Chemical Reaction Engineering II Notes 5 1Dr. A. Okullo

 When the tube or pipe is long enough and the fluid is not very viscous, the dispersion or tank-in-series model can be used to represent the flow in these vessels.  For viscous fluids, you have laminar flow with its characteristic parabolic velocity profile.  Because of the high viscosity, there is a slight radial diffusion between faster and slower fluid elements. 2Dr. A. Okullo

 In the extreme we have pure convection model; this assumes that each element of fluid slides past its neighbour with no interaction by molecular diffusion. Thus the spread in residence time is caused only by velocity variations (Figure 1). 3 Figure 1: Flow of fluids according to the convection model Dr. A. Okullo

 How to tell from theory which model to use  The first question to ask is, "Which model should be used in a given situation?"  The chart, adapted from Ananthakrishnan (1965) tells what regime you are in and which model to use. Figure 2.  Just locate the point which corresponds to the fluid being used (Schmidt number), the flow conditions (Reynolds number), and vessel geometry (L/d t ). Your system should not be in turbulent flow. This chart only works for laminar flows. In this chart, is the reciprocal of the Bodenstein number. It measures the flow contribution made by molecular diffusion. It is NOT the axial dispersion number, D/ud t, except in the pure diffusion regime. 4Dr. A. Okullo

 The pure diffusion regime is not a very interesting regime because it represents very slow flow.  Gases are likely to be in the dispersion regime, not the pure convection regime. Liquids can be in one regime or another. Very viscous liquids like polymers are likely to be in the pure convection regime. If your system falls in no- man’s- land, calculate the reactor behaviour based on the two bounding regimes and try to average it. Numerical solution is impractical here. It is very important to use the correct model since the RTD curves are different for different regimes (see figure 3). 5Dr. A. Okullo

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 The sharpest way of experimentally distinguishing between models comes by noting how a pulse or sloppy input pulse of tracer spreads as it moves downstream in a flow channel. For example, consider the flow, as shown in Figure 4, The dispersion or tanks-in-series models are both stochastic models; thus, we see that the variance grows linearly with distance or;  The convective model is a deterministic model; 8 (1) Dr. A. Okullo

 The spread of tracer grows linearly with distance;  Whenever you have measurements of  at 3 points use this test to tell which model to use. Just see if (in the figure 4). 9 (2) (4) Dr. A. Okullo

 The following figure 5 shows how you introduce and measure tracers in this system. 10Dr. A. Okullo

11Dr. A. Okullo

 The shape of the response curve is strongly influenced by the way tracer is introduced into the flowing fluid and how it is measured.  You may inject or measure the tracer in two main ways, as shown in Fig. 5. We therefore have four combinations of boundary conditions each with its own particular E curve (Figure 6). These E curves are shown in Fig. 7. As may be seen in Fig. 15.7, the E, E*, and E** curves are quite different, one from the other. 12Dr. A. Okullo

 E is the proper response curve for reactor purposes; it is the curve treated in Note 1, it represents the RDT in the vessel.  E* and *E are identical always, so we will call them E* from now on. One 13Dr. A. Okullo

 One correction for the planar boundary condition will transform this curve to the proper RTD.  E** requires two corrections-one for entrance, one for exit-to transform it to a proper RTD.  It may be simpler to determine E* or E** rather than E. However, remember to transform these measured tracer curves to the E curve before calling it the RTD. Let us see how to make this transformation 14Dr. A. Okullo

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 For pipes and tubes with their parabolic velocity profiles the various pulse response curves are found to be as follows: 16 (4) Dr. A. Okullo

 Where;  Two different ways of measuring the output curves are displayed in Figure 8. Note the simple relationship between 17Dr. A. Okullo

 Thus we can write;  Step Response Experiment and the F curve  When we do the step experiment by switching from one fluid to the other, we obtain the C step curve from which we should be able to find the F curve. However, this input always represents the flux input, while the output can be either planar or flux. Thus we only have two combinations, as shown in Fig. 8 With these two combinations of boundary conditions their equations and graphs are given in Eq. 6 and Fig. 9 for graphs. 18 (5) Dr. A. Okullo

19 (6) Dr. A. Okullo

 Also each F curve is related to its corresponding E curve. Thus at any time t 1 or  1 ;  The relationship is similar between E and F. 20 (7) Dr. A. Okullo

 Single n th order Reactions  In the pure convection regime (negligible molecular diffusion) each element of fluid follows its own streamline with no intermixing with neighbouring elements.  In essence this gives macro-fluid behaviour. The conversion expression is therefore; 21Dr. A. Okullo

22 For Zero order reaction of a Newtonian in Laminar flow in a pipe, integration of eq. 8 gives; (8) Dr. A. Okullo

 For first-order reaction of a Newtonian in laminar flow in a pipe;  Where ei(y) is the exponential integral.  For second-order reaction of a Newtonian in laminar flow in a pipe; 23 (9) (10) (11) Dr. A. Okullo

 Notes: 1)- Test for RTD curves. Proper RTD curves must satisfy the material balance checks (calculated zero and first moments should agree with the measured values.  The E curves of this topic (Note 5), for non- Newtonians and all shapes of channels, all meet this requirement. All the E* and E** curves of this topic do not; however, their transforms to E do. 24 (12) Dr. A. Okullo

2)- The variance of all the E curves of this topic is finite but infinite for all E* and E** curves. Be sure to know which curve you are dealing with.  In general the convection model E curve has a long tail. This makes the measurement of its variance unreliable. Thus  2 is not a useful parameter for convection models and is not presented here.  The breakthrough time  0 is probably the most reliably measured and most useful descriptive parameter for convection models, so it is widely used. 25Dr. A. Okullo

3)- Comparison with plug flow for nth-order reaction is shown in Fig. 10.  This graph shows that even at high XA convective flow does not drastically lower reactor performance. This result differs from the dispersion and tanks-in-series models. This result differs from dispersion and tank- in-series models seen earlier. 26Dr. A. Okullo

 Multiple Reactions in Laminar flow  For a two-step first-order irreversible reaction like  Because laminar flow represents a deviation from plug flow, the amount of intermediate formed will be somewhat less than for plug flow (figure 11). 27Dr. A. Okullo

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 The disappearance of A is given by the eqn. (10) and the formation and disappearance of R is given by a more complicated eqn. not shown here. The previous graph fig.11 is a result of the solutions to the equations. It shows that LFR gives a little less intermediate than does the PFR, about 20% of the way from PFR to MFR (CSTR). This can be generalised to more complex reaction systems. 30Dr. A. Okullo