Diffusion mixing length

If there is a temperature gradient within the fluid, the eddies will be responsible for heat transfer and an eddy thermal diffusivity Ep may be defined in a similar way. It is suggested that, since the mechanism of transfer of heat by eddies is essentially the same as that for transfer of momentum, Eh is related to mixing length and velocity gradient in a similar manner. 

Whereas the kinematic viscosity fx/p, the thermal diffusivity k/Cpp, and the diffusivity D are physical properties of the system and can therefore be taken as constant provided that physical conditions do not vary appreciably, the eddy coefficients E, Eh, and ED will be affected by the flow pattern and will vary throughout the fluid. Each of the eddy coefficients is proportional to the square of the mixing length. The mixing length will ... 

Confined flows typically exhibit laminar-flow regimes, i.e. rely on a diffusion mixing mechanism, and consequently are only slowly mixed when the diffusion distance is set too large. For this reason, in view of the potential of microfabrication, many authors pointed to the enhancement of mass transfer that can be achieved on further decreasing the diffusional length scales. By simple correlations based on Fick s law, it is evident that short liquid mixing times in the order of milliseconds should result on decreasing the diffusion distance to a few micrometers. 

With injection mixers (Figures 10.52b,c), in which the one fluid is introduced into the flowing stream of the other through a concentric pipe or an annular array of jets, mixing will take place by entrainment and turbulent diffusion. Such devices should be used where one flow is much lower than the other, and will give a satisfactory blend in about 80 pipe diameters. The inclusion of baffles or other flow restrictions will reduce the mixing length required. 

Prandtl s mixing length theory, the basis of which is outlined in Section 2.9, predicts that the three eddy diffusivities are equal. It is important to appreciate that these eddy diffusivities are not genuine physical properties of the fluid their values vary with position in the flow, as illustrated in Example 1.10. 

Figure 8.24 The advection-diffusion model (Craig, 1974) in a water column of depth Z, mixing length lm, and scavenging length /s. Concentrations [left, equation (8.8.8)] and fluxes [right, equation (8.8.9)] in the water column for the IJZ values labeled on the curves.

Prandtl s mixing length hypothesis (Prandtl, 1925) was developed for momentum transport, instead of mass transport. The end result was a turbulent viscosity, instead of a turbulent diffusivity. However, because both turbulent viscosity and turbulent diffusion coefficient are properties of the flow field, they are related. Turbulent viscosity describes the transport of momentum by turbulence, and turbulent diffusivity describes the transport of mass by the same turbulence. Thus, turbulent viscosity is often related to turbulent diffusivity as... 

Assuming that the mixing length for the concentration field is equal to that of the velocity field, the turbulent diffusivity... 

The turbulent diffusivity concept involves a local (point) description of the transfer process, which is physically reasonable as long as the mixing length is... 

THE MIXING LENGTH RATIO, EDDY DIFFUSIVITY, AND ACOUSTIC WAVES... 

The Mixing Length Ratio, Eddy Diffusivity and Acoustic Waves... 

Classical theory of turbulent diffusion assumes some sort of constant mixing length, whereby... 

The bended micro channels had a width of 180 pm and a depth of 25 pm and a reduced length (25 mm) compared with the mini channels [151]. The flow in such channels was characterized at two very low Re (1.0 and 0.1) and compared with the flow in straight channels under some hydrodynamic conditions. In all four cases, undisturbed laminar flow was found. Mixing was only detectable at Re = 0.1 owing to diffusion mixing at a much prolonged residence time. At Re = 1, no mixing could be detected. 

The data also show that mixing is probably better than expected for diffusion as the only mechanism [33], This is to be expected, since owing to shearing of the layers it is likely that secondary motion is induced. Further, the measurements show that the FTOL and the mixing length are not necessarily equal. 

Transfer by molecular diffusion is discussed in Section 12.2 and the concept of the mixing length in Section 12.3.2. By analogy with kinetic theory, the eddy kinematic viscosity, E, is given by ... 

A comparison of Eqs. (5.15) and (5.38) shows that the eddy diffusivity and mixing length are related by ... 

The more usual approach to speeding up FCM methods for multi-body flow / dispersion calculations such as this is to use the continuum approach. This requires some estimates of turbulence parameters such as mixing lengths or eddy diffusivities for the canopy (on the neighbourhood scale), but these characteristics cannot be deduced from turbulence closure models, although they may be estimated from detailed calculations of flow/dispersion around a few typical obstacles - not a straightforward or accurate process (see Moulinec et al., 2003 [436]). 

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