Plug flow, mixing model residence-time distribution

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

As predicted from the model, the residence time distribution curve shown in Fig. 3.11 exhibits the feature of perfect mixing with a certain lag time, that is, the feature of plug flow-perfect mixing flow in series. All the experiments yield the results of RTD exhibiting the same feature. 

Figure 6.50 presents the cumulative residence time distribution for a tube with a Newtonian model and for a shear thinning fluid with power law indices of 0.5 and 0.1. Plug flow, which represents the worst mixing scenario, is also presented in the graph. A Bingham fluid, with a power law index of 0, would result in plug flow. 

Tubular reactor advantages include their well-defined residence time distributions, turbulent mixing reactants, ease of obtaining and applying kinetic data, efficient use of reactor volume, and mechanical simplicity. However, great care must be taken when applying the correct flow model (e.g., plug... 

Real reactors deviate more or less from these ideal behaviors. Deviations may be detected with residence time distributions (RTD) obtained with the aid of tracer tests. In other cases a mechanism may be postulated and its parameters checked against test data. The commonest models are combinations of CSTRs and PFRs in series and/or parallel. Thus, a stirred tank may be assumed completely mixed in the vicinity of the impeller and in plug flow near the outlet. 

The determination of volumetric mass transfer coefficients, kLa, usually requires additional knowledge on the residence time distribution of the phases. Only in large diameter columns the assumption is justified that both phases are completely mixed. In tall and smaller diameter bubble columns the determination of kLa should be based on concentration profiles measured along the length of the column and evaluated with the axial dispersed plug flow model ( 5,. ... 

The most commonly used model of a mixed vessel is the fractional tubularity (delay-lag) model in which some part of the reactor is taken as exhibiting plug-flow conditions and contributing a delay and the rest of the reactor is taken as perfectly mixed (uniform concentrations) contributing a first-order lag (/( ,). The delay and lag in series are taken as describing the reactor residence time distribution (RTD). The delay-lag representation was validated using both CFD analysis and experimental residence time distributions (Walsh, 1993). 

The step-function and exponential residence-time distributions of Figure 4.5 can be modeled by two different types of flow systems. For the step-function response we have already alluded to the model of plug flow through a tube, whieh is, indeed, a standard model for this response. The exponential response, deseribed previously as the result of the equality of the internal- and exit-age distributions, requires a bit more thought. In the following we will derive the equations for the mixing models and then the corresponding reactor models for these two limits. 

The dispersion model assumes that the residence-time distribution of a real tubular reactor can be regarded as the superimposition of the plug flow that is characteristic of the ideal tubular reactor and diffusionlike axial mixing, characterized by an axial dispersion coefficient which has the same dimensions as, but can be much larger than, the molecular diffusion coefficient. The following effects can contribute to the axial mixing ... 

In terms of this modified Damkdhler number, the well-mixed dense phase behaves exactly as it should—namely, like a stirred tank. It may be shown that if f(t) is the residence time distribution of the bubble phase [in our plug flow model it is 5(t - 1)] the exponential e " in Equations 19 and 20 need only be replaced by the Laplace transform of f(t) and Tr playing the role of the transform variable. 

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