Transport properties Prandtl numbers

The physics of the problem under study is assumed to be governed by the compressible form of the Favre-filtered Navier-Stokes energy and species equations for an ideal gas mixture with constant specific heats, temperature-dependent transport properties, and equal diffusion coefficients. The molecular Schmidt, Prandtl, and Lewis numbers are set equal to 1.0, 0.7, and 1.43, respectively [17]. 

The introduction of heat capacity into the relationships for thermal conductivity and the Prandtl number gives us an opportunity to make a clarification regarding these two quantities. Thermal conductivity is a true heat transport property it describes the ability of a material to transport heat via conduction. Heat capacity, on the other hand, is a thermodynamic quantity and describes the ability of a material to store heat as energy. The latter, while not technically a transport property, will nonetheless be described in this chapter for the various materials types, due in part to its theoretical relationship to thermal conductivity, as given by Eq. (4.35) and (4.36), and, more practically, because it is often used in combination with thermal conductivity as a design parameter in materials selection. 

There is a natural draw rate for a rotating disk that depends on the rotation rate. Both the radial velocity and the circumferential velocity vanish outside the viscous boundary layer. The only parameter in the equations is the Prandtl number in the energy equation. Clearly, there is a very large effect of Prandtl number on the temperature profile and heat transfer at the surface. For constant properties, however, the energy-equation solution does not affect the velocity distributions. For problems including chemistry and complex transport, there is still a natural draw rate for a given rotation rate. However, the actual inlet velocity depends on the particular flow circumstances—there is no universal correlation. 

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. 

When organizing a chapter of thermophysical properties with limited space, some difficult decisions have to be made. Since this is a handbook for heat transfer practitioners, emphasis has been placed on transport rather than thermodynamic properties. The primary exception has been the inclusion of densities and isobaric specific heats, which are needed for the calculation of Prandtl numbers and thermal diffusivities. 

Typical transport properties for the above systems are listed in Table 16.18. Note that they are high-Schmidt number (analogous to Prandtl number) fluids. 

J9A,mix in the expressions for 5c and Sc represents a diffusivity instead of a molecular transport property, one must replace a, mix by the thermal diffusivity 0 (= kidpCp, where p = density, Cp = specific heat, and kjc = thermal conductivity) to calculate the analogous heat transfer boundary layer thickness Sj and the Prandtl number [i.e., Pr = d/p)ja. In the creeping flow regime, where g 9) = I sine. 

The Prandtl number is simply the ratio of kinematic viscosity (t /p) to thermal diffu-sivity (a). Physically, the Prandtl number represents the ratio of the hydrodynamic boundary layer to the thermal boundary layer in the heat transfer between fluids and a stationary wall. In simple fluid flow, it represents the ratio of the rate of impulse transport to the rate of heat transport, ft is determined by the material properties for high viscosity polymer melts, the number is of the order of 10 to 10 . 

The basic similarity hypothesis states simply that the turbulent transport processes of momentum, heat and mass are caused by the same mechanisms, hence the functional properties of the transfer coefficients are simiiar. The different transport coefficients can thus be related through certain dimensionless groups. The closure problem is thus shifted and henceforth consist in formulating sufficient parameterizations for the turbulent Prandtl Pr )- and Schmidt (Sct) numbers. 

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