Material transport turbulent

J. B. Opfell and B. H. Sage, Turbulence in Thermal and Material Transport Robert E. Treybal, Mechanically Aided Liquid Extraction... 

VI. Combined Thermal and Material Transport in Turbulent Flow. 278... 

The basic nature of the turbulent exchange process is not yet well enough known to allow accurate prediction of behavior without recourse to experiment. Correlation of the growing body of experimental knowledge in this field, however, offers the possibility of evaluating time-averaged point values of thermal and material transport for many conditions of industrial interest. It is the purpose of this discussion to present some of the more elementary considerations of the nature of turbulent flow with particular emphasis upon thermal and material transport. 

The recent advances in turbulence theory have been so rapid and the experimental investigations sufficiently extensive that it is beyond the scope of the present discussion to consider these matters in detail. However, before the influence of turbulence upon thermal and material transport is considered, it is worth while to describe briefly some of the characteristics of turbulent flow from an experimental point of view. The description will be limited to the characteristics of steady uniform flow... 

Figure 4 depicts the variation of the root-mean-square longitudinal fluctuation as a function of position in a floiving stream. These data were taken from the recent experimental investigation by Laufer (L3) and illustrate the complexity of behavior encountered in a steady, uniformly flowing turbulent stream. It is to be expected that fluctuations of temperature and composition are encountered in turbulent streams involving thermal or material transport. 

Fallis (FI) considered thermal transport in transitional and turbulent boundary flows and supplied a reasonable analysis of this difficult problem which is in agreement with the work of Eber (El) and the theory of Eckert and Drewitz (E2). Callaghan (Cl) contributed to the analogies between thermal and material transport in turbulent flow with particular emphasis upon the behavior near and in the boundary layer. The effect... 

Sherwood was one of the early workers to recognize the importance of turbulence (S15, S16, S17) in material transport. He summarized the progress in this field some years ago (S13) and contributed additional experimental work (L7, M2, M3). Kirkwood and Crawford (K7) set forth the relationships for transport in homogeneous phases with particular emphasis upon the interrelation of material and thermal flux. These contributions have laid a satisfactory basis for work in the field which has been well summarized from a macroscopic standpoint by Sherwood and Pigford (S14). 

The influence of turbulence was studied somewhat earlier by Bag-otskaya (Bl). Local coefficients of transport of water into an air jet were evaluated experimentally by Spielman and Jakob (S20). These essentially macroscopic studies supplement the investigations reported elsewhere in this discussion and contribute to the background of experimental information concerning material transport in fluid systems. 

Such expressions can be extended to permit the evaluation of the distribution of concentration throughout laminar flows. Variations in concentration at constant temperature often result in significant variation in viscosity as a function of position in the stream. Thus it is necessary to solve the basic expressions for viscous flow (LI) and to determine the velocity as a function of the spatial coordinates of the system. In the case of small variation in concentration throughout the system it is often convenient and satisfactory to neglect the effect of material transport upon the molecular properties of the phase. Under these circumstances the analysis of boundary layer as reviewed by Schlichting (S4) can be used to evaluate the velocity as a function of position in nonuniform boundary flows. Such analyses permit the determination of material transport from spheres, cylinders, and other objects where the local flow is nonuniform. In such situations it is not practical at the present state of knowledge to take into account the influence of variation in the level of turbulence in the main stream. 

Deissler (D3) made an interesting analysis of material transport in turbulent flow based upon the Reynolds analogy. Under these circum-... 

At present analytical solutions of the equations describing the microscopic aspects of material transport in turbulent flow are not available. Nearly all the equations representing component balances are nonlinear in character even after many simplifications as to the form of the equation of state and the effect of the momentum transport upon the eddy diffusivity are made. For this reason it is not to be expected that, except by assumption of the Reynolds analogy or some simple consequence of this relationship, it will be possible to obtain analytical expressions to describe the spatial variation in concentration of a component under conditions of nonuniform material transport. 

Material transport is usually associated with thermal transport except in situations involving homogeneous phases which can be treated as ideal solutions (L4). For this reason it is necessary to consider the behavior of combined thermal and material transport in turbulent flow. The evaporation of liquids under macroscopic adiabatic conditions is a typical example of such a phenomenon. Under such circumstances the behavior in the boundary layer is similar to that found in the field of aerodynamics in a blowing boundary layer (S4). However, it is not... 

Little detailed experimental information is available on the value of eddy transport properties under conditions of simultaneous thermal and material transport. If it is assumed that the Reynolds analogy is applicable, it follows that the eddy diffusivity and eddy conductivity are equal and independent of cross linking. Such an assumption is probably not true since it is to be expected that a substantial part of the eddy transport is associated with molecular transport particularly as the eddies become small in accordance with Kolmogoroff s (K10) principle. For this reason it is to be expected that temperature gradients in turbulent streams will influence to some extent the material transport in the same... 

Satterfield (S2, S3) carried out a number of interesting macroscopic studies of simultaneous thermal and material transfer. This work was done in connection with the thermal decomposition of hydrogen peroxide and yielded results indicating that for the relatively low level of turbulence experienced the thermal transport did not markedly influence the material transport. However, the results obtained deviated by 10 to 20 from the commonly accepted macroscopic methods of correlating heat and material transfer data. The final expression proposed by Satterfield (S3), neglecting the thermal diffusion effect (S19) in the boundary layer, was written as... 

At present the development of more effective basic correlations of thermal and material transport in turbulent shear flow rests primarily upon an extension of the understanding of the mechanics of turbulence. Howarth and K rm n (Kl, K4) and Batchelor (B6) contributed materially to the knowledge of isotropic, homogeneous turbulence, but the prediction of the behavior in shear flow still must be based on experiment (L3) even for steady, uniform flow. The absence of a basic understanding of the growth and decay of turbulence (K5) prevents a microscopic analysis of thermal and material transport under nonuniform or unsteady conditions. 

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