Liquid metals, Prandtl numbers

The Prandtl numbers of ideal gases lie between around 0.6 and 0.9, so that their thermal boundary layer is only slightly thicker than their velocity boundary layer. Liquids have Prandtl numbers above one and viscous oils greater than 1000. The thermal boundary layer is therefore thinner than the velocity boundary layer. By the presumptions made, the solution is only valid if 5T/6 < 1. This means that the solution is good for liquids, approximate for gases but cannot be applied to fluids with Prandtl numbers Pr [Pg.318]

The analogy has been reasonably successful for simple geometries and for fluids of very low Prandtl number (liquid metals). For high-Prandtl-number fluids the empirical analogy of Colburn [Trans. Am. Tn.st. Chem. Ting., 29, 174 (1933)] has been veiy successful. A J factor for momentum transfer is defined asJ =//2, where/is the friction fac tor for the flow. The J factor for heat transfer is assumed to be equal to the J factor for momentum transfer... 

Fortunately, most gases and liquids fall within this category. Liquid metals are a notable exception, however, since they have Prandtl numbers of the order of 0.01. 

Equation (5-44) is applicable to fluids having Prandtl numbers between about 0.6 and 50. It would not apply to fluids with very low Prandtl numbers like liquid metals or to high-Prandtl-number fluids like heavy oils or silicones. For a very wide range of Prandtl numbers, Churchill and Ozoe [9] have correlated a large amount of data to give the following relation for laminar flow on an isothermal flat plate ... 

Let us first consider the simple flat plate with a liquid metal flowing across it. The Prandtl number for liquid metals is very low, of the order of 0.01. so that the thermal-boundary-layer thickness should be substantially larger than the hydrodynamic-boundary-layer-thickness. The situation results from the high values of thermal conductivity for liquid metals and is depicted in Fig. 6-15. Since the ratio of 8/8, is small, the velocity profile has a very blunt shape over most of the thermal boundary layer. As a first approximation, then, we might assume a slug-flow model for calculation of the heat transfer i.e., we take... 

Suppose the fluid is highly conducting, such as a liquid metal. In this case, the thermal-boundary-layer thickness will be much greater than the hydrodynamic thickness. This is evidenced by the fact that the Prandtl numbers for liquid metals are very low, of the order of 0.01. For such a fluid, then, we might approximate the actual fluid behavior with a slug-flow model for energy transport in the thermal boundary layer, as outlined in Sec. 6-5. We assume a constant velocity profile... 

Table 2.5 Thermal conductivities, heat capacities, and Prandtl numbers of some liquid metals at atmospheric pressure...

Liquid metals constitute a class of heat-transfer media having Prandtl numbers generally below 0.01. Heat-transfer coefficients for liquid metals cannot be predicted by the usual design equations applicable to gases, water, and more viscous fluids with Prandtl numbers greater than 0.6. Relationships for predicting heat-transfer coefficients for liquid metals have been derived from solution of Eqs. (5-38a) and (5-38b). By the momentum-transfer-heat-transfer analogy, the eddy conductivity of heat is = k for small IVp,. Thus in the solu-... 

As explained earlier, with respect to the heat and mass transfer analogies, the Schmidt number is the Prandtl number analogue. Both dimensionless numbers can be appreciated as dimensionless material properties (they only contain transport media properties). For gases, the Sc number is unity, for normal liquids it is 600-1800. The refined metals and salts can have a Sc number over 10 000. 

It i.s named after l.udwig Prandtl, who introduced the concept of boundary layer in 1904 and made significant contributions to boundary layer theory. The Prandtl numbers of fluids range from less than 0.01 for liquid metals to more than 100,000 for heavy oils (Table 6-2). Note tliat the Prandtl number is in the order of 10 for water. 

The Prandtl numbers of gases are about 1, which indicates that both momentum and heat dissipate tlirough tine fluid at about the same rale. Heat diffuses very quickly in liquid metals (Pr < 1) and very slowly in oils (Pr > 1) relative to momentum. Consequently the thermal boundary layer i.s much thicker for liquid melals and much thinner for oils relative to the velocity boundary layer. 

Liquid metals such as mercury have high thermal conductivities, and are commonly used in applications that require high heat transfer rates. However, they have very small Prandtl numbers, and thus the thermal boundary layer develops much faster than the velocity boundary layer. Then we can assume the velocity in the thermal boundary layer to be constant at the free stream value and solve the energy equation. It gives... 

It is desirable to have a single correlation that applies to all fluids, including liquid metals. By curve-fitting existing data, Churchill and Ozoe (1973) pro posed the following relation which is applicable for all Prandtl numbers and is claimed to be accurate tp 1%,... 

During laminar flow in a tube, the magnitude of the dimensionless Prandtl number Pr is a measure of the relative growth of the velocity and thermal boundary layers. For fluids with Pr = I, such as gases, the two boundary layers essentially coincide with each other. For fluids with Pr > I, such as oils, the velocity boundary layer outgrows the thermal boundary layer. As a result, the hydrodynamic entry length is smaller than the thermal entry length. The opposite is tnie for fluids with Pr < 1 such as liquid metals. 

The relations given so far do not apply to liquid metals because of their very low Prandtl numbers. For liquid metals (0.004 < Pr < 0.01), the following relations are recommended by Sleicher and Rouse (1975) for 10 < Re < 10 ... 

The Prandtl numbers of gases (such as H2 and Ar) commonly used in the CVD processes are around 0.7. Accordingly, the velocity boundary layer is just slightly thinner than that of the thermal boundary layer. For liquid metals (e.g. mercury) with small Prandtl numbers and low viscosities, the thickness of the velocity boundary layer is much thinner that of the thermal boundary layer. For oils with large Prandlt numbers and high viscosities, the thickness of the thermal boundary layer is one order less than that of the velocity boundary layer, as shown in Figure 2.20. 

The range of the Prandtl number is narrower than that of the Schmidt number. In gases such as air, Pr — 1, and in liquids like water, Pr 10. In extremely viscous liquids like glycerin, the Prandtl number is of the order of 103. Liquid metals (sodium, lithium, mercury, etc.) are characterized by low Prandtl numbers 5 x 10-3 < Pr < 5 x 10 2. 

In the molecular thermal conduction layer, adjacent to the tube wall, the deviation of the average temperature T from the wall temperature Ts satisfies the linear dependence (3.3.10). In the logarithmic layer, the average temperature can be estimated using relations (3.3.11), which are valid for liquids, gases, and liquid metals within a wide range of Prandtl numbers, 6 x 10-3 < Pr < 104 [209,212,289],... 

The relationship between the thicknesses of the two boundary layers at a given point along the plate depends on the dimensionless Prandtl number, defined as Cpfijk. When the Prandtl number is greater than unity, which is true for most liquids, the thermal layer is thinner than the hydrodynamic layer, as shown in Fig. 12.1a. The Prandtl number of a gas is usually close to 1.0 (0.69 for air, 1.06 for steam), and the two layers are about the same thickness. Only in heat transfer to liquid metals, which have very low Prandtl numbers, is the thermal layer much thicker than the hydrodynamic layer. 

Prandtl numbers encountered in practice covers a wide range. For liquid metals it is of the order 0.01 to 0.04. For diatomic gases it is about 0.7, and for water at 70 C it is about 2.5. For viscous liquids and concentrated solutions it may be as large as 600. Prandtl numbers for various gases and liquids are given in Appendixes 17 and 18. 

The eddy diffusitives for momentum and heat, and Ejj, respectively, are not properties of the fluid but depend on the conditions of flow, especially on all factors that affect turbulence. For simple analogies, it is sometimes assumed that and jf are both constants and equal, but when determined by actual velocity and temperature measurements, both are found to be functions of the Reynolds number, the Prandtl number, and position in the tube cross section. Precise measurement of the eddy diffusivities is diflScult, and not all reported measurements agree. Results are given in standard treatises. The ratio Sh/sm also varies but is more nearly constant than the individual quantities. The ratio is denoted by i/f. For ordinary liquids, where Np > 0.6, is close to 1 at the tube wall and in boundary layers generally and approaches 2 in turbulent wakes. For liquid metals is low near the wall, passes through a maximum of about unity at j/r X 0.2, and decreases toward the center of the pipe. ... 

