1. Reactors

2. Mixing Model Conservative and Instantaneous Signal
A signal is a tracer that injected into the influent
to a reactor
Conservative means it does not react
Instantaneous means the tracer is introduced into the
influent instantaneously

3. Ideal reactors Mixed Batch Plug Flow Completely Mixed Flow Reactor
Fully mixed, no inflow, no out flow
Plug Flow
Continuous flow, but no mixing
Completely Mixed Flow Reactor
Continuous flow with perfect mixing

4. Mixed Batch Reactor Conservative Instantaneous Signal Added
Signal added at time zero, mixed instantaneously
This figure is called a C-distribution
A plot of C vs t or C/C0 vs t

5. Plug Flow Reactor
No longitudinal mixing, all elements of water enter and exit in
same order. The time spent in the reactor, the residence time,
is equal for all elements:
t = V/Q
V = volume
Q = flow rate
Another way to think of the residence time is the time to
fill the reactor. This is actually true for any type of reactor.

6. Plug Flow Reactor
C-distribution Curve

7. Plug Flow Reactor F-distribution Curve
F is the fraction of that has left
the reactor at any time t:
F = (A0 –AR)/A0
A0 = the mass of tracer added to the reactor
AR = the mass of tracer remaining in the reactor

8. Completely Mixed Flow Reactor
Abbreviated CMF. Often called Completely Stirred Tank
Reactor (CSTR) or Complete Mix reactor
Rate of
Signal
Accumulated in
out
Generated = - +
Signal introduced instantaneously so the rate in = 0
Also, no generation (conservative tracer)
Rate of
Signal
Accumulated
out = -
C0 = A0/V
Where: A0 = mass of signal in the reactor at t = 0
V = volume of the reactor
C0 = concentration in reactor at t = 0

9. Where: A = mass of tracer in the reactor at any time t
After the tracer has been introduced (instantaneously) the concentration
changes with time because water without tracer keeps entering the reactor and:
C = A/V
Where: A = mass of tracer in the reactor at any time t
V = volume of the reactor
C = concentration at time t
Now:
Rate of
Signal
Accumulated
out = -
Rate of
Signal
Accumulated
= - C Q = -(A/V) Q
So: dA/dt = -(A/V) Q

10. lnA - lnA0 = - (Q/V) t A/A0 = e-(Q/V)t t = V/Q So: A/A0 = e-t / t
dA/dt = -(A/V) Q
dA/A = Q/V dt A0 A t
lnA - lnA0 = - (Q/V) t
A/A0 = e-(Q/V)t
t = V/Q
So: A/A0 = e-t / t
Or: C/C0 = e-t / t

11. C/C0 = e-t / t
When t = t,:
C/C0 = e-1 = 0.368

12. Completely Mixed Flow Reactor
F-distribution Curve

13. CSTR’s in Series The rest of the discussion is section 5.1.4 in your
N reactors in series
Each reactor has a volume of V0
Total volume of the system, V = n x V0
The rest of the discussion is section in your
text is incorrect. The following derivation is the
right way to do it!

14. Mass balance on the first reactor:
(V/n) dC1/dt = Q C0 – Q C1
Or: dC1/dt = (nQ/V) (C0 – C1)
Separate variables and integrate:
C1 = C0 e-n(Q/V)t = C0 e-n t / to where to = V/Q (residence time
of the whole system)

15. Mass balance on the second reactor:
(V/n)dC2/dt = Q C1 - Q C2
Or: dC2/dt + (nQ/V)C2 = (nQ/V) C1
Since: C1 = C0 e-n t / to
dC2/dt + (nQ/V)C2 = (nQ/V) C0 e-n t / to
The solution to this equation is:
C2 = C0n (t / t0) e-n t / to

16. Ci = (C0/(i – 1)!) [n(t/t0)]i-1 e-n(t/to)
By extension:
Ci = (C0/(i – 1)!) [n(t/t0)]i-1 e-n(t/to)
Use this equation to predict concentration coming
from any reactor in the series at any time t
Or: Ci/C0 = (1/(i – 1)!) [n(t/t0)]i-1 e-n(t/to)
Use this equation to plot a C-distribution curve
Note: These equations are slightly different from the
equations in the book. The equations in the book are incorrect.
(Equations 5.2 and 5.3)

17. F-distribution curves for a series of CSTR’s may also be
Developed. The equation is in your text on page It is correct.
The F-distribution for n= infinity is identical to the f-distribution
for a plug flow reactor

19. In the real world there is no such thing as a perfect plug
flow reactor or CSTR
Causes:
Imperfect mixing
Short circuiting
Dead Space
Arbitrary Flow Reactors