Residence time distribution second-order

Example 15.14 Solve Zwietering s differential equation for the residence time distribution corresponding to two stirred tanks in series. Use second-order kinetics with ai ki = 5. 

Part (c) in Example 15.15 illustrates an interesting point. It may not be possible to achieve maximum mixedness in a particular physical system. Two tanks in series—even though they are perfectly mixed individually—cannot achieve the maximum mixedness limit that is possible with the residence time distribution of two tanks in series. There exists a reactor (albeit semi-hypothetical) that has the same residence time distribution but that gives lower conversion for a second-order reaction than two perfectly mixed CSTRs in series. The next section describes such a reactor. When the physical configuration is known, as in part (c) above, it may provide a closer bound on conversion than provided by the maximum mixed reactor described in the next section. 

A vessel has a residence time distribution represented by the given equations. A second order reaction with C0 = 1 is conducted in this vessel and... 

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.

Figure 5. Possible extremes of transient response to step change of reactant input concentration for second order reaction and selected values of kccT. The flow has the same residence time distribution as two perfectly mixed vessels in sequence. Solid curves complete segregation. Broken...

These two types of deviations occur simultaneously in actual reactors, but the mathematical models we discuss 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 with the residence-time distribution function to predict conversion levels accurately for first-order reactions that occur isothermally (see Section 11.2.1). The second... 

The catalytic dehydration of isopropanol was studied under tiansirat conditimis in a catalytic microreactor. The reaction is characterised by educt inhibition and shows a pronounced stop-effect . Therefore, the average productivity under forced poiodic operation can be considerably higher compared to the maximal productivity obtainable at steady state. For high rates of the sorption processes and surface reactions involved, the timal cycle time for the forced concentration variations lies in the order of seconds. As microreactors are characterized by low mass storage capacity and narrow residence time distribution, they are particularly suitable for periodic operation at relatively high fiequencies. Tis could be demonstrated in the present study. 

These models are usually applied to continuous mixed reactors. In order to be able to calculate the conversion in the exit, an assumption has to be made concerning the residence time distribution (see section 7.2.1). In the first and third models it was assumed that the mixed reactor has the residence time distribution (RTD) of a perfectly mixed CSTR. In the second model a measured RTD-function was included... 

In Chapter 3 two ideal types of continuous flow reactors were presented the plug flow reactor and the perfectly mixed reactor. In the first one all fluid elements have the same residence time. In the second one all fluid elements entering the reactor are mixed instantaneously. The consequence is that two volume elements entering the reactor at the same moment, may leave the reactor at different moments. In fact the residence times of different volume elements may vary between zero and infinite, though a large fraction has a residence time on the order of the mean residence time. In a well mixed reactor there is a large residence time distribution, in a plug flow reactor there is none. 

The liquid-phase, second-order reaction 2A —> R will be run in a continuous, agitated reactor that has the same residence time distribution as an ideal CSTR. At the operating conditions of the reactor, kCpjjX = 2.0, where k is the second-oiderrateconstantatreactortemperature, Cao is the inlet concentration of A, and r is the spacetime. 

Qader, et al., (4) reported work on hydrogenation in a dilute phase free fall reactor at temperatures in the order of 515°C, pressures of 2000 psi and with a heavy dose of catalyst, 15% stannous chloride by weight of coal. Up to 75% conversion was reported with a product distribution of 43% oil, 32% gas and 25% char. The residence time of the coal feed particles was estimated to be in the order of seconds, however, no measurement was made and aromatics were reported after further hydrorefining in a second stage hydrogenation. 

Because pit) = , the right side of Eq. (141) is a double integral over both X and t, and this can be reconducted to a double-label formalism if the second label is regarded as distributing over residence times (Aris, 1989). Of course, one expects Eq. (141) to hold for any degree of micromixing if the intrinsic kinetics are first order, but this remains to be demonstrated. 

Exercise 9.9.4. Show that the distribution function of residence times for laminar flow in a tubular reactor has the form 2z /Zp, where tp is the time of passage of any fluid annulus and the minimum time of passage. Diffusion and entrance effects may be neglected. Hence show that the fractional conversion to be expected in a second order reaction with velocity constant k is 2B[1 + j lnu5/(5 + 1)] where B = akt n and a is the initial concentration of both reactants. (C.U.)... 

Even in a large fluid bed reactor, the average residence time of the gas is quite short, i.e. a few seconds or tens of seconds. The distribution of gas residence times has been studied by Danckwerts et who concluded that it corresponds much more closely to piston flow than to complete mixing. It is possible to pulse the air supply in order to compress the range of gas residence times and to increase the average residence time. This technique also allows a wider range of particle sizes to be accepted than would otherwise be possible in a fluid-bed process. 

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