Danckwerts reactor

A number of successful devices have been in use for finding mass-transfer coefficients, some of which are sketched in Fig. 23-29, and all of which have known or adjustable interfacial areas. Such laboratoiy testing is reviewed, for example, by Danckwerts (Ga.s-Liquid Reac-tion.s, McGraw-Hih, 1970) and Charpentier (in Ginetto and Silveston, eds., Multiphase Chemical Reactor Theory, De.sign, Scaleup, Hemisphere, 1986). 

Two lists of gas/liquid reactions of industrial importance have been compiled recently. The literature survey by Danckwerts (Gas-Liquid Reactions, McGraw-Hill, 1970) cites 40 different systems. A supplementary list by Doraiswamy and Sharma (Heterogeneous Reactions Fluid-Lluid-Solid Reactions, Wiley, 1984) cites another 50 items, and indicates the most suitable land of reactor to be used for each. Estimates of values of parameters that may be expec ted of some types of gas/liquid reac tors are in Tables 23-9 and 23-10. 

The term macromixing refers to the overall mixing performance in a reactor. It is usually described by the residence time distribution (RTD). Originally introduced by Danckwerts (1958), this concept is based on a macroscopic lumped population balance. A fluid element is followed from the time at which it enters the reactor (Lagrangian viewpoint - observer moves with the fluid). The probability that the fluid element will leave the reactor after a residence time t is expressed as the RTD function. This function characterises the scale of mixedness in a reactor. 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find = 0, which is the definition of a closed system. See... 

As a result, there is a jump discontinuity in the temperature at Z=0. The condition is analogous to the Danckwerts boimdary condition for the inlet of an axially dispersed plug-flow reactor. At the exit of the honeycomb, the usual zero gradient is imposed, i.e. 

Several age-distribution functions may be used (Danckwerts, 1953), but they are all interrelated. Some are residence-time distributions and some are not. In the discussion to follow in this section and in Section 13.4, we assume steady-flow of a Newtonian, single-phase fluid of constant density through a vessel without chemical reaction. Ultimately, we are interested in the effect of a spread of residence times on the performance of a chemical reactor, but we concentrate on the characterization of flow here. 

The boundary conditions for a closed-vessel reactor are analogous to those for a tracer in a closed vessel without reaction, equations 19.4-66 and -67, except that we are assuming steady-state operation here. These are called the Danckwerts boundary conditions (Danckwerts, 1953).1 With reference to Figure 19.18,... 

In this chapter, we consider process design aspects of reactors for multiphase reactions in which each phase is a fluid. These include gas-liquid and liquid-liquid reactions, although we focus primarily on the former. We draw on the results in Section 9.2, which treats the kinetics of gas-liquid reactions based on the two-film model. More detailed descriptions are given in the books by Danckwerts (1970), by Kastanek et al. (1993), and by Froment and Bischoff (1990, Chapter 14). 

The concept of segregation and its meaning to chemical reactors was first described by Danckwerts (1953). The intensity or degree of segregation is given the symbol I, which varies between one and zero. Shown in Fig. 1 is a tank with two components, A and B, which are separated into volume fractions, qA and l-qA this condition represents complete initial segregation (1 = 1). Stirring or... 

One of the early successes of the CRE approach was to show that RTD theory suffices to treat the special case of non-interacting fluid elements (Danckwerts 1958). For this case, each fluid element behaves as a batch reactor. 

At the same symposium, Danckwerts (D2) drew attention to the effect of incomplete mixing on homogeneous reactions. He introduced the concept of segregation, which indicates that in the same vessel there are clumps of fluid which have different concentrations, caused by incomplete mixing. The effect on the conversion of chemical reactors is again an increase in conversion for reactions of an order greater than 1 and a decrease in conversion when the order is less than 1. 

That notorious pair, the Danckwerts boundary conditions for the tubular reactor, provides a good illustration of boundary conditions arising from nature. Much ink has been spilt over these, particularly the exit condition that Danckwerts based on his (perfectly correct, but intuitive) engineering insight. If we take the steady-state case of the simplest distributed example given previously but make the flux depend on dispersion as well as on convection, then, because there is only one-space dimension,/= vAc — DA dddz), where D is a dispersion coefficient. Then, as the assumption of steady state eliminates... 

