Systems turbulent, diffusion

Validation and Application. VaUdated CFD examples are emerging (30) as are examples of limitations and misappHcations (31). ReaUsm depends on the adequacy of the physical and chemical representations, the scale of resolution for the appHcation, numerical accuracy of the solution algorithms, and skills appHed in execution. Data are available on performance characteristics of industrial furnaces and gas turbines systems operating with turbulent diffusion flames have been studied for simple two-dimensional geometries and selected conditions (32). Turbulent diffusion flames are produced when fuel and air are injected separately into the reactor. Second-order and infinitely fast reactions coupled with mixing have been analyzed with the k—Z model to describe the macromixing process. 

The generalized transport equation, equation 17, can be dissected into terms describing bulk flow (term 2), turbulent diffusion (term 3) and other processes, eg, sources or chemical reactions (term 4), each having an impact on the time evolution of the transported property. In many systems, such as urban smog, the processes have very different time scales and can be viewed as being relatively independent over a short time period, allowing the equation to be "spht" into separate operators. This greatly shortens solution times (74). The solution sequence is... 

The Gaussian Plume Model is the most well-known and simplest scheme to estimate atmospheric dispersion. This is a mathematical model which has been formulated on the assumption that horizontal advection is balanced by vertical and transverse turbulent diffusion and terms arising from creation of depletion of species i by various internal sources or sinks. In the wind-oriented coordinate system, the conservation of species mass equation takes the following form ... 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

In accordance with Pick s Law, diffusive flow always occurs in the direction of decreasing concentration and at a rate, which is proportional to the magnitude of the concentration gradient. Under true conditions of molecular diffusion, the constant of proportionality is equal to the molecular diffusivity of the component i in the system, D, (m /s). For other cases, such as diffusion in porous matrices and for turbulent diffusion applications, an effective diffusivity value is used, which must be determined experimentally. 

The slope of the lines presented in Figure 5 is defined as k(q/v). The q/v term defines the turnover of the tank contents or what is commonly referred to as the retention time. When q is increased, the liquid contacts the carbon more often and the removal of pesticides should increase, however, the efficiency term, k, can be a function of q. As the waste flow rate is increased, the fluid velocity around each carbon particle increases, thereby increasing system turbulence and compressing the liquid boundary layer. The residence time within the carbon bed is also decreased at higher liquid flow rates, which will reduce the time available for the pesticides to diffuse from the bulk liquid into the liquid boundary layer and into the carbon pores. From inspection of Table II, the pesticide concentration also effects the efficiency factor, k can only be determined experimentally and is valid only for the equipment and conditions tested. 

Chapters Turbulent Diffusion. Turbulent diffusion is an important transport mechanism in the atmosphere, oceans, lakes, estuaries, and rivers. In fact, most of the atmosphere and surface waters of the Earth are turbulent. If you are going to work in any of these systems, it will be important to have at least a working knowledge of turbulent diffusion. 

Other factors also come into play in laboratory systems. For example, McMurry and Rader (1985) have shown that particle deposition at the walls of Teflon smog chambers is controlled by Brownian and turbulent diffusion for particles with Dp 0.05 yxm and by gravitational settling for particles with Dp > 1.0 yxm. However, in the 0.05- to 1.0-yxm range, the deposition is controlled by electrostatic effects Teflon tends to... 

First a derivative is given of the equations of change for a pure fluid. Then the equations of change for a multicomponent fluid mixture are given (without proof), and a discussion is given of the range of applicability of these equations. Next the basic equations for a multicomponent mixture are specialized for binary mixtures, which are then discussed in considerably more detail. Finally diffusion processes in multicomponent systems, turbulent systems, multiphase systems, and systems with convection are discussed briefly. 

In natural systems (lakes, oceans, atmosphere) turbulent diffusion is usually anisotropic (i.e., much larger in the horizontal than vertical direction). There are two main reasons for that observation (1) the extension of natural systems in the horizontal is usually much larger than in the vertical. Thus, the turbulent structures (often called eddies) that correspond to the mean free paths of random motions often look like pancakes that is, they are flat along the vertical axis and mainly extended along the horizontal axes. (2) Often the atmosphere or the water body in a lake or ocean is density stratified (i.e., the density increases with depth). This compresses the eddies even further in the vertical. Gravitational forces keep the water parcels from moving too far away from the depth where they are neutrally buoyant, that is, where they have the same density as their environment. Thus, the anisotropic shape of the eddies results in turbulent diffusivities which differ in size along different spatial directions. 

