Integral Plug-Flow Reactors

In an integral PFR, the reactant conversion is significant Therefore, it is not valid to assume that the reaction rate is essentially constant in the direction of flow. Suppose that an ideal, isothermal PFR is operated with a constant feed composition at several different values of V/Faq (or W/Fao), and the fractional conversion, xa, is measured at each value of V/Faq. The resulting data will have the form shown in the following figure. 

We can work directly with this kind of data to test a postulated rate equation. This approach will be discussed later, in Section 6.3 of this chapter. However, it is also possible to calculate values of the reaction rate —rA at various values of xa, to obtain the type of data shown in Table 6-1. 

This equation shows that the value of the reaction rate at any value of Xa is equal to the slope of the curve in the figure above, taken at the specified value of X. This relationship is represented in the following figure. 

In other words, a value of the reaction rate at x can be obtained by taking the derivative of the Xa versus V/F q curve at x.  

If the xa versus V/Fao curve was obtained at a constant temperature, a subset of data similar to the first partofTable 6-1 can be generated by taking slopes at various values ofxA.Tbe values of Ca, Cb, etc. in Table 6-1 can be calculated from xa and the known feed compositiorL 

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Differential (flow) reactor Integral (plug flow) reactor Mixed flow reactor Batch reactor for both gas and solid... 

Figure 13.27 Heat-integrated plug-flow reactor...

This is the equation for a plug flow reactor. It can be derived directly from the rate equations with the aid of Laplace transforms. The sequences of second-order reactions of Figs. 7-5n and 7-5c required numerical integrations. 

Example 5 Application of Effectiveness For a second-order reaction in a plug flow reactor the Thiele modulus is ( ) = SVQ, and inlet concentration is C50 = 1.0. The equation will he integrated for 80 percent conversion with Simpsons rule. Values of T) are... 

This equation is the basic relation for the mean residence time in a plug flow reactor with arbitrary reaction kinetics. Note that this expression differs from that for the space time (equation 8.2.9) by the inclusion of the term (1 + SAfA) and that this term appears inside the integral sign. The two quantities become identical only when 5a is zero (i.e., the fluid density is constant). The differences between the two characteristic times may be quite substantial, as we will see in Illustration 8.5. Of the two quantities, the reactor... 

This equation differs from that for the plug flow reactor (8.2.9) in that for a CSTR the rate is evaluated at effluent conditions and thus appears outside the integral. 

Fig. 9. Graphical integration of the design equation for a plug-flow reactor.

In contrast to the design equations for batch and plug-flow reactors, eqns. (5) and (62), the design equation for the continuous stirred tank reactor does not contain an integral sign. Figure 14 shows [ A]o/r plotted... 

Now, for a constant-density first-order reaction, the integrated form of the design equation for a plug-flow reactor, eqn. (66), may be rewritten... 

For a first-order reaction which is not accompanied by any change in density, the design equation for a plug-flow reactor can be integrated to give... 

Batch or Plug Flow Reactors. Integration gives the performance equations for this system... 

This equation, again, cannot be integrated immediately as t is a function of the substrate concentration C., which changes when going from the entrace to the exit of the plug-flow reactor. The procedure to solve this equation is the calculation of the overall effectiveness factor at n substrate concentration in the interval C to Through this n (... 

If product inhibition occurs, either a stirred-tank reactor in batch or a plug-flow reactor should be used. In these two reactors, the product concentration increases with time. Alternatively a reactor with integrated product separation (membrane, solvent, etc.) is preferable. 

The distributed nature of the tubular plug flow reactor means that variables change with both axial position and time. Therefore the mathematical models consist of several simultaneous nonlinear partial differential equations in time t and axial position z. There are several numerical integration methods for solving these equations. The method of lines is used in this chapter.1... 

The parameters for PFRs include space time, concentration, volumetric flow rate, and volume. This reactor follows an integral reaction expression identical to the batch reactor except that space time has been substituted for reaction time. In the plug flow reactor, concentration can be envisioned as having a profile down the reactor. Conversion and concentration can be directly related to the reactor length, which in turn corresponds to reactor volume. 

The plug-flow reactor may be operated in the differential or the integral mode. In the differential mode (small conversion) the whole catalyst can be considered to be exposed to the same concentration of reactants. The influence of products is generally weak, except when the catalyst is extremely sensitive to one particular product.Thc plug-flow reactor operating in the dif-... 

In the integral mode (e.g. in a batch reactor), the evolution of the concentration can be measured as a function of time. For a plug flow reactor, it is measured as a function of residence time, cf. Section 4.1.2.4.2. 

In some cases it is possible to perform experiments at conversions low enough to allow neglect of the effect of conversion on the rate. In other words the rate is constant throughout the plug flow reactor. Equation (7.159) can then be integrated to ... 

These were calculated by numerical integration of Equation (8). SD must be less than SD if an ideal plug-flow reactor is to be designed and operated. 

We next turn to process feedback. W e mentioned earlier that a plug-flow reactor can be viewed as a string of small batch reactors. We also pointed out that the result of each batch is uniquely determined by the fresh feeds since the solution to the batch equations is a forward integration in time. A plug-flow7 reactor cannot by itself show output multiplicity or open-loop instability. This picture changes when we... 

We start by plotting the temperature rise in the reactor. This is done by integrating the steady-state differential equations that describe the composition and heat effects as functions of the axial position in the reactor. The adiabatic plug-flow reactor gives a unique exit temperature for a given feed temperature. This also means that we get a unique difference between the exit and feed temperatures. The temperature difference has to be less than or equal to the adiabatic temperature rise at a given, constant feed composition. Figure 5.20 show s the fractional temperature rise as a function of the reactor feed temperature for a typical system. 

This equation is a generalization of the Aris (1968) result given in Eq. (98) The apparent overall order is (a + l)/a times the intrinsic order of 2. Scaramella et al. (1991) have also analyzed a perturbation scheme around this basic solution, which, incidentally, lends itself to a solution via reduction to an integral Volterra equation of the same type as discussed in Section IV,C with regard to a plug flow reactor with axial diffusion. Scaramella et al. have also shown that if b x,y) can be expressed as the sum of M products of the type b x)B y) + B(x)b(y), which is... 

Petroleum refinery flowsketch, 26 PER (plug flow reactor), 55,558 comparison with CSTR, complex reactions, 569 volume ratio to CSTR, 571 Phase diagrams nitrotoluene isomers, 544 salt solutions, 526 use of example, 528 Phenol bv the chlorbenzene process, 34 Phosgene synthesis, 594 PhthMic anhydride synthesis, 593 PID (proportional-integral-derivative) controllers, 41, 42... 

We now separate the variables and integrate with the limit V = 0 when X = G to obtain the plug-flow reactor volume necessary to achieve a specified conversion X ... 

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