Mass Balance, Kinetics
& Reactors
Dr. Martin T. Auer
MTU Department of
Civil & Environmental Engineering
Environmental Response
…the question is not will a system will respond,
but rather when and to what extent.
(Cooke et al. 1999)
and, as engineers, we might add ‘at what cost’?
Lake and River Management
…the environmental engineering equivalent
of building a bridge to nowhere.
(Thomann and Mueller 1987, p. ix)
http://www.zen39641.zen.co.uk/ps/
Drinking Water Treatment
…the environmental engineering equivalent of
building a bridge that falls down.
(Thomann and Mueller 1987, p. ix)
http://www.jansenkiener.com/Bridge%20Engineering.htm
Reactor Analogs
Plug Flow
Reactor
Completely-Mixed
Flow Reactor
CMF Reactor
Control Volume
Zero Order Kinetics
Oxygen in Dollar Bay
Ct = -k∙t + C0
Dollar Bay
Dissolved Oxygen (mg/L)
12
10
8
6
4
2
0
0
30
60
90 120 150 180 210 240 270 300 330 360
Day of Year
Dollar Bay
Dissolved Oxygen (mg/L)
10
y = -0.1285x + 23.752
2
8
R = 0.9759
6
4
2
Zero Order
k = 0.13 mg∙L-1∙d-1
0
0
30
60
90 120 150 180 210 240 270 300 330 360
Day of Year
First Order Kinetics
Radioisotope Decay
lnCt = -k∙t + lnC0
Pb-210
Radioisotope Concentration
1.0
0.8
0.6
0.4
0.2
0.0
0
20
10
30
40
60
50
70
80
90
100
Time (yr)
Pb-210
Radioisotope Concentration
0
k = 0.036 yr-1
-1
t0.5 = 19.25 yr
-2
y = -0.036x + 6E-16
-3
2
R =1
-4
0
20
40
60
Time (yr)
80
100
Temperature and Kinetics
Rate Coefficient (d -1)
Theta Function
4
Q
3
1.08
2
1.04
1
1.00
0
0
5
10
15
20
25
30
35
Temperature (°C)
kT  k20  Q
(T  20)
40
Temperature and Kinetics
WWTP Nitrification
Effluent Ammonia Load (MT∙d-1)
OrgN  NH 3  NO2  NO3
J F M A M J J A S O N D
CMF Reactor
with first order decay
dC
V
 Q  Cin  Q  C  V  k  C
dt
time-variable or steady state application
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
Response Time
dC
V
 Q  Cin  Q  C  V  k  C
dt
assume no input,
dC
V
  Q C  V  k C
dt
divide by V, collect terms and integrate,
Ct  C0  e
1 
 + k t
 
assume SS when 95% completed, i.e.
0.05  C0  C0  e
1 
 + k t
 
Response Time
t95%
 ln 0.05

1
+k
t95% 

Wastewater Treatment
Grit removal, 0.5 hr
1°, 2° settling, 1-2 hr
Activated sludge, 4-8 hr
Anaerobic digestion, 15-30 d
Drinking Water Treatment
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)
Typical ‘fast’ k, 30 yr-1
Typical ‘slow’ k, 0.03 yr-1
3
1

+k
Time-Variable Response
100
FG IJ
C t  C ss1  e H K
80
1
  + k t
60
40
20
0
0
5
10
15
100
80
60
C t  C ss2
40
20
FG IJ I
F
H K
G
1 e
J
H
K
1
+ k t

0
0
5
10
15
100
80
60
F
GH
FG1 + kIJ t
FG IJ
H K
C t  C ss1  e H K + C ss2  1  e 
1
  + k t
40
20
I
JK
0
0
5
10
15
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
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
Train of Batch Reactors
Distance or TimeTime (yr)
PF-CMF Comparison: Reactor Efficiency
PF-CMF Comparison: Sensitivity to Spikes
PF Reactor – Application to Rivers
dD
 k1 ·L  k2 ·D
dt
k1 ·L0
 k1 t
 k2 t
 k2 t
Dt 
  e  e  + D0  e
k2  k1
x
x

k


k


k1·L0
1
2
U
Dx 
e
e U
k2  k1 
tcrit
x
 k2 

U
+
D

e

0

 k2   D0  (k2  k1 ) 
1

 ln    1 

k2  k1
k1  L0
 k1  

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.