The gradient-flux model to describe turbulent diffusion (Eq. 18-70) has the disadvantage that turbulent diffusivity, Ex, is scale dependent. As discussed in more detail in Chapter 22, in natural systems Ex increases with increasing horizontal scale of diffusion. This means that the speed with which two fluid parcels are separated by turbulence increases the further they are from each other. This is because turbulent structures (eddies) of increasing size become effective when the size of a diffusing patch becomes larger. Typical ranges of turbulent diffusivities in the environment are summarized in Table 18.4. 

Once the mathematical description of dispersion has been clarified, we are left with the task of quantifying the dispersion coefficient, Eiis. Obviously, Edh depends on the characteristics of the flow field, particularly on the velocity shear, dvx/dy and dvx /dz. As it turns out, the shear is directly related to the mean flow velocity vx. In addition, the probability that the water parcels change between different streamlines must also influence dispersion. This probability must be related to the turbulent diffusivity perpendicular to the flow, that is, to vertical and lateral diffusion. At this point it is essential to know whether the lateral and vertical extension of the system is finite or whether the flow is virtually unlimited. For the former (a situation typical for river flow), the dispersion coefficient is proportional to (vx )2 ... 

First, recall that the nondimensional Damkohler number, Da (Eq. 22-11 b), allows us to decide whether advection is relevant relative to the influence of diffusion and reaction. As summarized in Fig. 22.3, if Da 1, advection can be neglected (in vertical models this is often the case). Second, if advection is not relevant, we can decide whether mixing by diffusion is fast enough to eliminate all spatial concentration differences that may result from various reaction processes in the system (see the case of photolysis of phenanthrene in a lake sketched in Fig. 21.2). To this end, the relevant expression is L (kr / Ez)1 2, where L is the vertical extension of the system, Ez the vertical turbulent diffusivity, and A, the first-order reaction rate constant (Eq. 22-13). If this number is much smaller than 1, that is, if... 

Not only may mass diffusion occur on a molecular basis, but also in turbulent-flow systems accelerated diffusion rates will occur as a result of the rapid-eddy mixing processes, just as these mixing processes created increased heat transfer and viscous action in turbulent flow. 

After presenting some background information on theories of turbulent combustion in the following section, we shall discuss theoretical descriptions of turbulent diffusion flames (Section 10,2) and of turbulent premixed flames (Section 10,3), Although there are situations in which turbulence can tend to wash out the distinction between premixed and non-premixed systems, the best approaches to analyses of these two limits remain distinct today, and thus it is convenient to treat them separately. 

For complex chemistry, in many cases, a conserved scalar or a mixture fraction approach can be used, in which a single conserved scalar (mixture fraction) is solved instead of transport equations for individual species. The reacting system is treated using either chemical equilibrium calculations or by assuming infinitely fast reactions (mixed-is-reacted approach). The mixture fraction approach is applicable to non-premixed situations and is specifically developed to simulate turbulent diffusion flames containing one fuel and one oxidant. Such situations are illustrated in Fig. 5.6. The basis for the mixture fraction approach is that individual conservation equations for fuel and oxidant can be combined to eliminate reaction rate terms (see Toor, 1975 for more details). Such a combined equation can be simplified by defining a mixture... 

The development of the concepts and the calibration of these models require systematic use of fully computational models with long runs on big computing systems, and laboratory and field experiments. It is noted that FCM s are being speeded up by innovative approximations, in particular by modelling the effect of buildings in terms of a force (or source) so that the representation of their shape is not exact and by ignoring Reynolds stresses in the dynamical equations which are solved inexactly by allowing numerical diffusion (caused by approximate discretisation of the equations) to simulate the physical process of turbulent diffusion. 

In practical combustion systems, the predominant mode of heat transfer is usually not molecular conduction, but turbulent diffusion, except at the boundaries and the flame front. Conduction is the only mode of heat transfer through refractory walls, and it determines ignition and extinction behaviors of the flame. Turbulent diffusion, an apparent or pseudo conduction mechanism arising from turbulent eddy motions, will be discussed in Section 4.4. The relations from the theory of conduction heat transfer15-17 can be used to evaluate heat losses through furnace walls and load zones, and through the pipe walls inside boilers and heat exchangers, etc. 

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