Exponential residence times

Mixing Models. The assumption of perfect or micro-mixing is frequently made for continuous stirred tank reactors and the ensuing reactor model used for design and optimization studies. For well-agitated reactors with moderate reaction rates and for reaction media which are not too viscous, this model is often justified. Micro-mixed reactors are characterized by uniform concentrations throughout the reactor and an exponential residence time distribution function. 

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. 

Example 15.13 Solve Zwietering s differential equation for arbitrary reaction kinetics and an exponential residence time distribution. 

The exponential distribution given in Eq. (4.19) has a convenient analogy to reaction engineering. An exponential residence time distribution is obtained from a CSTR. Therefore, since an MSMPR crystallizer has an exponential residence time distribution, it should come as no surprise that the CSD has that form because it is caused by kinetic events, namely nucleation and growth. 

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. 

Basic assumptions in both models include (1) membrane/bulk and membrane/lnternal phases are immiscible, (2) local phase equilibrium between membrane and internal phases, (3) no internal circulation in the globule, (4) uniform globule size, (5) mass transfer is controlled by globule diffusion, (6) internal droplets are solute sinks with finite capacity, (7) reaction of solute in the internal phase is instantaneous, (8) no coalescence and redistribution of globules, and (9) a well-mixed tank with an exponential residence time distribution of emulsion globules. 

FIGURE 3.2.2 Concetration distributions, j (c), predicted by the population balance, model of Section 3.2.4 for different drop sizes (continuous lines) compared with predictions by model based on instantaneous breakage and exponential residence time distribution (dotted lines). Reprinted from Shah and Ramkrishna (1973) with permission from Elsevier Science. 

The classical experiment tracks the off-gas composition as a function of temperature at fixed residence time and oxidant level. Treating feed disappearance as first order, the pre-exponential factor and activation energy, E, in the Arrhenius expression (eq. 35) can be obtained. These studies tend to confirm large activation energies typical of the bond mpture mechanism assumed earlier. However, an accelerating effect of the oxidant is also evident in some results, so that the thermal mpture mechanism probably overestimates the time requirement by as much as several orders of magnitude (39). Measurements at several levels of oxidant concentration are useful for determining how important it is to maintain spatial uniformity of oxidant concentration in the incinerator. 

For systems following invariant growth the crystal population density in each size range decays exponentially with the inverse of the product of growth rate and residence time. For a continuous distribution, the population densities of the classified fines and the product crystals must be the same at size Accordingly, the population density for a crystallizer operating with classified-fines removal is given by... 

In an ideal continuously stirred tank reaclor (CSTR), the conditions are uniform throughout and the condition of the effluent is the same as the condition in the tank. When a batteiy of such vessels is employed in series, the concentration profile is step-shaped if the abscissa is the total residence time or the stage number. The residence time of individual molecules varies exponentially from zero to infinity, as illustrated in Fig. 7-2>e. 

Mixing, ideal or complete A state of complete uniformity of composition and temperature in a vessel. In flow, the residence time varies exponentially, from zero to infinity. 

An ideal plug flow reactor, for example, has no spread in residence time because the fluid flows like a plug through the reactor (Westerterp etal., 1995). For an ideal continuously stirred reactor, however, the RTD function becomes a decaying exponential function with a wide spread of possible residence times for the fluid elements. 

The mean residence time (MRT) gives one parameter for the multi-exponential elimination kinetics with more than one half-life. 

X 10 years old, this implies that the content of the reservoir today is about half of what it was when the Earth was formed. The probability density function of residence time of the uranium atoms originally present is an exponential decay function. The average residence time is 6.5 x 10 years. (The average value of... 

The other case assumes that the fluid particles are well mixed. Specifically, assume that they have an exponential distribution of residence times... 

A CSTR has an exponential distribution of residence times. The corresponding differential distribution can be found from Equation (15.7) ... 

The stagnant region can be detected if the mean residence time is known independently, i.e., from Equation (1.41). Suppose we know that f=lh for this reactor and that we truncate the integration of Equation (15.13) after 5h. If the tank were well mixed (i.e., if W t) had an exponential distribution), the integration of Equation (15.13) out to 5f would give an observed t of... 

This function is shown in Figure 15.9. It has a sharp first appearance time at tflrst = tj2. and a slowly decreasing tail. When t > 4.3f, the washout function for parabohc flow decreases more slowly than that for an exponential distribution. Long residence times are associated with material near the tube wall rjR = 0.94 for t = 4.3t. This material is relatively stagnant and causes a very broad distribution of residence times. In fact, the second moment and thus the variance of the residence time distribution would be infinite in the complete absence of diffusion. 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

The molecules in the system are carried along by the balls and will also have an exponential distribution of residence time, but they are far from perfectly mixed. Molecules that entered together stay together, and the only time they mix with other molecules is at the reactor outlet. The composition within each ball evolves with time spent in the system as though the ball was a small batch reactor. The exit concentration within a ball is the same as that in a batch reactor after reaction time tf,. 

We have just described a completely segregated stirred tank reactor. It is one of the ideal flow reactors discussed in Section 1.4. It has an exponential distribution of residence times but a reaction environment that is very different from that within a perfectly mixed stirred tank. 

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 ... 

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