film interface and, in particular, of diffusion resistance grows if noncondensable gases are present or chemical reactions proceed in vapor phase, which require that efforts be made to expel impurities from a vapor. Typical values of the heat transfer coefficient for condensation in the absence of noncondensable gases are from 8 × 104 to 105 W/m2K.