In this case the results from a batch process also hold for the continuous system, using the residence time t=z/u as a process time (or mean residence time L/u at the exit, for the yield calculation of the whole reactor). Plug flow is the best flow regime, not only from the point of view of simplicity, but above all for excellent performance, giving a high yield of reaction and a uniform composition of products. It is a difficult task for a process designer to approach this ideal flow regime by special modifications of continuous apparatuses (rectifying baffles etc.), and the assumption of plug flow is usually acceptable only for turbulent pipe flows. In the case that the residence times of particles flowing through an apparatus are different (departure from plug flow is characterised by the

see Fig. 8.14.

The reactor yield is calculated again according to the "batch procedure": the inlet stream is divided into several small streams which can be considered for the plug flow regions (in Fig.8.14 it was only 3 streams, with the residence times t

(8.60)

Terms are the ratios of mass flowrates of streams with residence time ti over the total flowrate. If we select the times t

(8.61)

This means that we can calculate the reaction progress (and therefore the yield) of a continuous reactor knowing the reaction progress in a batch reactor z(t) and the residence time distribution E(t). The distribution E(t) depends on the flowfield inside the apparatus and can be computed or measured; do you remember the example describing the application of radioisotopes for evaluating residence times in a basin for the waste water treatment process (Chapter 4, p.13)? The curve plotted by the radiation counter at the outlet of the basin was the residence time distribution E(t).

This model of fully segregated flow is accurate only for homogeneous reactions of the first overall order, but can also be used for heterogeneous reactions (solid particle - gas) with reasonable accuracy, Thın (1998). Reactions of a higher order than one (r>1) taking place inside the apparatus require a more precise description of the flow structure. The idea of isolated batch microreactors must be abandoned. How can I explain this qualitative difference in a simple way?

- Calculation of the reactor yield based on the assumption of segregated flow (macromixing) gives the same result as a calculation assuming maximum mixedness (micromixing at the molecular level) for a first order reaction. This conclusion does not hold for reactions of an order other than one.
- Chemical reactions of the first order are typically monomolecular, i.e., collisions of reactant A with molecules of the other species have no effect on the reaction rate. Therefore the penetration of "foreign" molecules into imaginary microreactors from surrounding streams are of little importance. This is not so for bimolecular reactions, which depend on the probabilities of collisions between molecules of different species; different concentrations of reactants in neighbouring streams and molecular mixing between these streams have a substantial influence on the reaction rate.

Remark: Maybe the figure is slightly misleading, because the particles drawn in Fig.8.14 are changing their colours continually; in the simple world of inorganic chemistry this is not usually so: a molecule of CO2 either exists or does not, particles should be either white or black, nothing in between. Only in this case it is possible to match the distribution of particles inside the CSTR with the distribution in the batch reactor at a particular time t and only then (at time t) will the rate of chemical reaction be the same in the batch and in the continuous reactor. The gradually changing state (colour) of particles is typical for bio-substances, composed of continually growing or decaying macromolecules; this is an unpleasant complication, because it is not possible to find an exact match between the distributions of reactants and products in a CSTR and a batch reactor. Therefore the data from a batch experiment or from rate equations like (8.24) should not be used for calculating continuous systems (at least theoretically). Fortunately, the situation is not as bad as it seems, and even the changes of bio-substances can be described in a classical way.

As an example of a calculation in a CSTR at steady state we shall suppose that only species A reacts, and that the reaction is of the r-th order. We shall write equations for the concentrations of species A in a batch reactor and in a CSTR:

(8.62,63)

where V is the volume of the CSTR and is the volumetric flowrate [m

(8.64,65)

Which reactor is better? If we select as a criterion of optimal design the maximum performance for a given reactor size (internal volume V), it is sufficient to compare (8.64) and (8.65) at a mean residence time , because at this time the volume of substance processed in the CSTR is V, the volume of the reactor. Relative decreases of concentrations [A]/[A]

