Turbulent reaction rate

The mean species source term, It, requires further examination. Because this term is usually highly nonlinear as illustrated in Equation 4.8 and its value directly controls the reaction progress, a proper closure model is necessary. Indeed, various methods are available from the literature.21-22 For the purpose of discussion, we consider the eddy breakup model of Magnussen and Hjertager.23 Note that this model is chosen for its simplicity rather than accuracy, much as the A - e model was selected for turbulence closure. In this model, a turbulent reaction rate is computed which is then compared with the kinetic rate. The smaller of the two is used as the reaction rate because it limits the reaction progress. The kinetic rate is simply the laminar reaction rate evaluated at the mean temperature, pressure, and concentrations ... 

To determine the turbulent reaction rate, conceptualize the reaction as a single-step process where the fuel and oxidant are converted to a product. Furthermore, visualize a turbulent flame as a collection of burning eddies. For a reaction to take place, the fuel and oxidant must be simultaneously available, and the reaction product must be removed properly. That is, the reaction can be limited by the availability of fuel, oxidant, or by the removal of product. Once the fuel and oxidant are mixed, combustion takes place, on average, within one eddy turnover time, k/e, so that the reaction rate can be written as... 

The development of combustion theory has led to the appearance of several specialized asymptotic concepts and mathematical methods. An extremely strong temperature dependence for the reaction rate is typical of the theory. This makes direct numerical solution of the equations difficult but at the same time accurate. The basic concept of combustion theory, the idea of a flame moving at a constant velocity independent of the ignition conditions and determined solely by the properties and state of the fuel mixture, is the product of the asymptotic approach (18,19). Theoretical understanding of turbulent combustion involves combining the theory of turbulence and the kinetics of chemical reactions (19—23). 

To analy2e premixed turbulent flames theoretically, two processes should be considered (/) the effects of combustion on the turbulence, and (2) the effects of turbulence on the average chemical reaction rates. In a turbulent flame, the peak time-averaged reaction rate can be orders of magnitude smaller than the corresponding rates in a laminar flame. The reason for this is the existence of turbulence-induced fluctuations in composition, temperature, density, and heat release rate within the flame, which are caused by large eddy stmctures and wrinkled laminar flame fronts. 

Ya.B. ZeFdovich, FizGoreniyaVzryva 7 (4), 463-76 (1971) CA 77, 64194 (1972) The influence of turbulence and nonturbulence is examined relative to a proplnt burning in a gas flow. Equations indicate exptl methods for determining the magnitudes of the thermal conductivity and viscosity under turbulent flow, and permit a study of thermal flow distribution and temps in a gas wherein an exothermic chem reaction occurs. Equations for non turbulent conditions can be used to calculate the distance from the surface of the proplnt to the zone of intense chem reaction and establish the relation of bulk burning rate to the vol reaction rate. 

This, in turn, reduced the number of polymer particles (the loci of reaction) and hence the reaction rate fell. However, this explanation is at variance with the results reported in Figure 12 where the molecular weight (weight-average) clearly increases with increasing Reynolds number. It seems more likely that the turbulent flow results could be explained by a decrease in the effective initiator concentration. This low concentration would also explain why there is no further reaction after a period of about one hour as contrasted with the batch reactions where the reaction is still proceeding after two to three hours. The current absence of corroborating evidence makes this explanation very tentative. 

The concept of turbulent flame stretch was introduced by Karlovitz long ago in [15]. The turbulent Karlovitz number (Ka) can be defined as the ratio of a turbulent strain rate (s) to a characteristic reaction rate (to), which has been commonly used as a key nondimensional parameter to describe the flame propagation rates and flame quenching by turbulence. For turbulence s >/ />, where the dissipation rate e and u, L and v... 

J.P. Dumont, D. Durox, and R. Borghi 1993, Experimental study of the mean reaction rates in a turbulent premixed flame. Combust. Sci. Technol. 89 219-251 (more informations through www.informaworld.com). 

