Prandtl’s number,

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

Lewis number (Le) Schmidt s number (Sc) Prandtl s number (Pr) Stefan number (Ste)... 

Gr Grashof s number defined in Equation 2.31 Pr Prandtl s number defined in Equation 2.32... 

The condition expressed by Equation 8.2 is usually fulfilled and channel flow occurs in the laminar range. This allows the problem to be treated analytically and the solutions presented in this chapter rely on this assumption. Accounting for the simultaneous development of the velocity profile compared with concentration/temperature fields requires numerical evaluation, and the importance of this effect is measured by Prandtl s number for heat transfer or by Schmidt s number for mass transfer ... 

One other measurement technique that has been used to measure Kl over a shorter time period, and is thus more responsive to changes in wind velocity, is the controlled flux technique (Haupecker et al., 1995). This technique uses radiated energy that is turned into heat within a few microns under the water surface as a proxy tracer. The rate at which this heat diffuses into the water column is related to the liquid film coefficient for heat, and, through the Prandtl-Schmidt number analogy, for mass as well. One problem is that a theory for heat/mass transfer is required, and Danckwert s surface renewal theory may not apply to the low Prandtl numbers of heat transfer (Atmane et al., 2004). The controlled flux technique is close to being viable for short-period field measurements of the liquid film coefficient. 

The simplest way to close equations (3.1.37) is to use the hypothesis that the turbulent Prandtl number for the examined process is a constant quantity. Then it readily follows from Eq. (3.1.39) that the turbulent diffusion coefficient is proportional to the turbulent viscosity >t = i /Pr,. By using the expression for vx borrowed from the corresponding hydrodynamic model, one can obtain the desired value of Dt. In particular, following Prandtl s or von Karman s model, one can use formula (1.1.21) or (1.1.22) for vx. 

Blasius steady-flow, laminar, flat-plate, boundary-layer solution is a numerical solution of his simplification of Prandtl s boundary-layer equations, which are a simplified, one-dimensional momentum balance and a mass balance. This type of solution is known in the boundary-layer literature as an exact solution. Exact solutions can be found for only a very limited number of cases. Therefore, approximate methods are available for making reasonable estimates of the behavior of laminar boundary layers (Prob. 11.8). 

For the curves in Fig. 11.7, the value of n that gives the best representation of the experimental curves varies from for the lowest Reynolds number to jq for the highest Reynolds number. Prandtl selected j as the best average, deducing Prandtl s power velocity distribution rule. This is not an exact rule, because if it were a general rule, then all the curves in Fig. 11.7 would be identical. Furthermore, it cannot be correct very near the wall of the tube, because there it predicts that dVIdy is infinite and hence that the shear stress is infinite. Nonetheless, it is widely used because it is simple and, as we will see in Sec. 11.5, because it gives useful results. 

It is possible to find more complex correlations for the velocity distribution in a pipe which do not have the limitations of Prandtl s power rule. In Fig. 11.7 the Reynolds number appears as a parameter in the velocity distribution plot. In trying to produce a universal velocity distribution rule, it seems logical to change the coordinates in Fig. 11.7 so that the Reynolds number enters either explicitly or implicitly in one of the coordinates, in the hope of getting all the data onto one curve. 

In Eq. 11.34 the velocity in the Reynolds number and the velocity in the expression for the friction factor are average pipe velocities. From Prandtl s 7 power rule it can be shown that for circular pipes (Prob. 11.10) this average velocity is 0.817 times the maximum velocity. Prandtl rounded this to 0.8. 

It can be shown that the Reynolds number represents the ratio of inertial forces to viscous forces in the flow field. Flows at sufficiently low Re therefore behave as if highly viscous, with httle to no fluid acceleration possible. At the opposite extreme, high Re flows behave as if lacking viscosity. One consequence of this distinction is that very high Reynolds number flow fields may at first seem to contradict the no-slip condition, in that they seem to slip along a solid boundary exerting no shear stress. This dilemma was first resolved in 1905 with Prandtl s introduction of the boundary layer, a thin region of the flow field adjacent to the boundary in which viscous effects are important and the no-slip condition is obeyed [19-21]. 

American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. 

For sources, units, and remarks, see Table 2-228. v = specific volume, mVkg h = specific enthalpy, kj/kg s = specific entropy, kJ/(kg-K) c = specific beat at constant pressure, kJ/(kg-K) i = viscosity, 10 Pa-s and k = tberni conductivity, VW(m-K). For specific beat ratio, see Table 2-200 for Prandtl number, see Table 2-369. 

Thermal conductivity, W/(m-K) Temperature, K Viscosity, 10 Pa-s Temperature, K Prandtl number, dimensio Temperature, K nless... 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

From Tolmin s theory and experimental data (e.g., Reichardtthe relationship between velocity profile and temperature profile in the jet cross-section can be expressed using an overall turbulent Prandtl number Pr = v /a, where Vf is a turbulent momentum exchange coefficient and a, is a turbulent heat exchange coefficient ... 

Y.-S. Lee, C.-H. Chun. Experiments on the oscillatory convection of low Prandtl number hquid in Czochralski configuration for crystal growth with cusp magnetic field. J Cryst Growth 180 411, 1997. 

Above a Reynolds number of around 2000, the condensate film becomes turbulent. The effect of turbulence in the condensate film was investigated by Colburn (1934) and Colburn s results are generally used for condenser design, Figure 12.43. Equation 12.51 is also shown on Figure 12.43. The Prandtl number for the condensate film is given by ... 

L/pj-A)(S/psA), liquid-solids velocity ratio, dimensionless Number of heat-transfer stages, dimensionless = hdp/kg, Nusselt number, dimensionless Pressure drop, gm-wt/cm2 = Cpu kg, Prandtl number, dimensionless = dpiipj U, Reynolds number, dimensionless S Mass velocity of solids, gm/cirf sec... 

In this equation S includes heat of chemical reaction, any interphase exchange of heat, and any other user-defined volumetric heat sources. At is defined as the thermal conductivity due to turbulent transport, and is obtained from the turbulent Prandtl number... 

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