Turbulence Prandtl number, turbulent

Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. 

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 ... 

The problems that arise when experiments are carried out in a greatly reduced scale can be overcome if the Reynolds number is high and the flow pattern is governed mainly by fully developed turbulence. It is possible to ignore the Reynolds number, the Schmidt number, and the Prandtl number because the structure of the turbulence and the flow pattern at a sufficiently high level of velocity will be similar at different supply velocities and therefore independent of the Reynolds number. The transport of thermal energy and mass by turbulent eddies will likewise dominate the molecular diffusion and will therefore also be independent of the Prandtl number and the Schmidt number. 

Figure 10-67B. Correlation of McAdams representing the condensing film coefficient on the outside of vertical tubes, integrated for the entire tube length. This represents the streamline transition and turbulent flow conditions for Prandtl numbers 1 and 5. Do not extrapolate Prandtl numbers, Pr beyond 5. (Used by permission Engineering Data Book II 1984, Wolverine Tube, Inc.)...

EjEh is termed the Turbulent Prandtl Number and E/Ep the Turbulent Schmidt Number. 

In this table the parameters are defined as follows Bo is the boiling number, d i is the hydraulic diameter, / is the friction factor, h is the local heat transfer coefficient, k is the thermal conductivity, Nu is the Nusselt number, Pr is the Prandtl number, q is the heat flux, v is the specific volume, X is the Martinelli parameter, Xvt is the Martinelli parameter for laminar liquid-turbulent vapor flow, Xw is the Martinelli parameter for laminar liquid-laminar vapor flow, Xq is thermodynamic equilibrium quality, z is the streamwise coordinate, fi is the viscosity, p is the density, <7 is the surface tension the subscripts are L for saturated fluid, LG for property difference between saturated vapor and saturated liquid, G for saturated vapor, sp for singlephase, and tp for two-phase. 

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 ... 

Here, Prt is the turbulent Prandtl number for temperature or enthalpy and ji is the thermal expansion coefficient,... 

In a system with both heat and mass transfer, an extra turbulent factor, kx, is included which is derived from an adapted energy equation, as were e and k. The turbulent heat transfer is dictated by turbulent viscosity, pt, and the turbulent Prandtl number, Prt. Other effects that can be included in the turbulent model are buoyancy and compressibility. 

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... 

The RNG model provides its own energy balance, which is based on the energy balance of the standard k-e model with similar changes as for the k and e balances. The RNG k-e model energy balance is defined as a transport equation for enthalpy. There are four contributions to the total change in enthalpy the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is fixed and user-defined the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). 

Particle Nusselt number Nu — V - kf Peclet number Pe — pv°fpdp kf Prandtl number Pr — r kf Turbulent Prandtl number Prt — ... 

Predicted Nusselt numbers for turbulent flow with constant wall heat flux (John Wiley and Sons from Bird et al., 1964). Abbreviations Nu = Nusselt number Re = Reynolds number Pr = Prandtl number. 

In a supersonic gas flow, the convective heat transfer coefficient is not only a function of the Reynolds and Prandtl numbers, but also depends on the droplet surface temperature and the Mach number (compressibility of gas). 154 156 However, the effects of the surface temperature and the Mach number may be substantially eliminated if all properties are evaluated at a film temperature defined in Ref. 623. Thus, the convective heat transfer coefficient may still be estimated using the experimental correlation proposed by Ranz and Marshall 505 with appropriate modifications to account for various effects such as turbulence,[587] droplet oscillation and distortion,[5851 and droplet vaporization and mass transfer. 555 It has been demonstrated 1561 that using the modified Newton s law of cooling and evaluating the heat transfer coefficient at the film temperature allow numerical calculations of droplet cooling and solidification histories in both subsonic and supersonic gas flows in the spray. 

The gas film coefficient is dependent on turbulence in the boundary layer over the water body. Table 4.1 provides Schmidt and Prandtl numbers for air and water. In water, Schmidt and Prandtl numbers on the order of 1,000 and 10, respectively, results in the entire concentration boundary layer being inside of the laminar sublayer of the momentum boundary layer. In air, both the Schmidt and Prandtl numbers are on the order of 1. This means that the analogy between momentum, heat, and mass transport is more precise for air than for water, and the techniques apphed to determine momentum transport away from an interface may be more applicable to heat and mass transport in air than they are to the liquid side of the interface. 

In the consideration of thermal transport the molecular Prandtl number of the fluid is important (Ml). However, in turbulent flow it is necessary to consider additional Prandtl numbers. The conventional molecular Prandtl number is defined by... 

The Reynolds analogy is equivalent to setting the turbulent Prandtl number as defined in Eq. (24) equal to unity. Figure 13 shows the effect of Reynolds number upon a space average value of the turbulent Prandtl number (C3, P3, S7). Information presented in Fig. 13 is open to uncertainty since it is based upon measurements for air and represents only the space average value of this ratio throughout the turbulent portion of the stream. The turbulent Prandtl number is undoubtedly a function of position as well as of the Reynolds number for a given stream (PI). 

The functions f5-f8 and ip5-ip8 may be determined from a system of two ordinary differential equations. In addition, the form of the functions /7, /8, Prandtl number. It, is interesting that along a laminar ascending flow, in both the plane and radially-symmetric cases, the Reynolds number (defined as ub/v) increases as x3/5 and x1/2, respectively.1 Consequently, at a sufficient height disruption of the laminar flow and transition to turbulent flow should take place. 

Prr then being termed the turbulent Prandtl number. In order to avoid confusion, Pr is then sometimes referred to as the molecular Prandtl number. 

Using Eqs. (5.33) and (5.39) it follows that according to the mixing length theory the turbulent Prandtl number is given by ... 

In applying the turbulence kinetic energy model it is common to assume that the turbulent Prandtl number, Prr, is constant. 

This is, of course, equivalent to assuming that the turbulent Prandtl number, Prj, is equal to 1. 

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