Prandtl mixing length number

Equation (6-37) represents the friction factor for Newtonian fluids in smooth tubes quite well over a range of Reynolds numbers from about 5000 to 105. The Prandtl mixing length theory and the von Karman and Blasius equations are referred to as semiempirical models. That is, even though these models result from a process of logical reasoning, the results cannot be deduced solely from first principles, because they require the introduction of certain parameters that can be evaluated only experimentally. 

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Despite the fact that equation (3.37) is applicable to all kinds of time-independent fluids, numerous workers have presented expressions for turbulent flow friction factors for specific fluid models. For instance, Tomita [1959] applied the concept of the Prandtl mixing length and put forward modified definitions of the friction factor and Reynolds number for the turbulent flow of Bingham Plastic fluids in smooth pipes so that the Nikuradse equation, i.e. equation (3.37) with n = 1, could be used. Though he tested the applicability of his method using his own data in the range 2000 < Reg(l — 4>f 3 — )< 10, the validity of this approach has not been established using independent experimental data. 

The Prandtl mixing length model, as well as the k-e model, coupled with (8) and (5), have given quite good results predicting turbulent diffusion fluxes is a number of cases however quite clear discrepancies of (5) have been emphasized in other important cases, and other models can now be proposed. For more details concerning turbulence models, a good basic book is the one of Tennekes and Lumley [1]. 

These models are usually categorized according to the number of supplementary partial differential transport equations which must be solved to supply the modeling parameters. The so-called zero-equation models do not use any differential equation to describe the turbulent quantities. The best known example is the Prandtl (19) mixing length hypothesis ... 

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

The first analytical study to predict the performance of tubes with straight inner fins for turbulent airflow was conducted by Patankar et al. [118]. The mixing length in the turbulence model was set up so that just one constant was required from experimental data. Expansion of analytical efforts to fluids of higher Prandtl number, tubes with practical contours, and tubes with spiraling fins is still desirable. It would be particularly significant if the analysis could predict with a reasonable expenditure of computer time the optimum fin parameters for a specified fluid, flow rate, etc. 

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