1. Reactor Theory: hydraulics
TVM 4145 Vannrenseprosesser / unit processes
Reactor Theory:
hydraulics
Prof. TorOve Leiknes

2. Principle of Mass balance:
System boundary
Reaction
Transport in
Transport out
Accumulation = In - Out + Generation
where: In – Out is net transport in the system
Generation is net production or consumption
Accumulation is net product left over

3. Mass balance principle applied to reactor theory:
Flow-sheet:
Equation form:
Reactor
Qi, Co
Qe, Ce
V, C
Where rc is reaction rate
Reaction rate; rc
where: k is the rate coefficient
C is the concentration in the reactor
n is a constant.

4. Definition of reactor types:
PF:
Plug flow reactor
NO - Stempelstrømning
CMB:
Completely mixed batch reactor
NO - Ideell blanding
CMFR:
Completely mixed flow
NO - Ideell blanding

5. CMB: Assumptions: closed system reactor volume is constant
homogeneous mixing

6. With a first order reaction and reduction of the concentration:
CMB:
With a first order reaction and reduction of the concentration:

7. CMFR: Assumptions: Q, Co Q, C V, C, rc closed system
reactor volume is constant
homogeneous mixing
Q, Co
Q, C
V, C, rc

8. CMFR
develop theoretical basis for a simplified system
consider the system under steady state conditions
assume a simple 1.order reaction for removal of a substance
i.e. the process kinetics is:
Solve the equation with regard to effluent concentration:

9. What is the goal?

10. CMFR in series: Assume steady state and 1.order reaction for rc:
V1, C1, rc1
V2, C2, rc2
V3, C3, rc3
Q Co
Q C1
Q C2
Q C3
Assume steady state and 1.order reaction for rc:
Reactors er equal, ki=k, Vi/Q=th:

11. What happens when V ≠ constant?
Change in volume:
HRT

12. Important / useful equations:
1. Calculation of effluent concentration:
2. Average hydraulic retention time (HRT):
For one reactor:
Total for n reactors:
3. Total retention time necessary
for desired reduction:

13. PF: Assumptions no mixing in the reactor Q
constant velocity through the reactor
retention time = theoretical retention time
no displacement of liquid elements
concentration gradient in along length Q v
Consider an element in the reactor: x
Q, C
Q, C + C A

14. Assume steady-state to analyze the PF reactor:vx x x
x = L
Assume 1.order reaction and steady-state:

15. Comparison of reactor types:
Average retention time, th:
Reaction order
0.order
1.order
2.order
n.order
(n1)
CMFR:
PF:

16. How will the reactors behave?
”Tracer” studies where an inert substance (colour, salt) are used to analyze the hydraulic conditions through the reactor.
Time
Ideal completely mixed
Ideal plug flow C
C (gitt volum) Q
Reactor
Measure C in the outlet

17. Co = concentration distributed in tank volume
t/th
C/Co
1.0
Co = concentration distributed in tank volume
C = concentration measured in outlet
th = theoretical hydraulic retention time (V/Q)
t = measured time
tmean = center of mass, (1.moment)
tmode = average of max concentration
tmdian= time for å measure 50% of tracer in outlet
Analysis of central tendencies:
none: times coincide
short-circuits: (tmean -tmedian)/ tmean or (tmean -tmedian)/ tmean
dead-zones: th- tmean ; in practice tmean < th
Short circuits given by:
Distribution given by:
t10/t90 or expressed by the variance (s2)

18. Retention time (th) as a function of recycling:Q rQ
Q + rQ V
recycling
fraction
Average retention time is calculated by considering an element(Dq):
average retention time:

19. A fraction of the fraction is recycled:
Total retention time for the element:
This is a geometric series:

20. PFD: Dispersion model Fick’s lov: Where:vx x L
Dispersion In
Bulk flow In
Bulk flow Out
Dispersion Out
 reaksjon
Fick’s lov:
Where:
D is the dispersion coefficient
u is average water velocity in x

21.  - dispersion large, CMFR
The equations is made dimensionless by:
z = x/L og  = tu/L
where
Is the dispersion number If
0 - dispersion negligible, PF
 - dispersion large, CMFR

22. Dispersion number is determined by ”Tracer studies”
use an inert / stable substance which is easy to measure, but with no reaction
introduce substance as a pulse with known concentration
measure the outlet concentration of the substance over time
plot concentration against time, commonly normalized (C)
graphs and plots are available for different reaction orders
(i.e. Levenspiel & Bischoff)

23. Interpretation of dispersion number:
Based on tracer study

24. Example:

25. Applied in biological systems:
Mass balance principle applies
Reaction kinetics for biological systems must be used
Activated sludge processes:
Reactor
Sedimentation
Q, So, Xo
Q(1+r)
Qe, Se, Xe
Inflow
Outflow
V, S, X
Return
sludge
Excess
sludge
Qr, Xr
Qw, Xw =Xr

26. Definitions: Mass balance on cell mass: (steady state)
Hydraulic retention time:
Average cell mass retention time:
Mass balance on cell mass: (steady state)

27. Solutions/adjustments of equations:
From reaction kinetics:
Change in cell mass:
Change in substrate concentration: