Prandtl number of gases

Most investigators have been content with correlations of the Nusselt with the Reynolds and Prandtl numbers, or with the Reynolds number alone. The range of numerical values of the Prandtl number of gases is small, and most of the investigations have been conducted with air whose Pr = 0.72 at 100°C. The effect of Pr is small on Figure 17.36(c), and is ignored on Figure 17.36(b) and in some of the equations of Tables 17.17 and 17.18. 

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. 

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. 

In the more dignified terms of dimensionless groups, the Schmidt and Prandtl numbers of gases are equal ... 

The first type is of interest only when considering fluids of low Prandtl number, and this does not usually exist with normal plate heat exchanger applications. The third is relevant only for fluids such as gases which have a Prandtl number of about one. Therefore, let us consider type two. 

The average Nusselt number is not very sensitive to changes in gas velocity and Reynolds number, certainly no more than (Re)I/3. The Sherwood number can be calculated with the same formula as the Nusselt number, with the substitution of the Schmidt number for the Prandtl number. While the Prandtl number of nearly all gases at all temperatures is 0.7 the Schmidt number for various molecules in air does depend on temperature and molecular type, having the value of 0.23 for H2, 0.81 for CO, and 1.60 for benzene. 

Equation (3.58) which, it will be recalled, was deduced without solving the energy equation, gives results which agree reasonably well with the exact results for gases with Prandtl numbers near one. If the Prandtl number of air is assumed to be 0.7, then the value of Nux given by Eq. (3.58) is about 12% greater than the true value. 

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. 

Table 2.4 Thermal conductivities, heat capacities, and Prandtl numbers of some gases and liquids...

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]

Quite recently Sparrow et al. [3.16] communicated an equation valid for Prandtl numbers of the order of 1 for gases and the higher Prandtl numbers for liquids. It is... 

Most liquids have higher Prandtl numbers than gases because the viscosity is generally two or more orders of magnitude higher than for gases, which more... 

These equations are valid only for Prandtl numbers of 1.0 or greater, since the derivation assumes a thermal boundary layer no thicker than the hydrodynamic layer. However, they can be used for gases with 0.7 with little error. The equations are also restricted to cases where the Nusselt number is fairly large, say, 10 or higher, since axial conduction, which was neglected in the derivation, has a significant elfect for thick boundary layers. 

This is the usual form of the Reynolds analogy equation. It agrees fairly well with experimental data for most gases, which have Prandtl numbers of about unity, provided the temperature drop T — T is not large. 

The Colburn equation (Colburn, 1931) also applies to liquids and gases and is almost idra-tical to the Dittus-Boelter equation, but is usually displayed in a j-factor form in terms of a Stanton number, = hJGCp. It is considered valid to a Prandtl number of 160 ... 

Table 2.2 Thermal Conductivities, Heat Capacities, and Prandtl Numbers of Some Gases and Liquids...

The gas-phase wall heat-transfer coefficient can be evaluated by using the gas-phase Reynolds number and Prandtl number in Eq. (33). The thermal conductivities of liquids are usually two orders of magnitude larger than the thermal conductivities of gases therefore, the liquid-phase wall heat-transfer coefficient should be much larger than the gas-phase wall heat-transfer coefficient, and Eq. (34) simplifies to... 

The convective mass transfer coefficient hm can be obtained from correlations similar to those of heat transfer, i.e. Equation (1.12). The Nusselt number has the counterpart Sherwood number, Sh = hml/Di, and the counterpart of the Prandtl number is the Schmidt number, Sc = p/pD. Since Pr k Sc k 0.7 for combustion gases, the Lewis number, Le = Pr/Sc = k/pDcp is approximately 1, and it can be shown that hm = hc/cp. This is a convenient way to compute the mass transfer coefficient from heat transfer results. It comes from the Reynolds analogy, which shows the equivalence of heat transfer with its corresponding mass transfer configuration for Le = 1. Fire involves both simultaneous heat and mass transfer, and therefore these relationships are important to have a complete understanding of the subject. 

The thermal conductivity of gases increases with temperature, but falls with increasing molecular weight. The dimensionless Prandtl number. 

Pr (Cp/Lt/X) is constant for most gases over wide ranges of temperature and pressure and this fact may be used to estimate the thermal conductivity at high temperatures. The Prandtl number is between 0.65 and 1.0, depending on the molecular complexity of the gas. 

This Prandtl-number expression is independent of temperature, since both the viscosity and conductivity expressions have the same temperature dependence. For monatomic gases, y as 5/3, so the expression shows Pr 0.67, which is close to that observed experimentally. For diatomic gases with y = 1.4, the expression yields Pr = 0.74, which is a bit high. 

In fact both the Prandtl number and the heat capacity are temperature-dependent. For gases, however, the dependency is relatively weak, especially for the Prandtl number. The heat capacity cp of air increases by about 30% between 300 K and 2000 K. Because of these temperature dependencies, it may be anticipated (e.g., from Eq. 3.144) that the viscosity and the thermal conductivity generally show slightly different temperature dependencies. 

PRANDTL NUMBER. A dimensionless number equal to the ratio of llie kinematic viscosity to the tlienuoiiielric conductivity (or thermal diffusivity), For gases, it is rather under one and is nearly independent of pressure and temperature, but for liquids the variation is rapid, Its significance is as a measure of the relative rates of diffusion of momentum and heat m a flow and it is important m the study of compressible flow and heat convection. See also Heat Transfer. 

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