Diffusion from turbulent pipe flow

Small particles in a turbulent gas dilfuse from one point to another as a result of the eddy motion. The eddy diffusion coefficient of the particles will in general differ from that of the carrier gas. An expression for the particle eddy diffusivity can be derived for a Stokesian particle, neglecting the Brownian motion. In carrying out the analysis, it is assumed that the turbulence is homogeneous and that there is no mean gas velocity. The statistical properties of the system do not change with time. Essentially what we have is a stationary, uniform turbulence in a large box. This is an approximate representation of the core of a turbulent pipe flow, if we move with the mean velocity of the flow. 

Dispersion is the combination of a nonuniform velocity profile and either diffusion or turbulent diffusion to spread the chemical longitudinally or laterally. Dispersion is something very different from either diffusion or turbulent diffusion, because the velocity profile must be nonuniform for dispersion to occur. The longitudinal dispersion of a pipe flow is illustrated in Figure 1.2. While there is diffusion of the chemical. 

When the pipe Reynolds number is greater than about 2100, the velocity boundary layer that forms in the entry region eventually turns turbulent as the gas passes down the pipe. The velocity profile becomes fully developed that is. the shape of the distribution ceases to change at about 25 to 50 pipe diameters from the entry. Small particles in such a flow are transported by turbulent and Brownian diffusion to the wall. In the sampling of atmospheric air through long pipes, wall losses result from turbulent diffusion. Accumulated layers of particles will affect heat transfer between the gas and pipe walls. 

The mean profiles of velocity, temperature and solute concentration are relatively flat over most of a turbulent flow field. As an example, in Figure 1.24 the velocity profile for turbulent flow in a pipe is compared with the profile for laminar flow with the same volumetric flow rate. As the turbulent fluxes are very high but the velocity, temperature and concentration gradients are relatively small, it follows that the effective diffusivities (iH-e), (a+eH) and (2+ed) must be extremely large. In the main part of the turbulent flow, ie away from the walls, the eddy diffusivities are much larger than the corresponding molecular diffusivities ... 

Flow dynamics predict that flow through a pipe is nonuniform with regard to velocity across the diameter of a pipe, for instance. The flow at pipe walls is assumed to be zero. In our idealized biochemical reactor, this concept is represented by a boundary layer in contact with the biofilm. It does not have, of course, a discrete dimension. Rather, it is represented as an area in the structure that has reduced flow and therefore different kinetics than what we would assume exist in a bulk liquid. The boundary layer is affected by turbulence and temperature and this is unavoidable to a degree. Diffusion within the boundary layers is controlled by the chemical potential difference based on concennation. Thus the rate of transfer of pollutant to the organisms is controlled by at least two physical chemical principles, and these principles differentiate an attached growth bioreactor from a suspended growth bioreactor. 

Dispersion coefficients are usually much greater than diffusion coefficients and cause much more rapid mixing than would ever be possible from molecular motion alone (Cussler, 1997). In particular, for turbulent flow in a pipe, the dispersion coefficient is given by... 

Mass transfer can be definnd simply as the movement of any identifiable species from one spatial location to another. Tha mechanism of movement can be macroscopic as in the flow of a fluid in a pipe (convection) or in the mechanical transport of solids by a conveyor belt. In addition, the transport of a panicolar species may be the result of madom molecular motion (molecular diffusion) or randum microscopic fluid motion (eddy or turbulent diffusion) in the presence of a composition gradient within a phase. This chapter is concerned primarily with mass transfer owing to molecular or microscopic processes. 

The problem addressed here is that of expressing mass transfer coefficients that apply to turbulent flow conditions in a tube in terms of appropriate dimensionless groups. To implement Step 1, both fluid mechanical and transport properties must be taken into account. The former determine the degree of turbulence or ability to form eddies, and hence the rate at which mass is transported to or from the tubular wall, whereas the transport parameter determines the rate at which mass is conveyed through the film adjacent to the interface. It is proposed to use velocity v, density p, and viscosity p as the fluid mechanical properties, as each of these parameters either promotes or resists the formation of eddies. Transport through the film is determined by only one parameter, the diffusivity of the conveyed species, hi addition to these factors, we expect pipe diameter to play a role because it determines the distance over which the mass is to be transported and plays a role as well in the degree of turbulence generated in the system. 

For turbulent flow in pipes the velocity profile can be calculated from the empirical power law design formula (1.360). Similar balance equations with purely molecular diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.359) might suffice. It is important to note that the difference between the cross section averaged ID axial dispersion model equations (discussed in the previous section) and the simplified 2D model equations (presented above) is that the latter is valid locally at each point within the reactor, whereas the averaged one simply gives a cross sectional average description of the axial composition and temperature profiles. 

---