Liquid metals are u ed for high-temperature heat transfer, espedally in nuclear reactors. Liquid mercury, sodium, and a mixture of sodium and potassium called NaK are commonly used as carriers of sensible heat. Mercury vapor is also used as a carrier of latent heat. Temperatures of 1500 F and above are obtainable by using such metals. Molten metals have good specific heats, low viscosities, and high thermal conductivities. Their Prandtl numbers are therefore very low in comparison with those of ordinary fluids. 

Stream differs from that in fluids of ordinary Prandtl numbers. In the usual fluid, heat transfer by conduction is limited to the viscous sublayer when is unity or more and occurs in the buffer zone only when the number is less than unity. In liquid metals, heat transfer by conduction is important throughout the entire turbulent core and may predominate over convection throughout the tube. 

The numerical results of the various Reynolds analogy factors are compared in Fig. 6.36 for laminar Prandtl numbers ranging from those characteristic of gases to those of oils and for Re = 107. Results for very low laminar Prandtl numbers, characteristic of liquid metals, are not shown because the assumptions for the velocity distributions in the various analyses are... 

For practical values of H and Prf, Eq. 14.33 was found to be near unity, indicating that acceleration and convection effects are negligible. Chen [34] included the effect of vapor drag on the condensate motion by using an approximate expression for the interfacial shear stress. He was able to neglect the vapor boundary layer in the process and obtained the results shown in Fig. 14.8. The influence of interfacial shear stress is negligible at Prandtl numbers of ordinary liquids (nonliquid metals, Pr< > 1). Chen [34] was able to represent his numerical results by the approximate (within 1 percent) expression ... 

The consequence of a large Schmidt number, common in liquids, is that convection dominates over diffusion at moderate and even relatively low Reynolds numbers (assuming consistent order of magnitude in the terms). In gases these effects are of the same order. On the other hand, heat transfer in low-viscosity liquids by convection and conduction are the same order since the Prandtl number is approximately 1. In highly viscous fluids where the Prandtl number is large, heat transfer by convection predominates over conduction, provided the Reynolds number is not small. The opposite is true for liquid metals, where the Prandtl number is very small, so conduction heat transfer is dominant. 

Convective Diffusion in Zone Refining of Low Prandtl Number Liquid Metals and Semiconductors... 

Several elementary aspects of mass diffusion, heat transfer and fluid flow are considered in the context of the separation and control of mixtures of liquid metals and semiconductors by crystallization and float-zone refining. First, the effect of convection on mass transfer in several configurations is considered from the viewpoint of film theory. Then a nonlinear, simplified, model of a low Prandtl number floating zone in microgravity is discussed. It is shown that the nonlinear inertia terms of the momentum equations play an important role in determining surface deflection in thermocapillary flow, and that the deflection is small in the case considered, but it is intimately related to the pressure distribution which may exist in the zone. However, thermocapillary flows may be vigorous and can affect temperature and solute distributions profoundly in zone refining, and thus they affect the quality of the crystals produced. 

The flow phenomena involved in zone refining will be discussed briefly. In particular we shall consider surface tension driven flow in a cavity containing a low Prandtl number, Pr, fluid (a low Pr number is typical of liquid metals and semiconductors). It will be shown that simplified models of such flow, which simulate the melt configuration in zone refining, predict multiple steady-state solutions to the Navier-Stokes equations exist over a certain range of the characteristic parameter. 

When strong temperature gradients exist, natural convection may be primarily induced thermally or both heat and mass transfer may play comparable roles. In these cases the situation is more complex, because the number of parameters increases. In liquid metals and semiconductors the Schmidt number, v/Dl, is several orders of magnitude greater than the Prandtl number, v/a, and this enables one to solve for the concentration profile in a rather general way without great difficulty as will be discussed next. 

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