So, which is less? You have two ways to get an answer: work or think; work involves using a calculator or a spreadsheet, think involves using your brain, e.g.: exp(kt)=1+kt+(kt)2/ 2+... (Taylor expansion) > 1+kt . Or simply: for kt->inf and kt=0 both expressions are equal, and for kt=1 it holds that e>2. Therefore the left side ( exp(-kt) ) is smaller than the right hand side (1/(1+kt) ) at any time t, and the concentration of species A is lower (and the yield higher) in the batch reactor. This is an important conclusion:

- a single CSTR,
- a series of CSTR, and
- a plug flow reactor

Note that there are two modifications of ideally mixed reactor CSTR,

Remark: Residence time distribution E(t) for a single CSTR is an exponential function (see e.g., Levenspiel (1956)), the time course of concentration of A in a batch reactor for r=1 is also an exponential function, see Eq. (8.64) and thus integral (8.61) can be integrated analytically

(8.66)

This is exactly the same result, obtained under the assumption of maximum mixedness, Eq.(8.65). This only supports the previous conclusion that the first order reactions (usually monomolecular reactions) give the same yield regardless of the segregated flow or maximum mixedness regime. Therefore it does not matter which method we select for evaluating CSTR series, assuming a first order reaction (micromixing is easier). As the volume of each CSTR is V/N (N-is number of vessels, V is reactor volume) the material balance of [A] species is (see 8.63)

(8.67)

and the concentration at the outlet of the last vessel [A]

(8.68)

If we increase N to infinity (dividing the actual reactor into an infinite number of infinitesimally small CSTRs, and preserving volume V) we obtain

(8.69)

and this is the conversion (8.64) of the plug flow reactor!

CSTR and plug flow reactors are basic elements of a kit for the construction of complex models. These models are suitable for flow descriptions in real reactors, and a series of CSTRs is such an example. Other examples are parallel series of CSTRs (e.g., a description of the parallel multiphase flows), models of short cut flows by plug flow units, dead zones or zones with poor interchange of mass with the mainstream, etc.

The computational procedure of a reactor described by these models remains, the only difference being in the more involved calculations of the residence time distribution E(t), which is needed in Eq. (8.61). An important class of combined models are models of recirculation. The simplest arrangement is in Fig.8.17.

The recirculation ratio f Î(0,1) is the ratio of flowrate in the recycle loop with respect to the flowrate through the apparatus. The interesting thing is: when increasing the recirculation ratio up to 1, the system tends to behave like a CSTR, like an ideally mixed vessel, regardless of the basic unit used (plug flow in Fig.8.17). This means that recirculation always worsens the performance (yield) of a system (there is nothing so bad as a CSTR). This conclusion is rather general:

Recirculation could be introduced in an university, too: e.g., 90% (f=0.9) of randomly selected students will be returned after graduation to the first semester. Then you will be sitting in a classroom with tired colleagues of forty, fifty, or sixty years of age, with beginners and with others who are repeating this course for the tenth time (these differences of age are characterised by the residence time distribution, E(40) is simply the number of your 40 year old classmates). What a ridiculous picture! Of course, there will also be a positive effect: The number of students will be increased, and also the university's income. Teaching staff will be instantaneously trained. There are always good specific reasons for recirculation: increasing f increases the flowrate through apparatuses, and thus some processes can be intensified - e.g. heat and mass transfer, cleaning of internal surfaces, etc. Recirculation of flue gases decreases the temperature of combustion, thus reducing the production of NOx. Recirculation of drying air in dryers saves energy and is required in cases when the dried material does not like dry air. Etc., etc. Be that as it may, have in mind that the necessity of recirculation indicates that there is something wrong in the basic system, and what is wrong can be improved (except bad character). Sometimes the outlet stream is separated, e.g., according to its composition and only the part which requires further processing is recirculated (this is something quite different, and selective recirculation eliminates most negative effects). This technique is used, for example, in some bio-reactors, where enzymes (catalysts) are separated by ultrafiltration and returned to the inlet of the reactor to promote bio-reaction - cultivation of biomass.

@TEC: 3. 3.2003 Change language to