Axial dispersion in tubular reactors

Exercise 9.9,2. Show how to use the graph in Fig. 9.22 for a reversible first order reaction. 

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 Peclet number of radial dispersion was found to be between 8 and 15. as we have noted above. However, the axial Peclet number is about 2. which shows that the axial eddy diffusion coefficient is anywhere from four to seven times the radial coefficient Er. A very simple model gives some indication why this should be so. The flow through a packed bed has been described 

Consider an isothermal first order irreversible reaction A 

However, if the axial dispersion can be described by an effective diffusion coefficient giving a diffusive flux of Eai—dc/dz), then a balance over the clement between z and z + dz will give  

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In Chapter 8, axial dispersion in tubular reactors was discussed. Typical industrial reactors have sufficiently high flow rates and reactor lengths so the effects of axial dispersion are minimal and can be neglected. A rule of thumb is that axial dispersion can be neglected if ... 

Table 3.2 Estimation of axial dispersion in tubular reactors [6].

Figure 3.15 Axial dispersion in tubular reactors (a) laminar flow and (b) turbulent flow. Gray area represents experimental results. (Adapted from [6], Figure 27.25 Copyright 2012, Wiley-VCH GmbH Co. KGaA.)...

An axially-dispersed, adiabatic tubular reactor can be described by the following mass and energy balances that are in dimensionless form (the reader should verify that these descriptions are correct) ... 

The dispersion in tubular reactors depends on the flow regime and is characterized by the Bodenstein number, the ratio of the axial diffusion time, tu,ax, in the reactor to the mean fluid residence time, x. 

The axial dispersion in the reactor is often expressed by the axial Peclet number, and the characteristic length, which equals the tube diameter for tubular reactors, and the particle diameter for packed-bed reactors. The Bodenstein number characterizing dispersion in the tubular reactor thus becomes the following ... 

For the design of tubular reactors an a priori estimation of the axial dispersion is indispensable. The dispersion in tubular reactors depends on the flow regime, characterized by the Reynolds number. Re, and the physical properties of the fluid, characterized by the Schmidt number. Sc. In addition, the presence of internal packings influences the flow behavior and, in consequence, the axial dispersion of the fluid. 

Axial and radial dispersion or non-ideal flow in tubular reactors is usually characterised by analogy to molecular diffusion, in which the molecular diffusivity is replaced by eddy dispersion coefficients, characterising both radial and longitudinal dispersion effects. In this text, however, the discussion will be limited to that of tubular reactors with axial dispersion only. Otherwise the model equations become too complicated and beyond the capability of a simple digital simulation language. 

Nonideal tubular reactor models, inclusion of interpellet axial dispersion in,... 

Note that setting one of the terms on the left side of the equation equal to zero yields either the batch reactor equation or the steady-state PFTR equation. However, in general we must solve the partial differential equation because the concentration is a function of both position and time in the reactor. We will consider transients in tubular reactors in more detail in Chapter 8 in connection with the effects of axial dispersion in altering the perfect plug-flow approximation. 

This bifurcation analysis is very valuable for every chemical, biochemical, and biomedical process that involves axial dispersion in a tubular reactor. 

The three ideal reactors form the building blocks for analysis of laboratory and commercial catalytic reactors. In practice, an actual flow reactor may be more complex than a CSTR or PFR. Such a reactor may be described by a residence time distribution function F(t) that gives the probability that a given fluid element has resided in the reactor for a time longer than t. The reactor is then defined further by specifying the origin of the observed residence time distribution function (e.g., axial dispersion in a tubular reactor or incomplete mixing in a tank reactor). 

In stirred tanks, the power input to agitate the tank will depend on the physical properties of the liquid. In tubular reactors, the axial dispersion in empty tubes may be estimated [e.g., Wen in Petho and Noble (eds.), Residence Time Distribution Theory in Chemical Engineering, Verlag Chemie, 1982] as... 

The other two methods are subject to both these errors, since both the form ofi the RTD and the extent of micromixing are assumed. Their advantage is that they permit analytical solution for the conversion. In the axial-dispersion model the reactor is represented by allowing for axial diffusion in an otherwise ideal tubular-flow reactor. In this case the RTD for the actual reactor is used to calculate the best axial dififusivity for the model (Sec. 6-5), and this diffusivity is then employed to predict the conversion (Sec. 6-9). This is a good approximation for most tubular reactors with turbulent flow, since the deviations from plug-flow performance are small. In the third model the reactor is represented by a series of ideal stirred tanks of equal volume. Response data from the actual reactor are used to determine the number of tanks in series (Sec. 6-6). Then the conversion can be evaluated by the method for multiple stirred tanks in series (Sec. 6-10). 

D.F. Leclerc, P.A. Bloxham, E.C. Toren Jr., Axial dispersion in coiled tubular reactors, Anal. Chim. Acta 184 (1986) 173. 

Evaluate the results obtained in problems 8 and 14, Chapter 4, in terms of the criterion developed by Meats for freedom from axial dispersion effects in tubular reactors [D.E. Meats, Ind. Eng. Chem. Proc. Design Devel., 10, 541 (1971)]. 

Measure the incremental conversion of ethanol per mass of catalyst and calculate the initial reactant product conversion rate with units of moles per area per time as a function of total pressure at the reactor inlet. One calculates this initial rate of conversion of ethanol to products via a differential material balance, unique to gas-phase packed catalytic tubular reactors that operate under plug-flow conditions at high-mass-transfer Peclet numbers. Since axial dispersion in the packed bed is insignificant. 

