Transcript Document

CE5504 Surface Water Quality Modeling
dc
V

dt
Lab 1. Modeling 101
CE5504 – Surface Water Quality Modeling
Surface Water Quality
Engineering
discovery
fate and
transport
integration
application
decision
support
Reactor Analogs
Plug Flow
Reactor
(rivers)
Fox River, Wisconsin
Completely-Mixed
Flow Reactor
(lakes, bays, nearshore)
Mille Lacs Lake, Minnesota
CE5504 – Surface Water Quality Modeling
X t  X 0  e( t )
The Reactor Analog for Lakes
Completely Mixed Flow Reactor
CMFR
CE5504 – Surface Water Quality Modeling
X t  X 0  e( t )
The Mass Balance
CMF Reactor Characteristics
• completely mixed (Cout = C)
• constant volume (Qin = Qout)
CE5504 – Surface Water Quality Modeling
X t  X 0  e( t )
The Mass Balance
Control Volume
• the system about which
the mass balance will be
computed.
CE5504 – Surface Water Quality Modeling
X t  X 0  e( t )
The Mass Balance
Kinetics
• growth
• decay
RXN
RXN
CE5504 – Surface Water Quality Modeling
X t  X 0  e( t )
The Mass Balance
12
Dissolved Oxygen (mg/L)
Kinetics
Dollar Bay
• 0 order reactions
10
8
6
4
2
0
0
30
60
90 120 150 180 210 240 270 300 330 360
Day of Year
 rate is not a function
of concentration
Dollar Bay
Ct = C0 - k∙t
k, mg·L-1·d-1
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
CE5504 – Surface Water Quality Modeling
X t  X 0  e( t )
Pb-210
Kinetics
1.0
Radioisotope Concentration
The Mass Balance
0.8
0.6
0.4
0.2
0.0
0
order reactions
 rate is a function of
concentration
lnCt = -k∙t + lnC0 or
Ct = C0·e-k·t
10
20
30
40
50
60
70
80
90
100
Time (yr)
Pb-210
0
Radioisotope Concentration
•
1st
-1
-2
y = -0.036x + 6E-16
-3
2
R =1
-4
k,
d-1
0
20
40
60
Time (yr)
CE5504 – Surface Water Quality Modeling
80
100
Writing the Mass Balance
dC
V
 Q  Cin  Q  C  V  k  C
dt
3
3
g
m
g
m
g
g
3
3 1
m  3 
 3
 3 m   3
m d
d m
d m
d m
CE5504 – Surface Water Quality Modeling
RXN
1st order
decay
At Steady State
0
dC
V
 Q  Cin  Q  C  V  k  C
dt
RXN
1st order
decay
Q  C  V  k  C  Q  Cin
C  (Q  V  k )  Q  Cin
Q
Css  Cin 
Q V  k
CE5504 – Surface Water Quality Modeling
At steady state, the source terms are
equal to the sink terms and there is
no net change in mass within the
control volume.
dC
V
 Q  Cin  Q  C  V  k  C
dt
Time Variable
(analytical solution)
Css,1
Css ,2
concentration
Q
Css ,1  Cin ,1 
Q V  k
Q
 Cin ,2 
Q V  k
Css,2
time
Ct  Css ,1  e
Q 
   k t
V

flushing out
CE5504 – Surface Water Quality Modeling
Q 

   k t 
 Css ,2  1  e  V  




building in
Time to Steady State Variable

 Css ,2  1  e


Q 
   k  t
V





concentration
Ct  Css ,1  e
Q 
   k  t
V

Css,1
Css,2
time
Noting that the hydraulic retention time,  = V/Q
Ct  Css ,1  e
1 
   k  t


1 

   k  t 
 Css ,2  1  e    




and (Chapra, Sec. 3.3)
 ln (1   )
t 
1
k

CE5504 – Surface Water Quality Modeling
or, for 95%
 ln 0.05
t95 
1
k

or
t95 
3
1

k
Variability in  and k
Lake
 (years)
Superior
179
Michigan
136
Ontario
8
Onondaga
Material
0.25
k (yr-1)
Organic C
Atrazine
PCB
Chloride
CE5504 – Surface Water Quality Modeling
36.5
1.0
0.05
0
t95 
3
1

k
Review
1. Can you see any limitations to the analogs in Slide 3?
2. Can you identify 2 additional source terms for lakes in Slide 4?
3. Discuss the completely mixed and constant volume assumptions in Slide 5.
4. Develop an example of an inappropriately-defined control volume in Slide 6.
5. Provide some additional examples of growth and decay in Slide 7.
6. Show how a system acts to bring itself to steady state; see Slide 10.
7. In lab -
a. Determine half-lives of selected chemical species.
b. Compare response times of lake/chemical couplets.
c. Calculation of steady state concentrations.
d. Time variable solutions: kinetics and step function response.
CE5504 – Surface Water Quality Modeling