So far, we have assumed that reactions take place in closed systems, that is to say, without mass transfer to and from the surroundings. While this may be valid for many cases, there are also examples where this is not the case. It is especially important for processing when foods are treated in continuous processes such as in heat exchangers. The topic is vast and we will only show the basic principles as to how the kinetics change when mass transfer comes into play. The reader who is interested in more details is referred to some literature references given at the end of this chapter.
1.0E + 08 1.0E + 09 1.9E + 09 2.8E + 09 3.7E + 09 4.6E + 09 5.5E + 09 6.4E + 09 7.3E + 09
8.00 9.00 10.00 11.00 12.00
log kchem
koverall
kdif = 7⫻ 109
FIGURE 4.32 Overall reaction rate constant as a function of chemical reaction rate constant, assuming a diffusion rate constant kdif¼ 7 3 109dm3mol1s1.
A reactor allowing mass transfer was already introduced in Figure 4.1, in which the average residence timet is
t ¼V
w (4:166)
Suppose next that we are able to introduce a certain amount of a component in the inlet of the CSTR as a pulse. Since the system is considered ideally mixed, the concentration is instantaneously equal in the system at t¼ 0. We can consider then how the concentration changes at the outlet of the system, supposing that the component is not subject to a reaction, so rA¼ 0 in Equation 4.1. The following mass balance should hold:
wincin woutcout Vdcreactor
dt ¼ 0 (4:167)
Because the reactor is ideally mixed, the exit stream has the same composition as that in the reactor, i.e., creactor¼ cout, hence:
The boundary conditions are that at t¼ 0 cout¼ c0(because the reactor is considered ideally mixed) and at t> 0: cin¼ 0. This leads to the following expression:
cout¼ c0exp t t
(4:170)
Figure 4.33 shows the change in concentration in this situation. It basically shows how a compound is washed out of a reactor due to mass transfer in the case that there is no further chemical change occurring.
The next question is what happens if the compound is subject to a chemical reaction. We need to know, of course, the kinetics of the reaction in order to calculate the change quantitatively. If we suppose a first-order decay reaction in A, assuming certain values for the parameters of interest, the equation becomes:
The change in the amount of component A is shown in Figure 4.34 for the conditions assumed, as well as the concentration of component B formed out of A. Both component A and B are washed out of the reactor. This result should be compared to the rate of change of A in a closed system as depicted in Figure 4.5, in which component B accumulates. Equation 4.171 shows that the chemical reaction will dominate the disappearance of the compound if k>> 1=t, while the physical removal due to mass transport will dominate if k 1=t. That gives the opportunity to direct the desired change via the flow ratew and the volume of the reactor V.
Now, consider the situation that an amount of A is constantly supplied at the entrance of the CSTR.
Then the concentration of A at the outlet will be building up and eventually reaches the concentration at the inlet if no reaction in the reactor takes place: this is the point where a steady state is reached. The equation describing this situation is
FIGURE 4.33 Change in concentration at the outlet of a CSTR when a compound A is added as a pulse at t ¼ 0 and when the compound is not subject to reaction. V¼ 10 dm3,w ¼ 2 dm3s1, [A]0¼ 1 mol dm3.
FIGURE 4.34 Schematic representation of a CSTR into which an amount of component A is introduced and for which afirst-order decomposition reaction is assumed. V ¼ 10 dm3,w ¼ 2 dm3s1, [A]0¼ 1 mol dm3, k¼ 0.1 s1.
A graphical representation is given in Figure 4.35. When a compound is also subject to a reaction in the reactor, the concentration changes because of mass transfer and the reaction. For a first-order decay reaction in a CSTR, the situation is described by the following equation:
cout¼ cin cinexp t t
exp k t
(4:173)
Assuming certain values for the parameters, this equation is depicted in Figure 4.36.
The result displayed in Figure 4.36 shows that eventually a steady-state situation is reached (which is not the same as equilibrium!). This situation is distinctly different from that in Figure 4.33 in which the concentration of A and B becomes zero eventually.
A different type of open system is a plugflow reactor. It is then assumed that material flows through, say, a pipe such that the residence time is the same for each element in the material and it is assumed that there is no mixing, i.e., no axial dispersion (see Figure 4.37).
If a reaction takes place inside the reactor, a concentration gradient will develop in the direction of the flow. We can set up a mass balance over an infinite small slice perpendicular to the direction of the flow having a volume dV. At the beginning of the slice the concentration of a component A is cA, at the end it will have changed to cAþ dcA. At steady state, when there is no change in the amount of component A, the following equation should hold (see Equation 4.1):
0¼ wcA w(cAþ dcA)þ rAdV (4:174)
This equation can be transformed to:
dcA
rA ¼dV
w (4:175)
0.1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 5 10 15 20
cout (mol dm−3)
t (s)
FIGURE 4.35 Pre-steady-state concentration change at the outlet of a CSTR reactor into which a continuous input of A is given from t¼ 0 onwards, without a chemical reaction taking place. V ¼ 10 dm3,w ¼ 2 dm3s1, [A]0¼ 1 mol dm3.
