Turbulent flow field

Cate etal. (2001) propose a method for the calculation of crystal-crystal collisions in the turbulent flow field of an industrial crystallizer. It consists of simulating the internal flow of the crystallizer as a whole and of simulating the motion of individual particles suspended in the turbulent flow in a small subdomain (box) of the crystallizer. 

For partieles to break-up in a turbulent flow field, fluid eddies responsible for break-up have to be of both less than the eritieal size and also possess suffieient disruptive energy. Eddies that are larger than the eritieal size tend to entrain... 

Turbulent flow reactors are modeled quite differently from laminar flow reactors. In a turbulent flow field, nonzero velocity components exist in all three coordinate directions, and they fluctuate with time. Statistical methods must be used to obtain time average values for the various components and to characterize the instantaneous fluctuations about these averages. We divide the velocity into time average and fluctuating parts ... 

Although Eq. (6-18) can be used to eliminate the stress components from the general microscopic equations of motion, a solution for the turbulent flow field still cannot be obtained unless some information about the spatial dependence and structure of the eddy velocities or turbulent (Reynolds) stresses is known. A classical (simplified) model for the turbulent stresses, attributed to Prandtl, is outlined in the following subsection. 

As a result, the turbulent-flow field in a stirred vessel may be far from isotropic and homogeneous. Some of the cornerstones of turbulence theory, however, start from the assumption that production and dissipation of turbulent kinetic energy balance locally. In many chemical engineering flows, this... 

The focus of RANS simulations is on the time-averaged flow behavior of turbulent flows. Yet, all turbulent eddies do contribute to redistributing momentum within the flow domain and by doing so make up the inherently transient character of a turbulent-flow field. In RANS, these effects of the full range of eddies are made visible via the so-called Reynolds decomposition of the NS equations (see, e.g., Tennekes and Lumley, 1972, or Rodi, 1984) of the flow variables into mean and fluctuating components. To this end, a clear distinction is required between the temporal and spatial scales of the mean flow on the one hand and those associated with the turbulent fluctuations on the other hand. 

In the Lagrangian approach, individual parcels or blobs of (miscible) fluid added via some feed pipe or otherwise are tracked, while they may exhibit properties (density, viscosity, concentrations, color, temperature, but also vorti-city) that distinguish them from the ambient fluid. Their path through the turbulent-flow field in response to the local advection and further local forces if applicable) is calculated by means of Newton s law, usually under the assumption of one-way coupling that these parcels do not affect the flow field. On their way through the tank, these parcels or blobs may mix or exchange mass and/or temperature with the ambient fluid or may adapt shape or internal velocity distributions in response to events in the surrounding fluid. 

These convective transport equations for heat and species have a similar structure as the NS equations and therefore can easily be solved by the same solver simultaneously with the velocity field. As a matter of fact, they are much simpler to solve than the NS equations since they are linear and do not involve the solution of a pressure term via the continuity equation. In addition, the usual assumption is that spatial or temporal variations in species concentration and temperature do not affect the turbulent-flow field (another example of oneway coupling). 

It makes sense to compare the implications (in terms of simulation times) of using FV vs. LB in simulating turbulent-flow fields in process devices. Hoekstra (2000) demonstrated the numerical implications of applying different numerical schemes in an industrial application. He compared the outcome of his RANS simulation for a gas cyclone with that of a LES carried out by Derksen and Van den Akker (2000). Table I presents a number of numerical features of the two types of simulations. 

The next question then is whether the processes taking place inside this turbulent-flow field can be modeled with confidence. We will now first consider the first question. 

As long as the interest is in fields of averaged velocity components and in overall mixing patterns, RANS-based simulations may suffice. Examples of such satisfactory simulation results are plentiful, e.g., Marshall and Bakker (2004) and Montante et al. (2006). When, however, the interest is in the details of the turbulent-flow field and in processes affected by these details, LES is the option to be preferred. From the findings reproduced above and from the validation studies of Derksen and Van den Akker (1999) and Derksen (2001) the general conclusion is that, as long as the spatial resolution is sufficient, LB LES deliver results in excellent agreement with experimental turbulence data. 

In view of secondary nucleation in crystallizers, Ten Cate et al. (2004) were interested in finding out locally about the frequencies of particle collisions in a suspension under the action of the turbulence of the liquid. To this end, they performed a DNS of a particle suspension in a periodic box subject to forced turbulent-flow conditions. In their DNS, the flow field around and between the interacting and colliding particles is fully resolved, while the particles are allowed to rotate in response to the surrounding turbulent-flow field. 

