Residence-time distributions laminar flow reactors

The polymerization time in continuous processes depends on the time the reactants spend in the reactor. The contents of a batch reactor will all have the same residence time, since they are introduced and removed from the vessel at the same times. The continuous flow tubular reactor has the next narrowest residence time distribution, if flow in the reactor is truly plug-like (i.e., not laminar). These two reactors are best adapted for achieving high conversions, while a CSTR cannot provide high conversion, by definition of its operation. The residence time distribution of the CSTR contents is broader than those of the former types. A cascade of CSTR s will approach the behavior of a plug flow continuous reactor. 

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.)... 

The drawback of randomly packed microreactors is the high pressure drop. In multitubular micro fixed beds, each channel must be packed identically or supplementary flow resistances must be introduced to avoid flow maldistribution between the channels, which leads to a broad residence time distribution in the reactor system. Initial developments led to structured catalytic micro-beds based on fibrous materials [8-10]. This concept is based on a structured catalytic bed arranged with parallel filaments giving identical flow characteristics to multichannel microreactors. The channels formed by filaments have an equivalent hydraulic diameter in the range of a few microns ensuring laminar flow and short diffusion times in the radial direction [10]. 

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... 

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. 

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. 

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. 

Figure 8-5 Residence time distribution in a laminar flow tubular reactor. The dashed curve indicates the p t) curve expected in laminar flow after allowing for radial diffusion, which makes p(t) closer to the plug flow.

A characteristic of micro-channel reactors is their narrow residence-time distribution. This is important, for example, to obtain clean products. This property is not imaginable without the influence of dispersion. Considering only the laminar flow would... 

In macroscopic reactors, knowledge of the velocity profile in the channel cross-section is a necessary and sufficient prerequisite to describe the material transport. In microscopic dimensions down to a few micrometers, diffusion also has to be considered. In fact, without the influence of diffusion, extremely broad residence time distributions would be found because of the laminar flow conditions. Superposition of convection and diffusion is called dispersion. Taylor [91] was among the first to notice this strong dominating effect in laminar flow. It is possible to transfer his deduction to rectangular channels. A complete fluid dynamic description has been given of the flow, including effects such as the influence of the wall, the aspect ratio and a chemical wall reaction on the concentration field in the cross-section [37]. 

A characteristic of micro channel reactors is their narrow residence-time distribution. This is important, for example, to obtain clean products. This property is not imaginable without the influence of dispersion. Just considering the laminar flow would deliver an extremely wide residence-time distribution. The near wall flow is close to stagnation because a fluid element at the wall of the channel is, by definition, fixed to the wall for an endlessly long time, in contrast to the fast core flow. The phenomenon that prevents such a behavior is the known dispersion effect and is demonstrated in Figure 3.88. 

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).

The fluidized bed reactor can also handle fast, complex reactions, with mixing and temperature control being especially good when stirring is provided. Unfortunately, the extent of back mixing is difficult to assess so that the residence time distribution of the reactants in the reactor is uncertain. In addition, only small catalyst particles can be used, and attrition, with the consequent breakdown and loss of catalyst, is a problem. Finally, a catalyst bed is adequately fluidized over only a comparatively narrow range of flow rates. More information about kinetic reactors can be found in reviews [33,34,50], Applications of the basket-type mixed reactor to liquid-solid systems are discussed by Suzuki and Kawazo [62] and by Teshima and Ohashi [63], and the development of a laminar flow, liquid-solid reactor by Schmalzer et al. [64], In the latter reactor the wall is coated with a catalyst layer. 

After studying this chapter the reader will be able to describe the cumulative F(t), external age E(t), and internal age I(t) residence-time distribution functions and to recognize these functions for PFR, CSTR, and laminar flow reactions. 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. 

Fig. 6-7 Residence-time distribution in a laminar-flow tubular reactor (segregated flow)...

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. 

Calculate the residence-time distribution (RTD) for a tubular reactor undergoing steady, laminar flow (Hagen-Poiseuille flow). The velocity profile for Hagen-Poiseuille flow is 2, p. 51]... 

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. 

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