Besides the deposition on the wafers, the radial boundary conditions (equations 38a and b) account for deposition on the reactor wall and the support boat. The first term in equation 38b represents the deposition on the wafers, and the second term gives the deposition on the boat, a is the boat area relative to the tube area. The axial boundary conditions (equation 38c) are the usual Danckwerts boundary conditions. 

This latter formula, which is commonly used in the industrial practice, also holds for the case of an estimation of the mixture quality using random sampling. However, the main problem in this global analysis lays in the well-known (since the famous Danckwert s example in chemical reactors [23]) fact that two different structures can correspond to the same intensity of segregation criteria. Fig. 1 illustrates this idea, which has a very practical sense in the context of pharmaceutical blending. 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find Din = Dout = 0, which is the definition of a closed system. See Figure 9.8. The flux in the inlet pipe is due solely to convection and has magnitude Qi ain. The flux just inside the reactor at location z = 0+ has two components. One component, Qina(0+), is due to convection. The other component, —DAc[da/dz 0+, is due to diffusion (albeit eddy diffusion) from the relatively high concentrations at the inlet toward the lower concentrations within the reactor. The inflow to the plane at z = 0 must be matched by material leaving the plane at z = 0+ since no reaction occurs in a region that has no volume. Thus,... 

Mac Mullin and Weber [1] introduced the concept of the RTD in the analysis of chemical reactors, and Danckwerts [2] developed this concept further in his classical paper, which has since formed the basis of various investigations involving flow systems in chemical and biochemical reactors. Levenspiel [3], Levenspiel and Bischoff [4], Himmelblau and Bischoff [5], Wen and Fan [6], and Shinnar [7] have given extensive treatments of this subject. 

The age of an element of fluid is defined as the time elapsed since it entered the system. The fraction of fluid having ages between t and t + dt is (uc/m) dt, where u is the volumetric rate of flow of fluid through the system, m is the quantity of tracer injected, and c is the local concentration of tracer at time t after injection. Danckwerts [8] introduced the concept of a fluid element or point, meaning a small volume with respect to the reactor vessel size, but still large enough... 

The definitions of the degree of mixing presented above aim at a local characterization of the mixture homogeneity in the physical space. There also exist more indirect mixing indices. The segregation index J of Danckwerts (12) is one of the most famous ones. It applies to continuous reactors and relies upon the variance of age J = Var oip/Var a>where a is the age of a molecule,... 

In conclusion to this section, research in the RTD area is always active and the initial concepts of Danckwerts are gradually being completed and extended. The population balance approach provides a theoretical framework for this generalization. However, in spite of the efforts of several authors, simple procedures, easy to use by practitioners, would still be welcome in the field of unsteady state systems (variable volumes and flow rates), multiple inlet/outlet reactors, variable density mixtures, systems in which the mass-flowrate is not conserved, etc... On the other hand, the promising "generalized reaction time distribution" approach could be developed if suitable experimental methods were available for its determination. 

A one-dimensional one-phase dispersion model subject to the Danckwerts boundary conditions has been used for a description of the dynamics of a nonisothermal nonadiabatic packed bed reactor. The dimensionless governing equations are ... 

The methods for both semi-batch and continuous reactors are generally restricted to the 02-aqueous system because of the limitations of the analytical technique for the direct measurement of liquid-phase concentration. Novel agitated vessels for measurement of transient absorption rates are described by Danckwerts and Kennedy (1954), Oishi et al. (1965) and Govindam and Quinn (1964). 

As our first application, we consider the classical Taylor-Aris problem (Aris, 1956 Taylor, 1953) that illustrates dispersion due to transverse velocity gradients and molecular diffusion in laminar flow tubular reactors. In the traditional reaction engineering literature, dispersion effects are described by the axial dispersion model with Danckwerts boundary conditions (Froment and Bischoff, 1990 Levenspiel, 1999 Wen and Fan, 1975). Here, we show that the inconsistencies associated with the traditional parabolic form of the dispersion model can be removed by expressing the averaged model in a hyperbolic form. We also analyze the hyperbolic model and show that it has a much larger range of validity than the standard parabolic model. 

---