Turbulent kinetic energy derivation

To understand the principal idea of Deacon s model we have to remember the key assumption of the film model according to which a bottleneck boundary is described by an abrupt drop of diffusivity, for instance, from turbulent to molecular conditions (see Fig. 19.3a). Yet, theories on turbulence at a boundary derived from fluid dynamics show that this drop is gradual and that the thickness of the transition zone from fully turbulent to molecular conditions depends on the viscosity of the fluid. In Whitman s film model this effect is incorporated in the film thicknesses, 8a and 8W (Eq. 20-17). In addition, the film thickness depends on the intensity of turbulent kinetic energy production at the interface as, for instance, demonstrated by the relationship between wind velocity and exchange velocity (Figs. 20.2 and 20.3). 

Jensen Webb (Ref 43) examined the data predicting the extent of afterburning in fuel-rich exhausts of metal-modified double-base proplnt rocket motors so as to determine the amt of an individual metal which is required to suppress this afterburning. The investigatory means they used consisted of a series of computer codes. First, an equilibrium chemistry code to calculate conditions at the nozzle throat then a nonequilibrium code to derive nozzle plane exit compn, temp and velocity and, finally, a plume prediction code which incorporates fully coupled turbulent kinetic energy boundary-layer and nonequilibrium chemical reaction mechanisms. Used for all the code calcns were the theoretical environment of a static 300 N (67-lb) thrust std research motor operating at a chamber press of S.SMNm 2 (500psi), with expansion thru a conical nozzle to atm press and a mass flow rate... 

For steady-state, incompressible, and isothermal flows, the transport equation of the turbulent kinetic energy can be derived as (Problem 5.4)... 

The particle turbulent kinetic energy is governed by its own transport equation. Similarly to the derivation of the -equation, the p-equation is given by... 

In the many traditional methods of calculating turbulent flows, these turbulence terms are empirically defined, i.e., turbulence models that are almost entirely empirical are used. Some success has, however, been achieved by using additional differential equations to help in the description of these terms. Empiricism is not entirely eliminated, at present, by the use of these extra equations but the empiricism can be introduced in a more systematic and logical manner than is possible if the turbulence terms in the momentum equation are completely empirically described. One of the most widely used additional equations for this purpose is the turbulence kinetic energy equation and its general derivation will now be discussed. 

As was the case with the full equations, these contain beside the three mean flow variables u, v, and T (the pressure is, of course, by virtue of Eq. (2.157) again determined by the external in viscid flow) additional terms arising as a result of the turbulence. Therefore, as previously discussed, in order to solve this set of equations, there must be an additional input of information, i.e., a turbulence model must be used. Many turbulence models are based on the turbulence kinetic energy equation that was previously derived. When the boundary layer assumptions are applied to this equation, it becomes ... 

The turbulence kinetic energy equation for forced convection was derived in this chapter. Rederive this turbulence kinetic energy equation bv starting with the momentum... 

The turbulent kinetic energy equation was derived in Chapter 5 using the momentum equations and assuming buoyancy force effects were negligible. Re-derive this equation starting with momentum equations in which the buoyancy terms are retained. Assume a vertically upward flow and use the Boussinesq approximation. 

Earlier it was stated that the structure of a turbulent velocity field may be presented in terms of two parameters—the scale and the intensity of turbulence. The intensity was defined as the square root of the turbulent kinetic energy, which essentially gives a root-mean-square velocity fluctuation U. Three length scales were defined the integral scale /q, which characterizes the large eddies the Taylor microscale X, which is obtained from the rate of strain and the Kolmogorov microscale 1, which typifies the smallest dissipative eddies. These length scales and the intensity can be combined to form not one, but three turbulent Reynolds numbers Ri = U lo/v, Rx. = U X/v, and / k = U ly/v. From the relationship between Iq, X, and /k previously derived it is found that / ... 

