Laminar-flow reactors, residence-time

The pilot reactor is a tube in isothermal, laminar flow, and molecular diffusion is negligible. The larger reactor wiU have the same value for t and will remain in laminar flow. The residence time distribution will be unchanged by the scaleup. If diffusion in the small reactor did have an influence, it wiU lessen upon scaleup, and the residence time distribution will approach that for the diffusion-free case. This wiU hurt yield and selectivity. 

The molecule diffuses across the tube and samples many streamlines, some with high velocity and some with low velocity, during its stay in the reactor. It will travel with an average velocity near u and will emerge from the long reactor with a residence time close to F. The axial dispersion model is a reasonable approximation for overall dispersion in a long, laminar flow reactor. The appropriate value for D is known from theory ... 

For a few highly idealized systems, the residence time distribution function can be determined a priori without the need for experimental work. These systems include our two idealized flow reactors—the plug flow reactor and the continuous stirred tank reactor—and the tubular laminar flow reactor. The F(t) and response curves for each of these three types of well-characterized flow patterns will be developed in turn. 

In a laminar flow reactor (LFR), we assume that one-dimensional laminar flow (LF) prevails there is no mixing in the (axial) direction of flow (a characteristic of tubular flow) and also no mixing in the radial direction in a cylindrical vessel. We assume LF exists between the inlet and outlet of such a vessel, which is otherwise a closed vessel (Section 13.2.4). These and other features of LF are described in Section 2.5, and illustrated in Figure 2.5. The residence-time distribution functions E(B) and F(B) for LF are derived in Section 13.4.3, and the results are summarized in Table 13.2. 

Develop the E(t) profile for a 10-m laminar-flow reactor which has a maximum flow velocity of 0.40 m min-1. Consider t = 0.5 to 80 min. Compare the resulting profile with that for a reactor system consisting of a CSTR followed by a PFR in series, where the CSTR has the same mean residence time as the LFR and the PFR has a residence time of 25 min. Include in the comparison a plot of the two profiles on the same graph. 

A reaction with rate equation rc = C/(1+0,2C) is conducted in a laminar flow reactor. Evaluate the ratio of the mean laminar conversion to the plug flow conversion for a range of residence times. 

In fact, the plug-flow approximation is even better than this calculation indicates because of radial mixing, which wiU occur in a laminar-flow reactor. A fluid molecule near the wall will flow with nearly zero velocity and have an infinite residence time, while a molecule near the center will flow with velocity 2u. However, the molecule near the wall will diffuse toward the center of the tube, and the molecule near the center will diffuse toward the wall, as shown in Figure 8-7. Thus the tail on the RTD will be smaller, and the spike at t/2 will be broadened. We will consider diffusion effects in the axial direction in the next section. 

A graphical representation of the cumulative residence time distribution function is given in Figure 4.97 for a structured well, a laminar flow reactor and an ideal plug flow reactor assuming the same average residence time and mean velocity in each reactor. 

Obviously the characteristic distribution of the structured square, as expected, is much closer to the ideal plug flow reactor than to the laminar flow reactor. This desired behavior is a result of the channel walls, which are flow-guiding elements and pressure resistors to the flow at the same time. Two of the streamlines are projecting with a residence time of more than 0.4 s. These are the streamlines passing the area close to the wall of the distribution area, which introduces a larger resistance to these particles due to wall friction. This could, for example, be accounted for by a different channel width between the near wall channels and the central channels. 

Figure 4.97 Calculated cumulative residence time distribution function for a multi-channel well, a laminar flow reactor and a plug flow reactor [147] (by courtesy of VDI-Verlag GmbH).

Overview In this chapter we learn about nonideal reactors, that is, reactors that do not follow the models we have developed for ideal CSTRs, PFRs, and PBRs. In Pan I we describe how to characterize these nonideal reactors using the residence time distribution function (/), the mean residence time the cumulative distribution function Fit), and the variance a. Next we evaluate E t), F(t), and for idea) reactors, so that we have a reference proint as to how far our real (i.e., nonideal) reactor is off the norm from an ideal reactor. The functions (f) and F(r) will be developed for ideal PPRs. CSTRs and laminar flow reactors, Examples are given for diagnosing problems with real reactors by comparing and E(i) with ideal reactors. We will then use these ideal curves to help diagnose and troubleshoot bypassing and dead volume in real reactors. 

After studying this chapter the reader will be able to describe the cumulative F t) and external age (f) and residence-time distribution functions, and to recognize these functions for PER, CSTR, and laminar flow reactors. The reader will also be able to apply these functions to calculate the conversion and concentrations exiting a reactor using the segregation model and the maximum mixedness model for both single and multiple reactions. 

The PFR and CSTR models encompass the extremes of the residence-time distributions shown in Figure 4.3 however the batch reactor and the laminar-flow reactor, both of which we have already mentioned in this chapter, are also types exhibiting a well-defined mixing behavior. The batch reactor is straightforward, since it is simply represented by the perfect mixing model with no flow into or out of the system, and has been treated extensively in Chapter 1. 

