1. Governing equation for concentration as a function of x and t for a pulse input for an “open” reactor.

2. D d or Pe is usually unknown so we need to determine these parameters by injecting a conservative substance and measure its concentration as a function of time and/or distance in the reactor. Two choices: -synoptic sampling at some time sample at different x values. -single point sampling (fixed x) at different t values

3. For synoptic sampling we get a symmetrical Guassian distribution for concentration no matter what D d or Pe exists in the reactor. This is shown in the following slide.

4. Pe = 333 Pe = 111 Pe = 56 Synoptic sampling

5. A Gaussian distribution has the form:

6. If we take and normalize it by multiplying it by A/M and let We get:

7. This is exactly the Gaussian distribution function so all we have to do is calculate the standard deviation to calculate D d as long as the concentration versus x is symmetrical – which it will be if the sampling is synoptic and the system is “open” The system can be considered “open “ if the concentration curve exhibits very low numbers at low and high x values. This is because the sampling is taking place at in a region of the reactor far from the boundaries of the reactor or, in fact, the boundary conditions are truly “open” in the sense that they exhibit the same D d as the reactor.

8. Now if the reactor is being sampled at a single point at multiple times the response curve ( C versus t) may be symmetrical if D d is low enough. In this case we can still use computation of the standard deviation. In this case we use:

9. As D d increases the response curve become skewed and therefore non-Gaussian and we need another method to determine D d. The following plot shows how the skewness develops with increasing D d (decreasing Pe).

10. t peak Pe = 67 Pe = 6.7 Pe = 3.3 time

11. Note that there is an increase in skewness so the Gaussian distribution is no longer appropriate. However, computation of  and are as before: But now you can’t use

12. Beside the increase in skewness also notice there is a shift in the position (x value) of the peak concentration, t peak as a function of Pe (D d ). It is possible to find the relationship between Pe or D d and this shift in t peak. by differentiating C(t) and setting it equal to zero to solve for x at the peak C. This can be done for a variety of Pe values to find:

13. This equation plots as: t peak /  Finding t peak can done graphically by plotting C(t). t peak occurs at the max C.

14. Given the possible skewness of the C vs. t profile for single point sampling, the mean predicted retention time,, will, in general, be greater than the time at which the peak concentration passes the monitoring point,  = V/Q (i.e., reactor volume divided by flow rate, or reactor length divided by advective velocity).

15. If D d (or Pe) is known is predicted by :

16. Integration of this expression is difficult but a numerical technique can be used to show the effect of Pe on the ratio of to .

17. This graph can be used determine Pe after computing and dividing by the theoretical retention time 

18. Finally, Pe can be determined for single point sampling at high dispersion using statistical methods even if the Gaussian conditions (symmetry) don’t exist. The standard deviation and t bar calculations still apply:

19. After some rather difficult math using Gamma functions it can be shown that for an open system: And for a closed system using numerical methods: