Differential Plug-Flow Reactors

For a plug flow reactor, differential volume moves along the length. The following equation may express the material balance for a plug flow reactor ... 

Plug Flow Reactor (PFR) The material balance over a differential vohime dV) is... 

After the rates have been determined at a series of reactant concentrations, the differential method of testing rate equations is applied. Smith [3] and Carberry [4] have adequately reviewed the designs of heterogeneous catalytic reactors. The following examples review design problems in a plug flow reactor with a homogeneous phase. 

Schematic representation of differential volume element of plug flow reactor.

Note that in this case the right side of equation 11.1.68 is zero for t = 0 and unity for t = 00. Figure 11.9 contains several F(t) curves for various values of n. As n increases, the spread in residence time decreases. In the limit, as n approaches infinity the F(t) curve approaches that for an ideal plug flow reactor. If the residence time distribution function given by 11.1.69 is differentiated, one obtains an... 

The volume of a plug flow reactor is represented by the differential equation,... 

Plug flow reactor Partial differential equations in one spatial variable (unsteady state)... 

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... 

Note that the difference between this material balance and that for the ideal plug flow reactors of Chapter 5 is the inclusion of the two dispersion terms, because material enters and leaves the differential section not only by bulk flow but by dispersion as well. Entering all these terms into Eq. 17 and dividing by S AZ gives... 

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

Fig. 16.2 Illustration of a cylindrical, straight-channel, plug-flow reactor. Also shown is the differential control volume by which one structures the governing-equation derivation.

We will consider a dispersed plug-flow reactor in which a homogeneous irreversible first order reaction takes place, the rate equation being 2ft = k, C. The reaction is assumed to be confined to the reaction vessel itself, i.e. it does not occur in the feed and outlet pipes. The temperature, pressure and density of the reaction mixture will be considered uniform throughout. We will also assume that the flow is steady and that sufficient time has elapsed for conditions in the reactor to have reached a steady state. This means that in the general equation for the dispersed plug-flow model (equation 2.13) there is no change in concentration with time i.e. dC/dt = 0. The equation then becomes an ordinary rather than a partial differential equation and, for a reaction of the first order ... 

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). 

The recycling reactor behaves similarly to the plug flow reactor, with one major difference, that the conversion range in the reactor is narrower than in the true plug flow reactor (Figure 8.10). This reactor type is also named the differential reactor. The performance equation is [1, 2, 6, 7]... 

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... 

In Chapter 3, the analytical method of solving kinetic schemes in a batch system was considered. Generally, industrial realistic schemes are complex and obtaining analytical solutions can be very difficult. Because this is often the case for such systems as isothermal, constant volume batch reactors and semibatch systems, the designer must review an alternative to the analytical technique, namely a numerical method, to obtain a solution. For systems such as the batch, semibatch, and plug flow reactors, sets of simultaneous, first order ordinary differential equations are often necessary to obtain the required solutions. Transient situations often arise in the case of continuous flow stirred tank reactors, and the use of numerical techniques is the most convenient and appropriate method. 

Plug Flow Reactor A plug flow reactor (PFR) is an idealized tubular reactor in which each reactant molecule enters and travels through the reactor as a plug, i.e., each molecule enters the reactor at the same velocity and has exactly the same residence time. As a result, the concentration of every molecule at a given distance downstream of the inlet is the same. The mass and energy balance for a differential volume between position Vr and Vr + dVr from the inlet may be written as partial differential equations (PDEs) for a constant-density system ... 

Finally, the mathematical model of a plug flow reactor consists of a system of ordinary differential equations with initial conditions. 

As has been seen in Sect. 3, the equations of mass, energy and momentum balances for batch and plug flow reactors generally constitute a system of ordinary differential equations, with initial values. It is convenient to write such a system in a compact vector form, viz. 

In eq 4 the rate is the time derivative of the conversion curve, which can be constructed from the observed conversion-time behavior by mathematical treatment, such as differentiation formulae or polynomial or spline interpolation, provided the product analysis is fast enough to follow the reaction. The same approach can be followed in principle for the plug flow reactor if data is collected at various space-time values. 

The plug flow reactor is increasingly being used under transient conditions to obtain kinetic data by analysing the combined reactor and catalyst response upon a stimulus. Mostly used are a small reactant pulse (e.g. in temporal analysis of products (TAP) [16] and positron emission profiling (PEP) [17, 18]) or a concentration step change (in step-response measurements (SRE) [19]). Isotopically labeled compounds are used which allow operation under overall steady state conditions, but under transient conditions with respect to the labeled compound [18, 20-23]. In this type of experiments both time- and position-dependent concentration profiles will develop which are described by sets of coupled partial differential equations (PDEs). These include the concentrations of proposed intermediates at the catalyst. The mathematical treatment is more complex and more parameters are to be estimated [17]. Basically, kinetic studies consist of ... 

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-... 

The major disadvantages of using a plug-flow reactor in the differential mode are ... 

Now, for a plug-flow reactor, the material balance is as follows for a differential volume at steady state ... 

Figure 4.29 shows a block diagram of a reactor with manipulated inputs U. other measured inputs W, and unknowm or unmeasured inputs N. We may assume that this reactor is more complicated than a simple plug-flow reactor or a CSTR. It may be more along the lines of the fluidized catalytic cracker that we showed in Fig. 4.4. The reactor can be described by a set of nonlinear differential equations as we have previously demonstrated. This results in a set of dynamic state variables X The state vector is often of high dimension and we normally only measure a subset of all the states. Y is the vector of all measurements made on the system.

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. 

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