Reactor plug flow

For a tubular (plug flow) reaetor, the eonditions at any point in the reaetor are independent of time, and the linear veloeity v of the reaeting mixture is the same at every point in a eross-seetion S perpendieular to the flow direetion and equal to (G/pS). The eom-position of the reaetion mixture depends on the distanee L from the inlet point. 

Introduction to Reactor Design Fundamentals for Ideal Systems 363 

364 Modeling of Chemioal Kinetios and Reaotor Design The following assumptions should be made  

Now eonsider the mass balanee for reaetant A over the eontrol volume S6l. 

The average rate of reaetion is rates at 1 and 1 + 61. That is. given by the arithmetie mean of the  

EXAMPLE 6.3 Plug flow reactor with a first-order sink 

Consider a plug flow reactor at steady state, with a first-order sink. What would be the concentration profile versus distance down the reactor  

Although there is no true plug flow, there are situations where plug flow is an appropriate assumption to make. For example, the gradients for dissolved oxygen 

EXAMPLE 6.4 Development of the Streeter-Phelps (1925) equation for DO sag below a point BOD source in a river (plug flow with a first-order and zero-order source/sink terms) 

For a differential steady-flow reactor, the species-based design equation, written for species j, is given by Eq. 4.2.10  

We use Eq. 2.3.11 to express the local molar flow rate of species j, Fj, in terms of extents of the independent reactions  

Note that here too, the summation on the left-hand side is over the independent reactions only, whereas the summation on the right-hand side is over all the reactions that take place in the reactor. We follow the same procedure as in the case of the ideal batch reactor. We first write the. summation on the right as two sums over dependent reactions and independent reactions. Next, we express the stoichiometric coefficient of species j in the kth-dependent reaction, (Sj)k, in terms of the stoichiometric coefficients of species j in the independent reactions, (Sj)m, using Eq. 2.4.9, and then switch the order of the. summations to obtain 

Equation 4.3.14 is the reaction-based, differential design equation for steady-flow reactors, written for the wth-independent reaction. As will be discussed below, to describe the operation of the reactor with multiple reactions, we have to write Eq. 4.3.14 for each of the independent reactions. 

For steady-flow reactors with a single chemical reaction, Eq. 4.3.14 reduces to 

Now consider the mass balance for reactant A over the control volume S8l. 

As 81 - 0, (l/2)[d(-rA)1/dl]8l becomes increasingly small and Equation 5-288 then becomes 

Consider the case where both volumetric flowrate u and density p are constant. A relationship between the mass flowrate G and the volumetric flow is  

As explained in Chapter 4, the plug-flow reactor differs from the batch reactor only with respect to the time coordinate. While for the batch reactor time elapsed since the commencement of reaction is directly used as a measure of this coordinate, in the plug-flow reactor it is replaced by the time required to traverse a given distance in the tubular reactor t = i/u, where is the distance and u the average velocity. Thus the rate equation now becomes 

Rate equations can be written for all the components of a complex reaction. For the ethylation reaction considered in Example 11.3, for instance, these will 

In this section we replace the CSTR by a plug-flow reactor and consider the conventional control structure. Section 4.5 presents the model equations. The energy balance equations can be discarded when the heat of reaction is negligible or when a control loop keeps constant reactor temperature manipulating, for example, the coolant flow rate. The model of the reactor/separation/recycle system can be solved analytically to obtain (the reader is encouraged to prove this)  

Similarly to the CSTR, case Eq. (4.14), X = 0 is a trivial solution satisfying the model equations irrespective of the Da value. However, it is unfeasible, corresponding to infinite flow rates. 

It can be shown that the nontrivial solution is feasible (positive flow rates) if, and only if  

Note that first inequality of Eq. (4.21) is identical to the feasibility constraint (4.13) characterizing the reactor/separation/recycle system involving a CSTR. 

For both the CSTR and PFR systems, at DaT = (z0 - z4)/z3 two different manifolds of steady states cross each other, in the combined space of state variables and parameters. According to the bifurcation theory, this is a transcritical bifurcation point Here, an exchange of stability takes place for Da DaT, the trivial solution 

In this section, you will solve the equations for an isothermal plug flow reactor. The first problem is very simple, and is patterned after a problem on the California Professional Engineers License Examination, according to Fogler (2005). Here it is modified. You take a reactor in which components A and C are fed in equimolar amounts, and the following reaction takes place  

You assume that the reaction takes place in the hquid phase and that the volumetric flow rate remains constant even when reaction occurs. The equations are Eq. (8.5) for each species. 

Step 1 The MATLAB program requires you to write a function that defines the righthand side. The input parameters to the function are the concentrations of all species. 

Step 2 You test this m-file by calling it with specific values for yj,j = 1, 2, 3 to ensure that it is correct. Using y(j) for yj, issue the following commands  

Step 3 Next, write a code that serves as the driver. This code must (1) set any constants (here they are just put into the function ratel for simplicity), (2) set the initial conditions and total reactor length, and (3) call the ode solver. 

Numerous reactions are performed by feeding the reactants continuously to cylindrical tubes, either empty or packed with catalyst, with a length which is 10 to 1000 times larger than the diameter. The mixture of unconverted reactants and reaction products is continuously withdrawn at the reactor exit. Hence, constant concentration profiles of reactants and products, as well as a temperature profile are established between the inlet and the outlet of the tubular reactor, see Fig. 7.1. This requires, in contrast to the batch reactor, the application of the law of conservation of mass over an infinitesimal volume element, dV, of the reactor. In contrast to a batch reactor the existence of a temperature profile does not allow us to consider the mass balances for the reacting components and the energy balance separately. Such a separation can only be performed for isothermal tubular reactors. 

In view of the high length-to-diameter ratio of most tubular reactors the flow through them can in most cases be described adequately as a so-called plug flow. It is assumed that  

The left-hand side of Eqn. (7.22) corresponds to the net flow of A out of the reactor element which, in the steady state, has to equal the net production rate of A, i.e. the right-hand side of (7.22). For a tubular reactor with a fixed catalytic bed, it is more convenient to relate the production rates to the catalyst mass, rather than to the reactor volume. Hence, the right-hand side of Eqn. 7.22 becomes  

Analogous to the batch reactor, a fractional conversion of a reactant A can be defined as  

Equations (7.25) or (7.26) are so-called continuity equations for A. Integration results in  

After combining terms, dividing by m, bringing the constant pressure inside the derivatives, and introducing the definition of the enthalpy hk = ek + pvk = ek + p/pk, the following equation emerges 

The specific enthalpy can be represented in terms of the temperature using the constant-pressure specific heat, 

Substituting these definitions and the species continuity equation, the energy equation for the constant-pressure system becomes 

In this case, of course, the volume V = m/p varies with time. 

Deriving the species-conservation equation begins with the conservation law for a flowing system 

The curves from the preceding graph show a clear optimum with respect to the holding time and parametric in the rate constant. 

The total differential of the concentration is equal to the rate of chemical reaction in this zone over some differential time dt. In Mathematica the total differential of f[x, y] is given as Dt[i[x, y]]  

Chapter 9 Continuous Stirred Tank and the Plug Flow Reactors 

Taking the total differential of the concentration and the rate, we obtain  

We can use the shortcut of dividing through by Dt[t], that is, the derivative of the time dt to obtain the following  

Pressure plays an important role in reversible gas phase reactions such as 

Assuming the ideal gas law to hold, the model for the reactor of uniform cross-section is given by 

The optimal control problem is to find the control function P z) that minimizes y at the reactor end z = L subject to Equation (1.11). The minimum y is the objective functional given by 

For irreversible first-order reactions in a PFR, Krambeck showed that the asymptotic kinetics Ra(C) = Cf whenever the feed contains a finite amount of unconvertible species. This is also true for reversible first-order reactions. The case where the feed may or may not contain unconvertibles was treated by Ho and Aris. ° The general treatment of the PCM starts with the expectation that the long-time behavior of the mixture should be governed by the most refractory part of the feed (as will be seen later, this is not always true). To find what goes on at large t, D k) and c/x) near = 0 can be expanded as follows  

This parameter characterizes the number of refractory reactant types and their concentrations. A small means a refiactory feed. For first-order reactions, h k) Ct4ok = hok near k=Q. Then C hoTif lfas, t implying an asymptotic power law of the form 

Rate of reaction, for first order with respect to A 

Substituting Equation 5.10 into Equation 5.7 at constant pressure 

Change in reaction temperature as a function of the reactor volume dT (-r K-AH 

Second method, the second form of the energy balance is 

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Multiple reactions in parallel producing byproducts. Consider again the system of parallel reactions from Eqs. (2.16) and (2.17). A batch or plug-flow reactor maintains higher average concentrations of feed (Cfeed) than a continuous well-mixed reactor, in which the incoming feed is instantly diluted by the PRODUCT and... 

In general terms, if the reaction to the desired product has a higher order than the byproduct reaction, use a batch or plug-flow reactor. If the reaction to the desired product has a lower order than the byproduct reaction, use a continuous well-mixed reactor. 

Keep both Cpeedi and Cpeed2 high (i.e., use a batch or plug-flow reactor). 

The series byproduct reaction requires a plug-flow reactor. Thus, for the mixed parallel and series system above, if... 

