Idealized flow models

In this chapter, we focus on the characteristics of the ideal-flow models themselves, without regard to the type of process equipment in which they occur, whether a chemical reactor, a heat exchanger, a packed tower, or some other type. In the following five chapters, we consider the design and performance of reactors in which ideal flow occurs. In addition, in this chapter, we introduce the segregated-flow model for a reactor as one application of the flow characteristics developed. 

Ideal flow models contain inherent assumptions about mixing behavior. In BMF, it is assumed that all fluid elements interact and mix completely at both the macroscopic and microscopic levels. In PF, microscopic interactions occur completely in any plane perpendicular to the direction of flow, but not at all in the axial direction. Fluid elements at different axial positions retain their identities as they progress through the vessel, such that a fluid element at one axial position never interacts with a fluid element at another position. 

The CRE approach for modeling chemical reactors is based on mole and energy balances, chemical rate laws, and idealized flow models.2 The latter are usually constructed (Wen and Fan 1975) using some combination of plug-flow reactors (PFRs) and continuous-stirred-tank reactors (CSTRs). (We review both types of reactors below.) The CRE approach thus avoids solving a detailed flow model based on the momentum balance equation. However, this simplification comes at the cost of introducing unknown model parameters to describe the flow rates between various sub-regions inside the reactor. The choice of a particular model is far from unique,3 but can result in very different predictions for product yields with complex chemistry. 

For ideal flow models such as perfect mixing flow, plug flow and all other ideal models, a combination of functions E(t) and F(t) can be obtained directly or indirectly using the model transfer function T(p). Before obtaining an expression for E(t) for the perfect mixing flow, we notice that the transfer function of a flow model is in fact the Laplace s transformation of the associated E(t) function ... 

As the main responsible for the changes in the material balance, the chemical reactor must be modelled accurately from this point of view. Basic flowsheeting reactors are the plug flow reactor (PFR) and continuous stirred tank reactor (CSTR), as shown in Fig. 3.17. The ideal models are not sufficient to describe the complexity of industrial reactors. A practical alternative is the combination of ideal flow models with stoichiometric reactors, or with some user programming. In this way the flow reactors can take into account the influence of recycles on conversion, while the stoichiometric types can serve to describe realistically selectivity effects, namely the formation of impurities, important for separations. Some standard models are described below. 

Daud, W.R.W., Non-ideal flow model of a top loading drum dryer, in Proceedings of the Sixth International Drying Symposium, 2, Versaflles, France, PA77, 1988. 

Itoh, N., Shindo, Y., Haraya, K. (1990). Ideal flow models for palladium membrane reactor. Journal of Chemical Engineering of Japan, 23, 420. 

The idealized flow models and axial dispersion model are used extensively to predict the performance of three-phase slurry reactors. For example, mixed flow model and axial dispersion models are used to predict the reactant conversion and product distribution for conversion of synthesis gas to liquid fuels using FT synthesis [52-57], methanol synthesis... 

Figure 14.3.6.A-1 illustrates the transient calculation of the meso-scale fluctuations around a statistically stationary state and shows the a posteriori timeaveraging period. Figure 14.3.6.A-2 shows radially averaged axial profiles of the concentration of A in the gas phase and in the liquid phase. The comparison with the conventional ideal flow models shows a behaviour typically between the extremes of plug flow and complete mixing, illustrating the importance of accounting for the details of the flow pattern.

Radially averaged axial profiles of the concentration of A In (a) the gas phase and (b) the liquid phase. Comparison with conventional Ideal flow models. 10% Inerts case. From van Baten and Krishna [2004]. 

The concept of dispersion is used to describe the degree of backmixing in continuous flow systems. Dispersion models have been developed to correct experimentally recorded deviations from the ideal plug flow model. As described in previous sections, the residence time functions E(t) mdF(t) can be used to establish whether a real reactor can be described by the ideal flow models (CSTR, PFR, or laminar flow) or not. In cases where none of the models fits the residence time distribution (RTD), the tanks-in-series model can be used, as discussed in Section 4.4. However, the use of a tanks-in-series model might be somewhat artificial for cases in which tanks do not exists in reality but only form a mathematical abstraction. The concept of a dispersion model thus becomes actual. 

The idealized flow situations depicted in Figs. 7.3-7.6 provide a good qualitative indication of the flow field within the system. There are other instances however, when a more quantitative description is needed. Such a quantitative description may be provided by combining these idealized flow models into composite mixed models. 

---