Mass Balance, Kinetics
& Reactors
Dr. Martin T. Auer
MTU Department of
Civil & Environmental Engineering
Reactor Analogs – Natural Systems
Plug Flow
Reactor
Completely-Mixed
Flow Reactor
Fox River
Wisconsin
Mille Lacs Lake
Minnesota
Reactor Analogs – Engineered Systems
Plug Flow
Reactor
Completely-Mixed
Flow Reactor
Resin-Based
Water Softener
Wastewater
Primary Clarifier
Soaking Rain
Dream Car
CMF Reactor
Control Volume
CMF Reactor
with first order decay
dC
V
 Q  Cin  Q  C  V  k  C
dt
Chloride in 9 Mile Creek
For many years, Allied Chemical and its ancestors
produced soda ash … a chemical used to soften water
and in the manufacture of glass, soap, and paper. The
raw materials were two locally abundant minerals:
CaCO 3 + NaCl  Na 2 CO 3 + CaCl 2
and the products were soda ash (Na2CO3) and calcium
chloride (CaCl2) waste. The wastes were deposited in
2000 acres of lagoons along the banks of 9 Mile Creek.
The waste continually leaks from the lagoons into the
creek, making the water highly ‘salty’.
Chloride in 9 Mile Creek
Cmb 
Cup  Qup + Cin  Qin
Qup + Qin
BATCH Reactor
with first order decay
dC
V
 Q  Cin  Q  C  V  k  C
dt
dC
  k C
dt
BATCH Reactor
with first order decay
Concentration
dC
  k C
dt
Ct  C0  e
Distance or TimeTime (yr)
 k t
Batch Reactor in Pipe
Batch Reactor in Pipe
Batch Reactor in Pipe
Batch Reactor in Pipe
Batch Reactor in Pipe
Batch Reactor in Pipe
Batch Reactor in Pipe
Batch Reactor in Pipe
Batch Reactor in Pipe
Batch Reactor in Pipe
Batch Reactor in Pipe
Batch Reactor in Pipe
Concentration
PFR = Train of Batch Reactors
Distance or TimeTime (yr)
To Water Quality
CMF Reactor
dC
V
 Q  Cin  Q  C  V  k  C
dt
and, at steady state
Q
Css  Cin 
Q +V  k
Change in Css
concentration
Css,1
Css,2
time
Time-Variable Response
FG IJ
C t  C ss1  e H K
concentration
1
 t + k t
C t  C ss2
FG IJ I
F
H
K
G

1 e
J
H
K
1
+ k t
t
F
GH
FG1 + kIJ t
F
1+ I
G
J
k
t
H K

C t  C ss1  e Ht K + C ss2  1  e t
time
I
JK
Response Time
t95%
 ln 0.05

1
+k
t95% 
3
1
t
t
+k
Rate Coefficients
‘fast’ k, 30 yr-1
‘slow’ k, 0.03 yr-1
Wastewater Treatment
Drinking Water Treatment
Grit removal, 0.5 hr
1°, 2° settling, 1-2 hr
Activated sludge, 4-8 hr
Anaerobic digestion, 15-30 d
Rapid mix, <1 min
Flocculator, 30 min
Disinfection, 15 min
Natural Systems
Onondaga Lake (0.25 yr)
Lake Ontario (8 yr)
Lake Michigan (136 yr)
Lake Superior (179 yr)
SS CMF
Application to Lakes
dP
V
 W  Q·P  V ·k ·P
dt
where W = Q∙Cin, i.e. the loading
SS CMF
Application to Lakes
dP
V
 W  Q·P  V ·k ·P
dt
v
k
 and
H
V
A
H
dP
V ·  W  Q·P  v·A·P
dt
W
@ SS , P 
Q + v·A
PF-CMF Comparison: Reactor Efficiency
PF-CMF Comparison: Sensitivity to Spikes
Mass Transport
CMF Reactor
PF Reactor
8
8
Lake
Huron
Saginaw
Bay
Advection and Diffusion
advection
alone
diffusion
alone
advection
plus
diffusion
Diffusion
PCBs
PCBs are a family of chemical compounds formed by the
addition of chlorine to biphenyl (C12H10). There are 10
substitution positions where chlorine may be added,
leading to a possible 209 unique chemical compounds
termed congeners.
3
2
2’
3’
4’
4
5
6
6’
5’
ClnH(10-n)
Congeners have been assigned numbers (1209) and
are also classified by the positions occupied by chlorine.
Referencing the substitution positions in the figure above,
three examples are:
Congener 1: 2-Chlorobiphenyl
Congener 101: 2,2’,4,5,5’-Pentachlorobiphenyl
Congener 209: Decachlorobiphenyl
Example 4.14 PCBs in Lake Superior
Dr. Perlinger’s research group
sampling on Lake Superior aboard
the U.S. EPA research vessel
Lake Guardian.
air
water