Exponential distribution of residence times

The other case assumes that the fluid particles are well mixed. Specifically, assume that they have an exponential distribution of residence times... 

A CSTR has an exponential distribution of residence times. The corresponding differential distribution can be found from Equation (15.7) ... 

The molecules in the system are carried along by the balls and will also have an exponential distribution of residence time, but they are far from perfectly mixed. Molecules that entered together stay together, and the only time they mix with other molecules is at the reactor outlet. The composition within each ball evolves with time spent in the system as though the ball was a small batch reactor. The exit concentration within a ball is the same as that in a batch reactor after reaction time tf,. 

We have just described a completely segregated stirred tank reactor. It is one of the ideal flow reactors discussed in Section 1.4. It has an exponential distribution of residence times but a reaction environment that is very different from that within a perfectly mixed stirred tank. 

The exponential distribution of residence times defines a well-stirred reactor. 

This exponential decay is typical of first-order processes as shown previously. Thus, there is an exponential distribution of residence times some molecules will spend little time in the reactor while others will stay very long. The mean residence time is ... 

It is reasonable to assume that the solids flow is fully backmixed with an exponential distribution of residence times. Based on this assumption and by writing an expression for the average activity of the leaving catalyst stream (which contains particles of all ages with their corresponding activities), the following equations are derived ... 

The continuous phase is well mixed but there is a dispersed phase. The particles in the dispersed phase behave as PFRs. They are in contact with the continuous phase but are isolated from each other and have an exponential distribution of residence times. This case is treated in Examples 11.17 and 15.12. 

Equations 11.54 and 11.55 apply to any distribution of particle residence times provided the linear consumption rate is constant. They do not require that the fluid phase is perfectly mixed, only that the consumption rate is strictly controlled by the surface reaction. For the special case of an exponential distribution of residence times per Equation 11.53, some calculus gives... 

A perfect mixer has an exponential distribution of residence times W t) = exp(—r/7). Can any other continuous flow system have this distribution Perhaps, surprisingly, the answer to this question is a definite yes. To construct an example, suppose the feed to a reactor is encapsulated. The size of the capsules is not critical. They must be large enough to contain many molecules but must remain small compared to the dimensions of the reactor. Imagine them as small ping-pong balls as in Figure 15.1 la. 

In a CSTR, each element of monomer feed has an equal chance of being withdrawn from the reactor at any instant regardless of the time it has been in the reactor. Therefore, in a CSTR, unlike in batch and tubular reactors, the residence time is variable. The contents of a well-stirred tank reactor show an exponential distribution of residence times of the type shown in Equation 10.15. 

The system is more complex in that for each reactant type, there is an exponential distribution of residence times among all the molecules of that reactant. No R C) can be found and RJiC) admits three possibilities. "" The case y < (refractory feeds) is similar to the PFR in that C at large t, which is dominated by the most refractory reactants. Specifically, RJiC) C. For 7 > 1, =1 and C Ht at large t (similar to that of its constituents)... 

For a segregated stirred tank, aout/ain = 0.299. This result is found by solving eq. (1-7) subject to an exponential distribution of residence times. The measured result, aout/ain = 0.38, is worse than any of the ideal reactors There are several possibilities ... 

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