Plug-flow reactor volume element

Figure 4.19 Plug flow reactor volume element.

Schematic representation of differential volume element of plug flow reactor.

It should be emphasized that for ideal tubular reactors, it is the total volume per unit of feed that determines the conversion level achieved. The ratio of the length of the tube to its diameter is irrelevant, provided that plug flow is maintained and that one uses the same flow rates and pressure-temperature profiles expressed in terms of reactor volume elements. 

In a plug-flow reactor, all the volume elements take the same time to pass through the reactor, but in a continuous stirred tank reactor, as a... 

In a plug flow reactor the composition of the fluid varies from point to point along a flow path consequently, the material balance for a reaction component must be made for a differential element of volume dV. Thus for reactant A, Eq. 4.1 becomes... 

Regarding reactor sizes, a comparison of Eqs. 5.4 and 5.19 for a given duty and for s = 0 shows that an element of fluid reacts for the same length of time in the batch and in the plug flow reactor. Thus, the same volume of these reactors is needed to do a given job. Of course, on a long-term production basis we must correct the size requirement estimate to account for the shutdown time between batches. Still, it is easy to relate the performance capabilities of the batch reactor with the plug flow reactor. 

It can be readily discerned that the reactor equation for the batch reactor (5.12) and the plug-flow reactor (5.13) are identical. In the former, the concentration changes with time, in the latter, with location. In contrast to the situation in the other two ideal reactors, the residence time T in a CSTR is only an average, as every volume element has a different residence time throughout the reactor. 

In a tubular reactor, the reactants are fed in at one end and the products withdrawn from the other. If we consider the reactor operated at steady state, the composition of the fluid varies inside the reactor volume along the flow path. Therefore, the mass balance must be established for a differential element of volume dV. We assume the flow as ideal plug flow, that is, that there is no back mixing along the reactor axis. Hence, this type of reactor is often referred to as Plug Flow Reactor (PFR). 

Plug flow reactor (PFR) for which defined volume elements flow through a column without mixing with other volume elements. A liquid element will travel from the inlet to the outlet for a period of time equal to the reactor volume (V) divided by the flow rate (Q). 

Fig.4.4-2 demonstrates a plug flow reactor containing a "dead water" element of volume Vd where the active part is of volume Vp. The system contains also two perfectly mixed reactors 1 and 2. A tracer in a form of a pulse is introduced into reactor 1 and is transferred by the flow Qi into reactor 2 where it accumulates. 

The effect of temperature and pressure variations on the residence time for an ideal tubular-flow (plug-flow) reactor can be evaluated by comparing an actual residence time Q with Qp. The actual time required for an element of fluid to pass through the volume of reactor dV is... 

For both types of reactors pilot plant studies have shown that if the operating pressure is doubled and the amount of gas fed to the reactor is also doubled (this combination results in no change in the gas residence time) the percentage conversion remains the same. This finding means that the production per reactor volume is doubled. All this is in keeping with kinetic model predictions. The basic rate equation used is that at any element inside a plug flow reactor the rate of conversion of CO to hydrocarbon products equals... 

Because of the plug-flow assumption, it is natural to take a thin disk for the reactor volume element as shown in Figure 4.19. The concentration does not change over the volume element because there is complete mixing in the radial and angular directions and because the axial distance Az is small. The element has volume AV = A Az in which Ac is the tube cross-sectional area. Writing Equation 4.2 in this situation gives... 

To derive an energy balance for the plug-flow reactor (PFR), consider the volume element in Figure 6.32. If we write Equation 6.5 for this element and neglect kinetic and potential energies and shaft work, we obtain... 

Plug flow is a simplified and idealized picture of the motion of a fluid, whereby all the fluid elements move with a uniform velocity along parallel streamlines. This perfectly ordered flow is the only transport mechanism accounted for in the plug flow reactor model. Because of the uniformity of o>nditions in a cross section the steady-state continuity equation is a very simple ordinary differential equation. Indeed, the mass balance over a differential volume element for a reactant A involved in a single reaction may be written ... 

The plug flow reactor also gives rise to an IVP. If we write down a mass balance equation for the thin element at the position z with a thickness of Az (see the control volume of Fig. 1.1b) and then allow the element thickness to approach zero, we obtain the following equations for the two reactions in series ... 

The RTD of ideal cascades with different numbers of tanks in series are given in Figures 3.6 and 3.7. With increasing subdivision of the entire reactor volume into ideally mixed individual elements, the residence time becomes more and more uniform and the RTD curves become more symmetrical. The RTD of the cascade of the total volume V approaches that of an ideal plug flow reactor of the same volume and becomes identical with this when N goes toward infinity. 

Element of volume dV the starting point for mathematical analysis of plug flow reactors... 

The ideal plug flow reactor PFR is a simplified picture of the motion of a fluid in a tubular reactor as it is assumed that all fluid elements move with a uniform velocity along parallel streamlines and thus have a fixed residence time r. Strictly speaking, this assumption breaks the hydrodynamic rule that the velocity is zero at the wall (no slip condition. Figure 3.2.22). The steady-state mass balance of a PFR for a constant volume reaction can be deduced from the one-dimensional mass balance for a differential small element with thickness Az in direction of flow ... 

In Chapter 3 two ideal types of continuous flow reactors were presented the plug flow reactor and the perfectly mixed reactor. In the first one all fluid elements have the same residence time. In the second one all fluid elements entering the reactor are mixed instantaneously. The consequence is that two volume elements entering the reactor at the same moment, may leave the reactor at different moments. In fact the residence times of different volume elements may vary between zero and infinite, though a large fraction has a residence time on the order of the mean residence time. In a well mixed reactor there is a large residence time distribution, in a plug flow reactor there is none. 

Figure 4.1. A differential volume element (dVr) in a tubular (or plug flow) reactor with F and f being the flow rate and fractional conversion, respectively, of the limiting reactant A.

FIGURE 5.3 Differential volume element for plug-flow reactor (Example 5.4). 

Consider a small element of volume, AV, of an ideal plug-flow tubular reactor, as shown in Fig. 4.6. 

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