Turbulent diffusional flux

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

The thermal motion of molecules of a given substance in a solvent medium causes dispersion and migration. If dispersion takes place by intermolecular forces acting within a gas, fluid, or solid, molecular diffusion takes place. In a turbulent medium, the migration of matter within it is defined as turbulent diffusion or eddy diffusion. Diffusional flux J is the product of linear concentration gradient dCldX multiphed by a proportionality factor generally defined as diffusion coefficient (D) (see section 4.11) ... 

The behavior of the boundary layer discussed in Section 2.2 can be used to obtain a quantitative expression for iV, the diffusional flux of solute 0 to the electrode surface. Let us use again the example from Section 2.2, namely that of a channel formed by two parallel solid electrodes. Figure 2.5 replaces the velocity profile of Fig. 2.1 by a concentration profile at the same location of the electrode for a fully developed turbulent flow. There are three regions a near horizontal portion corresponding to the fully mixed turbulent bulk ... 

Note that Eq. (3.5) does not have a diffusive flux of species k. Intuitively, one could argue that due to, for example, turbulent fluctuations in the liquid phase, the concentration of gas bubbles gets smeared out. Local high concentrations of bubbles are redistributed by the eddies. Thus, a diffusive term should be present in the mass balance, which is proportional to the gradient in the gas fraction. However, it is not and it should not be a rigorous mathematical treatment of the averaging process to derive the E—E equations does exclude such a diffusional term. 

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