But what is the correct choice a byproduct reaction calls for a continuous well-mixed reactor. On the other hand, the byproduct series reaction calls for a plug-flow reactor. It would seem that, given this situation, some level of mixing between a plug-flow and a continuous well-mixed reactor will give the best... 

Polymerization reactions. Polymers are characterized by the distribution of molecular w eight about the mean as well as by the mean itself. The breadth of this distribution depends on whether a batch or plug-flow reactor is used on the one hand or a continuous well-mixed reactor on the other. The breadth has an important influence on the mechanical and other properties of the polymer, and this is an important factor in the choice of reactor. 

Another possibility to improve selectivity is to reduce the concentration of monoethanolamine in the reactor by using more than one reactor with intermediate separation of the monoethanolamine. Considering the boiling points of the components given in Table 2.3, then separation by distillation is apparently possible. Unfortunately, repeated distillation operations are likely to be very expensive. Also, there is a market to sell both di- and triethanolamine, even though their value is lower than that of monoethanolamine. Thus, in this case, repeated reaction and separation are probably not justified, and the choice is a single plug-flow reactor. 

Plug-flow reactors have a decreasing concentration gradient from inlet to outlet, which means that toxic compounds in the feed remain undiluted during their passage along the reactor, and this may inhibit or kill many of the microorganisms within the... 

The first distinction to be drawn, as far as heat transfer is concerned, is between the plug-flow and continuous well-mixed reactor. In the plug-flow reactor shown in Fig. 13.1, the heat transfer can take place over a range of temperatures. The shape of the profile depends on... 

Figure 13.1 The heat transfer characteristics of plug-flow reactors.

Eigure 2 shows that even materials which are rather resistant to oxidation ( 2/ 1 0.1) are consumed to a noticeable degree at high conversions. Also the use of plug-flow or batch reactors can offer a measurable improvement in efficiencies in comparison with back-mixed reactors. Intermediates that cooxidize about as readily as the feed hydrocarbon (eg, ketones with similar stmcture) can be produced in perhaps reasonable efficiencies but, except at very low conversions, are subject to considerable loss through oxidation. They may be suitable coproducts if they are also precursors to more oxidation-resistant desirable materials. Intermediates which oxidize relatively rapidly (/ 2 / i — 3-50 eg, alcohols and aldehydes) are difficult to produce in appreciable amounts, even in batch or plug-flow reactors. Indeed, for = 50, to isolate 90% or more of the intermediate made, the conversion must... 

Du Pont uses a Hquid-phase hydrogenation process that employs a palladium —platinum-on-carbon catalyst. The process uses a plug-flow reactor that achieves essentially quantitative yields, and the product exiting the reactor is virtually free of nitroben2ene. 

Continuous-Flow Stirred-Tank Reactor. In a continuous-flow stirred-tank reactor (CSTR), reactants and products are continuously added and withdrawn. In practice, mechanical or hydrauHc agitation is required to achieve uniform composition and temperature, a choice strongly influenced by process considerations, ie, multiple specialty product requirements and mechanical seal pressure limitations. The CSTR is the idealized opposite of the weU-stirred batch and tubular plug-flow reactors. Analysis of selected combinations of these reactor types can be useful in quantitatively evaluating more complex gas-, Hquid-, and soHd-flow behaviors. 

Fig. 8. Combined flow reactor models (a) parallel flow reactors with longitudinal diffusion (diffusivities can differ), (b) internal recycle—cross-flow reactor (the recycle can be in either direction), comprising two countercurrent plug-flow reactors with intercormecting distributed flows, (c) plug-flow and weU-mixed reactors in series, and (d) 2ero-interniixing model, in which plug-flow reactors are parallel and a distribution of residence times dupHcates that...

There are essentially three types of coal gasifiers moving-bed or countercurrent reactors fluidized-bed or back-mixed reactors and entrained-flow or plug-flow reactors. The three types are shown schematically in Eigure 2. 

Example 5 Percent Approach to Equilibrium For a reversible reaction with rate equation r = L[A — (1 — A)Vl6], the size function kV,./V of a plug flow reactor will be found in terms of percent approach to equilibrium ... 

Often, complete mixing cannot be approached for economic reasons. Inactive or dead zones, bypassing, and limitations of energy input are common causes. Packed beds are usually predominantly used in plug flow reactors, but they may also have small mixing zones... 

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. 

Material and energy balances of a plug flow reactor are summarized in Table 7-7. 

TABLE 7-7 Material and Energy Balances of a Plug Flow Reactor (PFR)... 

Plug flow reactors with recycle exhibit some of the characteristics of CSTRs, including the possibility of multiple steady states. This topic is explored by Penmutter Stah dity of (%emical Reactors, Prentice-Hall, 1972). 

A reversible reaction A B is conducted in a plug flow reactor. The rate equation is... 

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