Integration over the whole reactor leads to:
ð
cA,out
cA,in
dcA
rA
¼ ð
V
0
dV
w (4:176)
Or alternatively:
t ¼V w¼
ð
cA,out
cA,in
dcA
rA (4:177)
[A]
[B]
0.1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 5 10 15 20
cout (mol dm−3)
t (s)
FIGURE 4.36 Pre-steady-state concentration changes at the outlet of a CSTR reactor into which a continuous input of A is given from t¼ 0 onwards and in which a first-order reaction A ! B takes place. V ¼ 10 dm3,w ¼ 2 dm3s1, [A]0¼ 1 mol dm3, k¼ 0.1 s1.
cA cA+dcA
x=0
t=0 x=L
t=τ
f , cA,in f , cA,out
FIGURE 4.37 Schematic drawing of a plug flow reactor.
The kinetics in a plugflow reactor are actually the same as in a batch reactor, except that the reaction time isfixed at the residence time t. This becomes apparent if we integrate the expression in Equation 4.175 for, e.g., a zero-order reaction, for which rA¼ k:
t ¼
The same exercise for afirst-order reaction (rA¼ kcA) yields:
t ¼
These equations are thus exactly the same as for a batch reactor, given earlier in this chapter. Figure 4.38 shows the change in a compound subject to afirst-order reaction in a plug flow reactor. If one knows the kinetics of a reaction and the initial concentration, one can calculate the outcoming concentration straightforwardly using the equations given in this chapter. Of course, one can vary the residence time by changing the volume V and theflow rate w. The residence time for a plug flow reactor is also given by Equation 4.166. Although this is also an idealized situation, it can be approximated quite closely if the flow inside the reactor is turbulent. In real life, however, there is a distribution of residence times that,
0
FIGURE 4.38 Change in the concentration of a compound A in a plug flow reactor subject to a first-order reaction.
incidentally, can be determined experimentally by introducing a tracer at the entrance of the reactor and following its concentration at the exit.
It is instructive to study the performance of the various reactor types after reaching steady-state conditions. The general kinetic equation for a CSTR at stationary conditions is
tCSTR ¼cA,out cA,in
rA (4:180)
The general kinetic equation for plugflow is
tPF¼ ð
cA,out
cA,in
dcA
rA (4:181)
This last equation is also valid for a batch reactor, as shown above.
For a zero-order reaction (rA¼ k) in a CSTR, it follows that under the condition that tCSTR< cA,in=k:
cA,out
cA,in ¼ 1 ktCSTR
cA,in (4:182)
(for the condition thattCSTR cA,in=k, cA,out=cA,in¼ 0). For a first-order reaction (r ¼ kcA) in a CSTR, it follows from Equation 4.177 that:
cA,out
cA,in
¼ 1
1þ ktCSTR
(4:183)
For a zero-order reaction in a plugflow reactor the equation is cA,out
cA,in ¼ 1 ktPF
cA,in (4:184)
Note that this equation is similar to the one for CSTR. And for afirst-order reaction in a plug flow reactor it follows:
cA,out
cA,in ¼ exp (ktPF) (4:185)
Since Equations 4.182 and 4.184 are the same it follows that the type of reactor does not matter for a zero-order reaction. This is not so for afirst-order reaction, as shown in Figure 4.39. A plug flow reactor gives a higher degree of conversion than a CSTR for equivalent residence timest, provided that we deal withfirst-order kinetics. One can of course exploit this phenomenon. It makes immediately clear that if one wants to reach a high conversion rate, that a plugflow reactor is much more efficient than a CSTR.
This is especially relevant for the killing of microorganisms (assuming for the moment that this can be described by afirst-order reaction), which would be very inefficient in a CSTR. Furthermore, one can design a reactor and its operating conditions (thereby effectively setting the residence timet) in such a
way that the desired effect is maximum. It is perhaps worth mentioning that a series of CSTRs is going to approach the performance of a plugflow reactor (for the same equivalent residence time).
There is much more to be said about kinetics in open systems. Here we only showed some basic principles to show the differences. The reader who is interested in more details is referred to some selected literature references at the end of this chapter. It is especially important to be aware of the phenomenon of residence time distribution. As a final remark we would like to point out that a very important difference between open and closed systems is that in closed systems thermodynamic equilibrium is the time-invariant condition, while for open, continuous systems the steady state is the time-invariant condition, and the reader should appreciate the differences between these two conditions as they are very relevant for foods and food processing.
4.9 Concluding Remarks
In this chapter kinetic models have been introduced and a connection has been made with thermo-dynamics. The treatment has been kept very general, but at the same time limited to cases that are of interest for foods. Therefore, we have illustrated the material as much as possible with relevant examples from food science. It has been discussed how rates can be expressed and it has been shown when reaction rates may become diffusion controlled. For reactions in solutions this does not seem to be the case very often, except perhaps for radical and photochemical reactions. Furthermore, the basis has been shown how to tackle kinetics in the case of transport of material in addition to a chemical reaction. This chapter forms the basis for further discussion of kinetic models throughout the book. In the next chapter, we will continue the discussion by focusing on the effects of temperature and pressure.
0.2
0 0.4 0.6 0.8 1
0 5 10 15 20 25 30
t (s) cA,out/cA,in
Plug flow reactor CSTR
FIGURE 4.39 Comparison of a first-order reaction having the same rate constant in a CSTR and a plug flow reactor.