Properly simulating a dissolution process of solid particles in a stirred vessel operated in the turbulent-flow regime urges for a very detailed simulation of the turbulent-flow field itself. Just reproducing the overall flow pattern by means of... 

One really may need an inherently transient LES to capture all these details. The finer the grid for such a LES, the more reliably the local transient conditions may be taken into account in reproducing this turbulent mass transfer process (while ignoring the issue of supplying the heat for the dissolution which may also depend on a proper representation of the turbulent-flow field). An additional important issue is how many particles have to be tracked for a proper representation of the transient spatial distribution of the particles over the vessel. 

The evolution of the two-phase turbulence depends on the initial random position of the particles, the motion of which modifies the turbulent-flow field directly. These DNS are therefore a nice example of two-way coupling between the two phases see Fig. 12. From these DNS, detailed knowledge can be derived as to the frequency of the particle-particle collisions and the forces involved... 

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 ... 

The transport rates fj will be determined by the turbulent flow field inside the reactor. When setting up a zone model, various methods have been proposed to extract the transport rates from experimental data (Mann et al. 1981 Mann et al. 1997), or from CFD simulations. Once the transport rates are known, (1.15) represents a (large) system of coupled ordinary differential equations (ODEs) that can be solved numerically to find the species concentrations in each zone and at the reactor outlet. 

As discussed in Chapter 2, a fully developed turbulent flow field contains flow structures with length scales much smaller than the grid cells used in most CFD codes (Daly and Harlow 1970).29 Thus, CFD models based on moment methods do not contain the information needed to predict x, t). Indeed, only the direct numerical simulation (DNS) of (1.27)-(1.29) uses a fine enough grid to resolve completely all flow structures, and thereby avoids the need to predict x, t). In the CFD literature, the small-scale structures that control the chemical source term are called sub-grid-scale (SGS) fields, as illustrated in Fig. 1.7. 

A general overview of models for turbulent transport is presented in Chapter 4. The goal of this chapter is to familiarize the reader with the various closure models available in the literature. Because detailed treatments of this material are readily available in other texts (e.g., Pope 2000), the emphasis of Chapter 4 is on presenting the various models using notation that is consistent with the remainder of the book. However, despite its relative brevity, the importance of the material in Chapter 4 should not be underestimated. Indeed, all of the reacting-flow models presented in subsequent chapters depend on accurate predictions of the turbulent flow field. With this caveat in mind, readers conversant with turbulent transport models of non-reacting scalars may wish to proceed directly to Chapter 5. 

In Section 5.1, we have seen (Fig. 5.2) that the molar concentration vector c can be transformed using the SVD of the reaction coefficient matrix T into a vector c that has Nr reacting components cr and N conserved components cc.35 In the limit of equilibrium chemistry, the behavior of the Nr reacting scalars will be dominated by the transformed chemical source term S. 36 On the other hand, the behavior of the N conserved scalars will depend on the turbulent flow field and the inlet and initial conditions for the flow domain. However, they will be independent of the chemical reactions, which greatly simplifies the mathematical description. 

One of the principal difficulties faced when employing Lagrangian micromixing models is the determination of tm based on properties of the turbulent flow fields. Researchers have thus attempted to use the universal nature of high-Reynolds-number isotropic turbulence to link tm to the turbulence time scales. For example, in the E-model (Baldyga and Bourne 1989) the engulfment rate essentially controls the rate of micromixing and is defined by... 

Chemical reaction in a turbulent flow field with uniform velocity gradient. The... 

Gibson, C. H. and W. H. Schwarz (1963a). Detection of conductivity fluctuations in a turbulent flow field. Journal of Fluid Mechanics 16, 357-364. 

The soot formation and its control was studied in an annular diffusion flame using laser diagnostics and hot wire anemometry [17, 18]. Air and fuel were independently acoustically forced. The forcing altered the mean and turbulent flow field and introduced coherent vortices into the flow. This allowed complete control of fuel injection into the incipient vortex shedding process. The experiments showed that soot formation in the flame was controlled by changing the timing of fuel injection relative to air vortex roll-up. When fuel was injected into a fully developed vortex, islands of unmixed fuel inside the air-vortex core led to... 

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