Residence-time distribution segregation

The combination of reac tor elements is facihtated by the concept of transfer functions. By this means the Laplace transform can be found for the overall model, and the residence time distribution can be found after inversion. Finally, the chemical conversion in the model can be developed with the segregation and maximum mixed models. 

Friis and Hamielec (48) offered some comments on the continuous reactor design problem suggesting that the dispersed particles have the same residence time distribution as the dispersing fluid and the system can be modeled as a segregated CSTR reactor. 

The concept of a well-stirred segregated reactor which also has an exponential residence time distribution function was introduced by Dankwerts (16, 17) and was elaborated upon by Zweitering (18). In a totally segregated, stirred tank reactor, the feed stream is envisioned to enter the reactor in the form of macro-molecular capsules which do not exchange their contents with other capsules in the feed stream or in the reactor volume. The capsules act as batch reactors with reaction times equal to their residence time in the reactor. The reactor product is thus found by calculating the weighted sum of a series of batch reactor products with reaction times from zero to infinity. The weighting factor is determined by the residence time distribution function of the constant flow stirred tank reactor. 

The monomer conversion in this seeded polymerization system is independent of the degree of segregation as long as an exponential residence time distribution function is maintained. 

The completely segregated stirred tank can be modeled as a set of piston flow reactors in parallel, with the lengths of the individual piston flow elements being distributed exponentially. Any residence time distribution can be modeled as piston flow elements in parallel. Simply divide the flow evenly between the elements and then cut the tubes so that they match the shape of the washout function. See Figure 15.12. A reactor modeled in this way is said to be completely segregated. Its outlet concentration is found by averaging the concentrations of the individual PFRs ... 

When the residence time distribution is known, the uncertainty about reactor performance is greatly reduced. A real system must lie somewhere along a vertical line in Figure 15.14. The upper point on this line corresponds to maximum mixedness and usually provides one bound limit on reactor performance. Whether it is an upper or lower bound depends on the reaction mechanism. The lower point on the line corresponds to complete segregation and provides the opposite bound on reactor performance. The complete segregation limit can be calculated from Equation (15.48). The maximum mixedness limit is found by solving Zwietering s differential equation. ... 

The limits for part (b) are at the endpoints of a vertical line in Figure 15.14 that corresponds to the residence time distribution for two tanks in series. The maximum mixedness point on this line is 0.287 as calculated in Example 15.14. The complete segregation limit is 0.233 as calculated from Equation (15.48) using/(/) for the tanks-in-series model with N=2 ... 

Suppose is a function of a alone and that neither dSt Ajda nor d Alda change sign over the range of concentrations encountered in the reactor. Then, for a system having a fixed residence time distribution. Equations (15.48) and (15.49) provide absolute bounds on the conversion of component A, the conversion in a real system necessarily falling within the bounds. If d S A/dc > 0, conversion is maximized by maximum mixedness and minimized by complete segregation. If d 0i A/da < 0, the converse is true. If cf- A/da = 0, micro-mixing has no effect on conversion. 

These two types of deviations occur simultaneously in actual reactors, but the mathematical models we will develop assume that the residence time distribution function may be attributed to one or the other of these flow situations. The first class of nonideal flow conditions leads to the segregated flow model of reactor performance. This model may be used... 

In the segregated flow model the contents of the volume elements of the fluid do not mix with one another as they move through the reactor. Each element may be considered as a small closed system that moves through the reactor. The different systems spend varying amounts of time in the reactor, giving rise to the measured residence time distribution func-... 

The chapter begins with a reiteration and extension of terms used, and the types of ideal flow considered. It continues with the characterization of flow in general by age-distribution functions, of which residence-time distributions are one type, and with derivations of these distribution functions for the three types of ideal flow introduced in Chapter 2. It concludes with the development of the segregated-flow model for use in subsequent chapters. 

A reactor has a residence time distribution like that of that of two equal completely mixed tanks in series. The rate equation is -dC/dt = 0.5C1-5. Inlet concentration is C0 = 1.2 lbmol/cuft and the feed rate is 10 Ibmol reactant/min. Conversion required is 95%. Find the reactor volume needed (a) assuming segregated flow (b) in a two stage CSTR. 

The integral for a batch reaction is tabulated in the first two columns. A potential reactor has the residence time distribution E(t) given in column 3. Find the outlet concentration in segregated flow. 

Generally speaking, the design of TBR requires knowledge of hydrodynamics and flow regimes, pressure-drop, hold-ups of the phases, interfacial areas and mass-transfer resistances, heat transfer, dispersion and back-mixing, residence time distribution, and segregation of the phases. 

The mixing pattern in an n-stage CSTR battery is intermediate between segregated and maximum mixed flow and is characterized by residence time distribution with variance o2 = 1/n. Conversion in the CSTR battery is found by solving n successive equations... 

Figure 17.3. Ratio of volumes of an n-stage CSTR battery and a segregated flow reactor characterized by a residence time distribution with variance a2 = 1/n. Second-order reaction.

Since the dimensionless time for a first-order reaction is the product of the reaction time t and a first-order rate constant k, there is no reason why k(x)t should not be interpreted as k(x)t(x), that is, the reaction time may be distributed over the index space as well as the rate constant. Alternatively, with two indices k might be distributed over one and t over the other as k x)t(y). We can thus consider a continuum of reactions in a reactor with specified residence time distribution and this is entirely equivalent to the single reaction with the apparent kinetics of the continuum under the segregation hypothesis of residence time distribution theory, a topic that is in the elementary texts. Three indices would be required to distribute the reaction time with a doubly-distributed continuous mixture. 

FIGURE 15.15 Extreme mixing models for an arbitrary residence time distribution (a) complete segregation (b) maximum mixedness. 

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