Pandit and co-workers have shown that scale-up may be possible on a more rational basis if cavitation is employed, and some data have been reported by Pandit and Mohalkar (1996), Mohalkar et al. (1999), Senthil et al. (1999), and Cains et al. (1998). A variety of reactors can be used, viz. the liquid whistle reactor, the Branson sonochemical reactor, the Pote reactor, etc. The principal factors affecting the efficiency of a hydrodynamic cavitation reactor are irreversible loss in pressure head and turbulence and friction losses in the reaction rates. 

Enhanced chemical reactivity of solid surfaces are associated with these processes. The cavitational erosion generates unpassivated, highly reactive surfaces it causes short-lived high temperatures and pressures at the surface it produces surface defects and deformations it forms fines and increases the surface area of friable solid supports and it ejects material in unknown form into solution. Finally, the local turbulent flow associated with acoustic streaming improves mass transport between the liquid phase and the surface, thus increasing observed reaction rates. In general, all of these effects are likely to be occurring simultaneously. 

CFD might provide a way of elucidating all these spatial variations in flow conditions, in species concentrations, in bubble drop and particle sizes, and in chemical reaction rates, provided that such computational simulations are already capable of reliably reproducing the details of turbulent flows and their dynamic effects on the processes of interest. This Chapter reviews the state of the art in simulating the details of turbulent flows and turbulent mixing processes, mainly in stirred vessels. To this end, the topics of turbulence and CFD both need a separate introduction. 

The factors that can affect the rate of heat transfer within a reactor are the speed and type of agitation, the type of heat transfer surface (coil or jacket), the nature of the reaction fluids (Newtonian or non-Newtonian), and the geometry of the vessel. Baffles are essential in agitated batch or semi-batch reactors to increase turbulence which affects the heat transfer rate as well as the reaction rates. For Reynolds numbers less than 1000, the presence of baffles may increase the heat transfer rate up to 35% [180]. 

Thus, due to limitations on the available computer memory, DNS of homogeneous turbulent reacting flows has been limited to Sc 1 (i.e., gas-phase reactions). Moreover, because explicit ODE solvers (e.g., Runge-Kutta) are usually employed for time stepping, numerical stability puts an upper limit on reaction rate k. Although more complex... 

As will be shown for the CD model, early mixing models used stochastic jump processes to describe turbulent scalar mixing. However, since the mixing model is supposed to mimic molecular diffusion, which is continuous in space and time, jumping in composition space is inherently unphysical. The flame-sheet example (Norris and Pope 1991 Norris and Pope 1995) provides the best illustration of what can go wrong with non-local mixing models. For this example, a one-step reaction is described in terms of a reaction-progress variable Y and the mixture fraction p, and the reaction rate is localized near the stoichiometric point. In Fig. 6.3, the reaction zone is the box below the flame-sheet lines in the upper left-hand corner. In physical space, the points with p = 0 are initially assumed to be separated from the points with p = 1 by a thin flame sheet centered at... 

Toor, H. L. and M. Singh (1973). The effect of scale on turbulent mixing and chemical reaction rates during turbulent mixing in a tubular reactor. Industrial and Engineering Chemistry Fundamentals 12, 448 451. 

To examine the effect of turbulence on flames, and hence the mass consumption rate of the fuel mixture, it is best to first recall the tacit assumption that in laminar flames the flow conditions alter neither the chemical mechanism nor the associated chemical energy release rate. Now one must acknowledge that, in many flow configurations, there can be an interaction between the character of the flow and the reaction chemistry. When a flow becomes turbulent, there are fluctuating components of velocity, temperature, density, pressure, and concentration. The degree to which such components affect the chemical reactions, heat release rate, and flame structure in a combustion system depends upon the relative characteristic times associated with each of these individual parameters. In a general sense, if the characteristic time (r0) of the chemical reaction is much shorter than a characteristic time (rm) associated with the fluid-mechanical fluctuations, the chemistry is essentially unaffected by the flow field. But if the contra condition (rc > rm) is true, the fluid mechanics could influence the chemical reaction rate, energy release rates, and flame structure. 

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