The laminar-flow reactor with segregation and negligible molecular diffusion of species has a residence-time distribution which is the direct result of the velocity profile in the direction of flow of elements within the reactor. To derive the mixing model of this reactor, let us start with the definition of the velocity profile. 

Using the residence-time distribution function for a laminar-flow reactor, compare the yield of B in n Type III reaction with that obtained in a PFR of the same average residence time. There is no B or C in the feed, and k = Ikj with kji = 1. 

Figure 3.8 Residence time distribution in a laminar flow reactor (without radial diffusion).

Each one of the fluid elements, which is a completely segregated cluster of fluid molecules, can be treated as a micro-batch reactor. The residence time 0 of a fluid element is taken as the batch reaction time to determine the conversion achieved in the fluid element. Consider a first-order reaction A—carried out in the laminar flow reactor, (-ni) = kCA is the kinetic rate equation. The rate of change of reactant concentration in a single fluid element (treated as a batch reactor) is given by... 

For the tubular reactor mean residence time 0 = x = 10.73. The conversion of a first-order reaction in a laminar flow reactor is... 

A second-order irreversible reaction A B with rate equation (-r, ) = fcCi and rate constant k = 0.1 m /kmol/min is carried out in a laminar flow reactor. The concentration of A in the feed solution is 5 kmol/m. The mean residence time of fluid in the reactor is 5.9 min. Calculate the conversion of A. 

Fig. 7.1. Measured hysteresis in our reaction system [1]. Plot of the redox potential of Br, Fpt, as a function of the flow rate coefficient fct (in units of reciprocal residence times, the time spent by a volume element in the laminar flow reactor (LFR)). Filled dots represent one of the stable stationary states (the oxidized state) and empty dots the other stable state, the reduced state. Prom [1]...

Roughly speaking, the first moment, t, measures the size of an RTD, while higher moments measure its shape. One common measure of shape is the dimensionless second moment about the mean, also known as the dimensionless variance, 0 (see Table 1-2). In piston flow, all particles have the same residence time, so = 0. This case is approximated by highly turbulent flow in a pipe. In an ideal continuous flow stirred tank reaction, 0 = 1. WeU-designed reactors in turbulent flow have a value between 0 and 1, but laminar flow reactors can have 0 > 1. 

The derivative of a step change is a delta function, and f(t) = 8(t — t). Thus, a piston flow reactor is said to have a delta distribution of residence times. The variances for these ideal cases are = 1 for a CSTR and 0 = 0 for a PFR, which are extremes for well-designed reactors in turbulent flow. Poorly designed reactors and laminar flow reactors with little molecular diffusion can have 0 values greater than 1. 

This situation changes in liquid-phase reactions if the cross-sectional dimension of the flow reactor is reduced up to several millimeters and the dimensions of microstmctured reactors are reached. Due to shorter radial diffusion path from the bulk to the edge and vice versa, the mean diffusion time becomes substantially shorter than the required mean residence time. As a result, the typical laminar flow profile is blurred by the radial diffusion and the flow profile is similar to the turbulent flow. The residence time distribution at the reactor exit becomes narrower and approaches that of the plug flow. 

Topics that acquire special importance on the industrial scale are the quality of mixing in tanks and the residence time distribution in vessels where plug flow may be the goal. The information about agitation in tanks described for gas/liquid and slurry reactions is largely apphcable here. The relation between heat transfer and agitation also is discussed elsewhere in this Handbook. Residence time distribution is covered at length under Reactor Efficiency. A special case is that of laminar and related flow distributions characteristic of non-Newtonian fluids, which often occiu s in polymerization reactors. 

RESIDENCE TIME DISTRIBUTION FOR A LAMINAR FLOW TUBULAR REACTOR... 

Polymerizations often give such high viscosities that laminar flow is inevitable. A t5rpical monomer diffusivity in a polymerizing mixture is 1.0 X 10 ° m/s (the diffusivity of the polymer will be much lower). A pilot-scale reactor might have a radius of 1 cm. What is the maximum value for the mean residence time before molecular diffusion becomes important What about a production-scale reactor with R= 10 cm ... 

This is the first reactor reported where the aim was to form micro-channel-like conduits not by employing microfabrication, but rather using the void space of structured packing from smart, precise-sized conventional materials such as filaments (Figure 3.25). In this way, a structured catalytic packing was made from filaments of 3-10 pm size [8]. The inner diameter of the void space between such filaments lies in the range of typical micro channels, so ensuring laminar flow, a narrow residence time distribution and efficient mass transfer. 

There will be velocity gradients in the radial direction so all fluid elements will not have the same residence time in the reactor. Under turbulent flow conditions in reactors with large length to diameter ratios, any disparities between observed values and model predictions arising from this factor should be small. For short reactors and/or laminar flow conditions the disparities can be appreciable. Some of the techniques used in the analysis of isothermal tubular reactors that deviate from plug flow are treated in Chapter 11. 

---