Fluctuating components of gas velocity are selected from a Gaussian distribution with variance derived from the local value of turbulent kinetic energy. Integration of the particle equation of motion yields the particle position at any given instant in time. 

It was shown by Taylor [159] that an analysis of the dissipation term occurring within the turbulent kinetic energy balance equation (derived later in this... 

A transport equation for the turbulent kinetic energy, or actually the momentum variance, can be derived by multiplying the equation for the fluctuating component v[, (1.389), by 2u, thereafter use the product rule of calculus to convert some of the terms in the provisional equation, and Anally time average the resulting equation [154]. 

Normally, as discussed earlier, in reactor modeling we are interested in the energy associated with the velocity fluctuations v only. The appropriate turbulent kinetic energy balance equation, the k equation, has therefore been derived via an equation for the Reynolds stress tensor. 

Nevertheless, in the first step in the model derivation a transport equation for the bubble induced turbulent kinetic energy was postulated ... 

To parameterize the new quantities occurring in these equations a few semi-empirical relations from the literature were adopted. The asymptotic value of bubble induced turbulent kinetic energy, fesia, is estimated based on the work of [3]. By use of the so-called cell model assumed valid for dilute dispersions, an average relation for the pseudo-turbulent stresses around a group of spheres in potential flow has been formulated. Prom this relation an expression for the turbulent normal stresses determining the asymptotic value for bubble Induced turbulent energy was derived ... 

In this context the number of particles depositing for given hydraulic conditions (C — Ceq) must be determined. This means that the characteristics of the particulates must be correlated to local parameters describing the turbulent flow field. This leads to a critical sedimentation velocity i s.cr for a particle. It is derived on the basis of an energy balance the potential energy loss attributable to settling in a non-turbulent system must equal the turbulent kinetic energy that must be imparted on the particle in a turbulent flow system to prevent sedimentation (29). 

The standard k-e model focuses on mechanisms that affect the turbulent kinetic energy. Robusmess, economy, and reasonable accuracy over a wide range of turbulent flows explain its popularity in industrial flow and heat transfer simulations. The RNG k-e model was derived using a rigorous statistical technique (called Re-Normalization Group theory). It is similar in form to the standard k-e model, but the effect of swirl on turbulence is included in the RNG mode enhancing the accuracy for swirling flows. 

Extensional fields are present when turbulent kinetic energy is produced by means of the burst-process. Donohue et al. [133 have shown that the number and intensities of bursts in a turbulent flow are reduced by the addition of polymers. A turbulent time scale, which was derived from measurements of Donohue et al. [133 and which is identical with equation (2), has been presented in C143, [153. The relaxation time derived by Bird... 

A remaining unclosed term in (12.5.1-7) or (12.5.1-10) is the turbulence dissipation rate e. A transport equation for the latter can be rigorously derived, but it is difficult to close. An alternative semi-empirical transport equation is commonly used. It is derived in analogy with the turbulent kinetic energy transport equation and takes the form ... 

Applying a turbulent energy cascade model, Ertesvag and Magnussen [2000] derived expressions for nY and ft in terms of the turbulent kinetic energy k and the turbulence dissipation rate s ... 

In Section 5-6.2, methods of plotting several common solution variables, such as velocity, stream function, and species concentration, were discussed. Plots of turbulent kinetic energy and dissipation are also of interest in turbulent flows, especially if other processes, such as chemical reactions, are to take place. In multiphase flows, the volume fraction of the phases is the most useful tool to assess the distribution of the phases in the vessel. In this section, three additional quantities are reviewed that are derived from the velocity field. These can provide a deeper understanding of the flow field than can plots of the velocity alone. 

Looking at a little slice of the process fluid as our system, we can derive each of the terms of Eq. (2.18). Potential-energy and kinetic-energy terms are assumed negligible, and there is no work term. The simplified forms of the internal ener and enthalpy are assumed. Diffusive flow is assumed negligible compared to bulk flow. We will include the possibility for conduction of heat axially along the reactor due to molecular or turbulent conduction. 

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