Prandtl analogy

V. Tubes, turbulent, smooth tubes, constant surface concentration, Prandtl analogy... 

This is known as the Taylor-Prandtl analogy. In order to apply it, the value of uju has to be known and the way in which this can be found will now be discussed. 

Using the Taylor-Prandtl analogy, determine the relation between the velocity and temperature profiles in the boundary layer. 

Modify the integral equation computer program to use the Taylor-Prandtl analogy. Use this modified program to determine the local Nusselt number variation for the situation described in Problem 6.6. 

The so-called Taylor-Prandtl analogy was applied to boundary layer flow in Chapter 6. Use this analogy solution to derive an expression for the Nusselt number in fully developed turbulent pipe flow. 

Nusselt s film condensation theory presumes a laminar film flow. As the amount of condensate increases downstream, the Reynolds number formed with the film thickness increases. The initially flat film becomes wavy and is eventually transformed from a laminar to a turbulent film the heat transfer is significantly better than in the laminar film. The heat transfer in turbulent film condensation was first calculated approximately by Grigull [4.14], who applied the Prandtl analogy for pipe flow to the turbulent condensate film. In addition to the quantities for laminar film condensation the Prandtl number appears as a new parameter. The results can not be represented explicitly. In order to obtain a clear representation, we will now define the Reynolds number of the condensate film... 

In Section 2.5.1 of Hewitt (1992), prepared by Gnielinski, a more accurate and more widely applicable correlation is given, which accounts for tube diameter-to-tube length ratio for 0 < DJL 1, and IS apphcable to Avide ranges of Reynolds and Prandtl numbers of 2,300 to 1,000,000 and 0.6 to 2,000, respectively. The correlation has a semitheoretical basis in the Prandtl analogy to skin friction in terms of the Darcy friction factor,/ , ... 

Von Karman further modified the Prandtl analogy by considering the buffer region in addition to the viscous sublayer and the turbulent core. These three regions are shown in the universal velocity profile in Fig. 3.10-4. Again, an equation is written for molecular diffusion in the viscous sublayer using only the molecular diffusivity and a Reynolds analogy equation for the turbulent core. Both the molecular and eddy diffusivity are used in an equation for the buffer layer, where the velocity in this layer is used to obtain an... 

Prandtl analogy (1925) provided further refinement to this model and is given as... 

The analogy has been reasonably successful for simple geometries and for fluids of very low Prandtl number (liquid metals). For high-Prandtl-number fluids the empirical analogy of Colburn [Trans. Am. Tn.st. Chem. Ting., 29, 174 (1933)] has been veiy successful. A J factor for momentum transfer is defined asJ =//2, where/is the friction fac tor for the flow. The J factor for heat transfer is assumed to be equal to the J factor for momentum transfer... 

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. 

For mass transfer, which is considered in more detail in Chapter 10, an analogous relation (equation 10.233) applies, with the Sherwood number replacing the Nusselt number and the Schmidt number replacing the Prandtl number. 

Kramers(581 carried out experiments on heat transfer to particles in a fixed bed and has expressed his results in the form of a relation between the Nussell, Prandtl and Reynolds numbers. This equation may be rewritten to apply to mass transfer, by using the analogy between the two processes, giving ... 

Taylor-Prandtl modification of Reynolds analogy for heat transfer and mass transfer... 

Taylor(4) and Prandtl(8 9) allowed for the existence of the laminar sub-layer but ignored the existence of the buffer layer in their treatment and assumed that the simple Reynolds analogy was applicable to the transfer of heal and momentum from the main stream to the edge of the laminar sub-layer of thickness <5. Transfer through the laminar sub-layer was then presumed to be attributable solely to molecular motion. 

Obtain the Taylor-Prandtl modification of the Reynolds Analogy for momentum and heat transfer, and give the corresponding relation for mass transfer (no bulk flow). 

Obtain the Taylor-Prandtl modification of the Reynolds analogy between momentum and heat transfer and write down the corresponding analogy for mass transfer. For a particular system, a mass transfer coefficient of 8,71 x 10 8 m/s and a heat transfer coefficient of 2730 W/m2 K were measured for similar flow conditions. Calculate the ratio of the velocity in the fluid where the laminar sub layer terminates, to the stream velocity. 

Derive the Taylor-Prandtl modification of the Reynolds analogy between heat and momentum transfer and express it in a form in which it is applicable to pipe flow. 

Obtain the Taylor-Prandtl modification of the Reynolds Analogy between momentum transfer and mass transfer (equimolecular counterdiffusion) for the turbulent flow of a fluid over a surface. Write down the corresponding analogy for heat transfer. State clearly the assumptions which are made. For turbulent flow over a surface, the film heat transfer coefficient for the fluid is found to be 4 kW/m2 K. What would the corresponding value of the mass transfer coefficient be. given the following physical properties ... 

Several analytical studies have sought to extend the application of the basic method of Chen. For fluids of Prandtl number different from unity, Bennett and Chen (1980) extended the analysis by a modified Chilton-Colburn analogy to give... 

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. 

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