Stirred reactor theory

In the discussion of premixed turbulent flames, the case of infinitely fast mixing of reactants and products was introduced. Generally this concept is referred to as a stirred reactor. Many investigators have employed stirred reactor theory not only to describe turbulent flame phenomena, but also to determine overall reaction kinetic rates [23] and to understand stabilization in high-velocity streams [62], Stirred reactor theory is also important from a practical point of view because it predicts the maximum energy release rate possible in a fixed volume at a particular pressure. 

Consider a fixed volume V into which fuel and air are injected at a fixed total mass flow rate m and temperature T0. The fuel and air react in the volume and the injection of reactants and outflow of products (also equal to rii) are so oriented that within the volume there is instantaneous mixing of the unbumed 

The stirred reactor may be compared to a plug flow reactor in which premixed fuel-air mixtures flow through the reaction tube. In this case, the unbumed gases enter at temperature T0 and leave the reactor at the flame temperature T. The system is assumed to be adiabatic. Only completely burned products leave the reactor. This reactor is depicted in Fig. 4.50. 

The volume required to convert all the reactants to products for the plug flow reactor is greater than that for the stirred reactor. The final temperature is, of course, higher than the stirred reactor temperature. 

It is relatively straightforward to develop the controlling parameters of a stirred reactor process. If ip is defined as the unbumed mass fraction, it must follow that the fuel-air mass rate of burning RB is 

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Now the great simplicity in stirred reactor theory is realizable. Since (F), (A), and Tr are constant in the reactor, the rate of conversion is constant. It is now possible to represent the mass rate of burning in terms of the preceding chemical kinetic expression ... 

Considering the wake of a flame holder as a stirred reactor may be inconsistent with experimental data. It has been shown [66] that as blowoff is approached, the temperature of the recirculating gases remains essentially constant furthermore, their composition is practically all products. Both of these observations are contrary to what is expected from stirred reactor theory. Conceivably, the primary zone of a gas turbine combustor might approach a state that could be considered completely stirred. Nevertheless, as will be shown, all three theories give essentially the same correlation. 

Stirred reactor theory was initially applied to stabilization in gas turbine combustor cans in which the primary zone was treated as a completely stirred region. This theory has sometimes been extended to bluff-body stabilization, even though aspects of the theory appear inconsistent with experimental measurements made in the wake of a flame holder. Nevertheless, it would appear that stirred reactor theory gives the same functional dependence as the other correlations developed. In the previous section, it was found from stirred reactor considerations that... 

In the discussion of premixed turbulent flames, the case of infinitely fast mixing of reactants and products was introduced. Generally this concept is referred to as a stirred reactor. Many investigators have employed stirred reactor theory... 

Stirred reactor theory reveals a fixed maximum mass loading rate for a fixed reactor volume and pressure. Any attempts to overload the system will quench the reaction. Attempts have been made to determine chemical kinetic parameters from stirred reactor measurements however, the usefulness of such measurements is limited. First, the analysis is based on the assumption that a hydrocarbon-air system can be represented by a simple one-step overall order kinetic expression. Recent evidence indicates that such an assumption is not realistic. Second, the analysis is based on the assumption of complete instantaneous mixing, which is impossible to achieve experimentally. 

Essenhigh, R.H. An Introduction to Stirred Reactor Theory Applied to Design of Combustion Chambers, in Palmer and Beer Combustion Technology pp. 389-391, Academic Press, New York, NY 1974. 

K. K. Boon, "A Flexible Mathematical Model for Analy2ing Industrial P. F. Furnaces," M.S. thesis. University of Newcasde, AustraUa, Sept. 1978. R. H. Essenhigh, "A New AppHcation of Perfectly Stirred Reactor (P.S.R.) Theory to Design of Combustion Chambers," TechnicalEeport FS67-1 (u), Peimsylvania State University, Dept, of Euel Science, University Park, Pa., Mar. 1967. 

Three ideal reactor types are relevant from reactor theory [15], the two continuous flow types, the plug flow reactor (PFR) and continuous flow stirred tank reactor (CSTR), and the well-stirred batch reactor. The... 

Baldi, G., Hydrodynamics and Gas-Liquid Mass Transfer in Stirred Slurry Reactors, Ch. 14 in Multiphase Chemical Reactors Theory, Design, Scale-up (A. Gianetto and P. L. Silveston, eds.). Hemisphere Publ. Co., New York, 1986. 

There are approaches to analyses of turbulent combustion that, although not deductively based on the Navier-Stokes equations, nevertheless appeal to concepts of coherent structures [68], [69]. We shall not have space here to present these approaches and must refer instead to reviews [18], [27], [40]. These methods share some aspects in common with age theories of stirred reactors [19], theories that we also shall forego discussing for the sake of brevity. Instead, we shall consider a promising approach to the theoretical analysis of turbulent diffusion flames. 

Varma, A., and R. Aris, Stirred pots and empty tubes, Chemical Reactor Theory A Review, L. Lapidus and N. R. Amundson, eds., Prentice-Hall, Englewood Cliffs. NJ (1977) Chapter 2. 

Of particular interest in the context of stirred tank reactor theory is the case of steadily rotating axes of reference DQ/Dt = 0) without translating acceleration Dyo/Dt). Equation (7.109) can then be written as ... 

The stability of the uniform stationary states in system (3,4) to small spatially uniform perturbations (SUPs) was studied in detail by the theory of perfectly stirred reactors (see reference 9). It will be recalled that the system... 

Other inadequacies of Kolmogorov s theory in the manner of its use by Shinnar and Church (1960) and Brian et al. (1969) to correlate solid-liquid mass transfer in stirred reactors have been dealt in some detail by Levins and Glastonbury (1972a). Stewart and Townsend (1951) have estimated the value of Re to be 1500 for the existence of the inertial subrange. Taylor (1935) estimated that for the region outside the impeller, the value of ROj. for conditions typical of stirred reactors is -144. This is much smaller than that required by Stewart and Townsend s estimated value of 1500. Therefore, the validity of Equation 6.8 outside the impeller discharge stream is doubtful. Further, the use of in Equation 6.10 for Re is not justified by... 

In precipitation processes that are carried out in stirred reactors, particles in the size range of 0.1 to 10 xm will generally aggregate under the influence of flow. In laminar flow, the velocity gradient causes particles to "overtake" others. When the distance between their trajectories is sm ler than their diameter, the particles will eventually collide, and mostly aggregate. This mechanism has been described in a classical paper by Von Smoluchowski (1917). According to his theory, the aggregation rate is express as numbers of "doublets" formed per m and per s, from spherical particles with sizes d and rf, and volume concentrations n and n ... 

The aim of this paper is to provide well estedslished experimental data, and to propose a new and tentative interpretation of micromixing in stirred reactors, with the hope of bridging the gap between phenomenological models and classical results of the theory of turbulence. 

These results constitute a first attempt at a comprehensive theory of micromixing in stirred reactors based on experimental facts. 

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