Prandtl number, for gases

The Prandtl number for gases varies only slightly with temperature. 

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. 

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... 

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. 

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... 

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 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. 

Thus only a small error is introduced when this expression is applied to gases. The only serious deviations occur for molten metals, which have very low Prandtl numbers. If h is the heat transfer coefficient, then ... 

Selected values of specific heat, density, viscosity, thermal conductivity and Prandtl s number for some gases. 

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. 

Considerations analogous to those for velocity apply to scalar fields as well, and lengths analogous to /k have been introduced for these fields. They differ from /k by factors involving the Prandtl and Schmidt numbers, which differ relatively little from unity for representative gas mixtures. Therefore, to a first approximation for gases, Zk may be used for all fields and there is no need to introduce any new corresponding lengths. 

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. 

A comparison of Eqs. (3.52) and (3.53) and also of their boundary conditions as given in Eqs. (3.24) and (3.54) respectively, shows that these equations are identical in all respects. Therefore, for the particular case of Pr equal to one, the distribution of 9 through the boundary layer is identical to the distribution of uJu ). In this par-ticular case, therefore, Fig. 3.4 also gives the temperature distribution and the two boundary layer thicknesses are identical in this case. Now many gases have Prandtl numbers which are not very different from 1 and this relation between the velocity and temperature fields and the results deduced from it will be approximately correct for them. 

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. 

No real fluids exist for which this is true, but, as mentioned earlier, most gases have Prandtl numbers that are close to unity. 

Equation (5-114) is called the Reynolds analogy for tube flow. It relates the heat-transfer rate to the frictional loss in tube flow and is in fair agreement with experiments when used with gases whose Prandtl numbers are close to unity. (Recall that Pr = 1 was one of the assumptions in the analysis.)... 

Notice that the difference between the adiabatic wall temperature and the actual wall temperature is used in the definition so that the expression will yield a value of zero heat flow when the wall is at the adiabatic wall temperature. For gases with Prandtl numbers near unity the following relations for the recovery factor have been derived ... 

We may note that the original correlation for gases omitted the Prandtl number term in-Eq. (6-17) with little error because most diatomic gases have... 

For gases the Prandtl number ratio may be dropped, and fluid properties are evaluated at the film temperature. For liquids the ratio is retained, and fluid properties are evaluated at the free-stream temperature. Equations (6-19) and (6-20) are in agreement with results obtained using Eq. (6-17) within 5 to 10 percent. 

Properties are evaluated at the film temperature, and it is expected that this relation would be primarily applicable to calculations for free convection in gases. However, in the absence of more specific information it may also be used for liquids. We may note that for very low values of the Grashof-Prandtl number product the Nusselt number approaches a value of 2.0. This is the value which would be obtained for pure conduction through an infinite stagnant fluid surrounding the sphere. 

For gases, the preceding equations for the turbulent regime can be simplified because the Prandtl number (cfi/k) and viscosity for most gases are approximately constant. Assigning the values c/x/k =... 

Under mechanical equilibrium on a molecular scale, the exchange of momentum proceeds faster than the exchange of mass and heat for liquids. On the other hand, the molecular exchange of momentum, matter, and heat is on the same order as gases. The rate of exchange of transport processes is measured by the Schmidt number Sc and the Prandtl number Pr. Usually, the assumption of mechanical equilibrium in gases for heat and mass transfer is not reliable. 

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