MASS TRANSFER PECLET NUMBERS BASED ON INTERPELLET AXIAL DISPERSION IN PACKED CATALYTIC TUBULAR REACTORS... 

Gas chromatography is a separation technique based on the fact that different components in the mixture exhibit different average residence times due to interactions with the porons packing material. These interactions can be classified as intrapellet diffusion and the column operates similar to a packed catalytic tubular reactor. The important mass transfer mechanisms are convection and diffusion. Hence, it is important to calculate the mass transfer Peclet number that represents an order-of-magnitude ratio of these two mass transfer rate processes. Intrapellet diffusion governs residence times, and interpellet axial dispersion affects the degree to which the output curve is broadened. For axial dispersion in packed columns and packed catalytic tubular reactors. 

For liquid-phase reactions, a single PFR or CSTR reactor is often used. For a single reaction at isothermal conditions, the volume of a PI is smaller than that of a CSTR for the same conversion and temperature. However, for (1) autocatalytic reactions, where the reaction rate depends on the concentration of a product, or (2) autothermal reactions, where the feed is cold, but the reaction is highly exothermic, the volume of a CSTR can be smaller than a PFR, such that axial dispersion in a tubular reactor may actually be beneficial. In general, a... 

The RTD in chemical reactors is a crucial parameter for process performances and product yield and selectivity. The RTD in tubular reactors can be described by the so-called dispersion model [7]. This model suggests that the RTD can be considered as the result of piston flow with the superposition of axial dispersion. The dispersion is considered by means of an effective axial dispersion coefficient... 

In Chapter 2, the design of the so-called ideal reactors was discussed. The reactor ideahty was based on defined hydrodynamic behavior. We had assumedtwo flow patterns plug flow (piston type) where axial dispersion is excluded and completely mixed flow achieved in ideal stirred tank reactors. These flow patterns are often used for reactor design because the mass and heat balances are relatively simple to treat. But real equipment often deviates from that of the ideal flow pattern. In tubular reactors radial velocity and concentration profiles may develop in laminar flow. In turbulent flow, velocity fluctuations can lead to an axial dispersion. In catalytic packed bed reactors, irregular flow with the formation of channels may occur while stagnant fluid zones (dead zones) may develop in other parts of the reactor. Incompletely mixed zones and thus inhomogeneity can also be observed in CSTR, especially in the cases of viscous media. 

An idealized continuous flow MSR can be described by plug flow behavior, meaning that all molecules at the reactor outlet exhibit an identical residence time. Non-idealities in tubular reactors can be characterized using the axial dispersion... 

Radial Dispersion of Mass In contrast to axial dispersion in a tubular fixed bed reactor, mass transfer in the radial direction should be maximized to prevent... 

For turbulent flow in pipes the velocity profile can be calculated from the empirical power law design formula (1.360). Similar balance equations with purely molecular diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.359) might suffice. It is important to note that the difference between the cross section averaged ID axial dispersion model equations (discussed in the previous section) and the simplified 2D model equations (presented above) is that the latter is valid locally at each point within the reactor, whereas the averaged one simply gives a cross sectional average description of the axial composition and temperature profiles. 

This section has based scaleups on pressure drops and temperature driving forces. Any consideration of mixing, and particularly the closeness of approach to piston flow, has been ignored. Scaleup factors for the extent of mixing in a tubular reactor are discussed in Chapters 8 and 9. If the flow is turbulent and if the Reynolds number increases upon scaleup (as is normal), and if the length-to-diameter ratio does not decrease upon scaleup, then the reactor will approach piston flow more closely upon scaleup. Substantiation for this statement can be found by applying the axial dispersion model discussed in Section 9.3. All the scaleups discussed in Examples 5.10-5.13 should be reasonable from a mixing viewpoint since the scaled-up reactors will approach piston flow more closely. 

Determine the yield of a second-order reaction in an isothermal tubular reactor governed by the axial dispersion model with Pe = 16 and kt = 2. 

Water at room temperature is flowing through a 1.0-in i.d. tubular reactor at Re= 1000. What is the minimum tube length needed for the axial dispersion model to provide a reasonable estimate of reactor performance What is the Peclet number at this minimum tube length Why would anyone build such a reactor ... 

This example models the dynamic behaviour of an non-ideal isothermal tubular reactor in order to predict the variation of concentration, with respect to both axial distance along the reactor and flow time. Non-ideal flow in the reactor is represented by the axial dispersion flow model. The analysis is based on a simple, isothermal first-order reaction. 

Fig. 2.4p shows three types of post-column reactor. In the open tubular reactor, after the solutes have been separated on the column, reagent is pumped into the column effluent via a suitable mixing tee. The reactor, which may be a coil of stainless steel or ptfe tube, provides the desired holdup time for the reaction. Finally, the combined streams are passed through the detector. This type of reactor is commonly used in cases where the derivatisation reaction is fairly fast. For slower reactions, segmented stream tubular reactors can be used. With this type, gas bubbles are introduced into the stream at fixed time intervals. The object of this is to reduce axial diffusion of solute zones, and thus to reduce extra-column dispersion. For intermediate reactions, packed bed reactors have been used, in which the reactor may be a column packed with small glass beads. 

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