Plug flow reactor velocity gradients

Tubular Reactors. The simplest model of a tubular reactor, the plug-flow reactor at steady state is kinetically identical to a batch reactor. The time variable in the batch reactor is transformed into the distance variable by the velocity. An axial temperature gradient can be imposed on the tubular reactor as indicated by Gilles and Schuchmann (22) to obtain the same effects as a temperature program with time in a batch reactor. Even recycle with a plug flow reactor, treated by Kilkson (35) for stepwise addition without termination and condensation, could be duplicated in a batch reactor with holdback between batches. 

The plug flow reactor is probably the most commonly used reactor in catalyst evaluation because it is simply a tube filled with catalyst that reactants are fed into. However, for catalyst evaluation, it is difficult to measure the reaction rate because concentration changes along the axis, and there are frequently temperature gradients, too. Furthermore, because the fluid velocity next to the catalyst is low, the chance for mass transfer limitations through the film around the catalyst is high. Eq. (3) is the reactor performance equation for a plug flow reactor. 

In practice, the fluid velocity profile is rarely flat, and spatial gradients of concentration and temperature do exist, especially in large-diameter reactors. Hence, the plug-flow reactor model (Fig. 7.1) does not describe exactly the conditions in industrial reactors. However, it provides a convenient mathematical means to estimate the performance of some reactors. As will be discussed below, it also provides a measure of the most efficient flow reactor—one where no mixing takes place in the reactor. The plug-flow model adequately describes the reactor operation when one of the following two conditions is satisfied ... 

Equations 14.2-3 and 14.2-4 bear a striking resemblance to the mass and energy balances for a batch reactor, Eqs. 14.1-13 and 14. There is, in fact, good physical reason why these equations should look very much alike. Our model of a plug-flow reactor, which neglects diffusion and does not allow for velocity gradients, assumes that each element of fluid travels through the reactor with no interaction with the fluid elements before or after it Therefore, if we could follow a small fluid element in a tubular reactor, we would find that it had precisely the same behavior in time as is found in a batch reactor. This similarity in the physical situation is mirrored in the similarity of the descriptive equations. 

Tubular reactors, which may be open or packed with catalyst, are considered ideal if there is plug flow of fluid and there are no radial gradients of temperature, concentration, or velocity. In plug-flow reactors, or PFRs, there are axial gradients of concentration and perhaps also axial gradients of temperature and pressure, but in the ideal PFR there is no axial diffusion or conduction. 

In a plug flow reactor all fluid elements move along parallel streamlines with equal velocity. The plug flow is the only mechanism for mass transport and there is no mixing between fluid elements. The reaction therefore only leads to a concentration gradient in the axial flow direction. For steady-state conditions, for which the term IV is zero the continuity equation is a first-order, ordinary differential equation with the axial coordinate as variable. For non-steady-state conditions the continuity equation is a partial differential equation with axial coordinate and time as variables. Narrow and long tubular reactors closely satisfy the conditions for plug flow when the viscosity of the fluid is-low. 

Plug flow reactors are subject to the same sorts of instabilities as batch reactors (since the kinetics are essentially those of a batch reactor). In addition, the problem of reduced heat transfer due to polymer build-up on the inside surface of the reactor is compounded by the fact that polymer near the wall may be more viscous (due to low temperature or high conversion). The resultant velocity gradient will result in continuing build-up on the walls as the viscous polymer near the walls has a longer residence time than the less viscous material near the center of the tube. This effect will lead to reduced heat transfer, and an even greater tendency toward thermal instability. 

Assume the plug-flow reactor approximation, for which the radial concentration gradient is ignored while the flow is taken to be equal to the average velocity. 

The modeling methodology is shown in Figure 6.17.7 for the example of a discontinuous slurry reactor. First, the concentration profiles within the catalyst particles are calculated. This information is then coupled (for each time step) with the change of concentrations in the bulk phase (Cj t)- The link between both procedures, that is, between the bulk phase and the porous catalyst particles, is the concentration gradient of each reactant at the external particle surface. Note that this calculation is also applicable for a continuous plug flow reactor simply by using the residence time t (= x/u) instead of the reaction time, whereby x represents the axial coordinate x in a tubular reactor and u the fluid velocity. 

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. 

Example 2.5. Instead of fluid flowing down a pipe as in Example 2.2, suppose the pipe is a tubular reactor in which the same reaction A A B of Example 2.3 takes place. As a slice of material moves down the length of the reactor the concentration of reactant decreases as A is consumed. Density p, velocity v, and concentration can aU vary with time and axial position z. We stiU assume plug-flow conditions so that there are no radial gradients in velocity, density, or concentration. 

In Sect. 3.2, the development of the design equation for the tubular reactor with plug flow was based on the assumption that velocity and concentration gradients do not exist in the direction perpendiculeir to fluid flow. In industrial tubular reactors, turbulent flow is usually desirable since it is accompanied by effective heat and mass transfer and when turbulent flow takes place, the deviation from true plug flow is not great. However, especially in dealing with liquids of high viscosity, it may not be possible to achieve turbulent flow with a reasonable pressure drop and laminar flow must then be tolerated. 

The flow patterns, composition profiles, and temperature profiles in a real tubular reactor can often be quite complex. Temperature and composition gradients can exist in both the axial and radial dimensions. Flow can be laminar or turbulent. Axial diffusion and conduction can occur. All of these potential complexities are eliminated when the plug flow assumption is made. A plug flow tubular reactor (PFR) assumes that the process fluid moves with a uniform velocity profile over the entire cross-sectional area of the reactor and no radial gradients exist. This assumption is fairly reasonable for adiabatic reactors. But for nonadiabatic reactors, radial temperature gradients are inherent features. If tube diameters are kept small, the plug flow assumption in more correct. Nevertheless the PFR can be used for many systems, and this idealized tubular reactor will be assumed in the examples considered in this book. We also assume that there is no axial conduction or diffusion. 

Gas-phase reacdotis are carried out primarily in tubular reactors where the flow is generally turbulent. By assuming that there is no dispersion and ttiere are no radial gradients in either temperature, velocity, or concentration, we can model the flow in the reactor as plug-flow. Laminar reactors are discussed in Chapter 13 and dispersion effects in Chapter 14. The differential form of the design equation... 

The ideal tubular reactor presents the velocity constant in the cross section of the tube with a plug flow through the reactor. There is no velocity gradient in both radial and axial directions. However, the concentration varies throughout the reactor and thus the molar balance should be differential. The molar flow varies throughout the reactor. Initially, we will assume isothermal PFR reactor. 

The assumption of plug flow is not always correct. The plug flow assumes that the convective flow (flow by velocity q/A, = v, caused by a compressor or pump) is dominating over any other transport mode. In fact, this is not always correct, and it is sometimes important to include the dispersion of mass and heat driven by concentration and temperature gradients. However, the plug flow assumption is valid for most industrial units because of the high Peclet number. We will discuss this model in some detail, not only because of its importance but also because the techniques used to handle these two-point boundary-value differential equations are similar to that used for other diffusion-reaction problems (e.g., catalyst pellets) as well as countercurrent processes and processes with recycle. The analytical analysis as well as the numerical techniques for these systems are very similar to this axial dispersion model for tubular reactors. 

---