Environmental Engineering

Today in Environmental Engineering class I will have a short and simple timed exercise of a few questions.

I will arrive to class in about 1 hour, and will be provided the exercise by my Professor.

Once I receive the exercise, I will immediately upload images of the exercise so that may get straight to work and have as much time as possible.

There will be 1 hour 30 minutes time limit on the exercise which will be more than enough time to get through the short exercise of a few questions.

Attached in the file upload section of this post, you will find a few files uploaded there which I was provided by my professor which contains notes/cheat sheets to use as a reference while completing the exercise which contains handy and useful information such as equations and formulas needed in order to complete the exercise.

The last file you will find in the file upload section will be titled “Practice/Sample Exercise,” which will contain 2 sample problems that are very similar to ones that will be provided on the actual exercise. So please go over the file as it will be good practice but, be aware that the sample problems are not completely answered.

So, please take a look at the files/documents to make sure you feel confident with the topics and know you will complete the exercise to your best ability.

Please feel free to message me if you have any questions or concerns.

Thank you so much in advance!

Lakes and the Completely Mixed Model
I. Complete Mixing of Wastes in a Water Body
A. Use lake as system to model
•
•
•
•
•
•
•
•
•
•
• g) and the Great Lakes (e.g. Erie).
B. Conservation of Mass & Example Model Derivation
Qs = Flow rate into lake from stream (m3
/d)
Cs = Contaminant concentration in stream (g/m3
)
Qw = Flow rate into lake from waste source (m3
/d)
Cw = Contaminant concentration in waste source water (g/m3
)
C = Contaminant concentration in lake (g/m3
)
V = Volume of lake (m3
)
Additional
Waste Source
Stream in Stream out
LAKE
Qw, Cw
Qs
Cs
C V C
(Qs + Qw)
Simple way of modeling based on Conservation of Mass – true of many models.
• Mass Balance: – mass balance for amount of contaminant in lake.
Contaminant Mass In – Mass Out = Change of Contaminant Mass in Lake
• Calculate conservation of mass: What is the quantity/product “QC”?
(
of time:
In a given length of time t:
Mass In = [QsCs + QwCw] t
Mass Out = [Qs + Qw] C t
Change in Mass = V C
• Plug into Mass Balance Equation:
[QsCs + QwCw] t – [Qs + Qw] C t = V C
IN OUT CHANGE in contaminant mass in lake
• Manipulate Equation to get to useful form:
QsCs + QwCw − (Qs + Qw )Ct = VC
Divide both sides by t:
V ( )
C
t
QsCs QwCw Qs Qw C


= + − +
As t → 0, you get the instantaneous change in lake concentration:
V ( )
dC
dt
= QsCs + QwCw − Qs + Qw C
* I don’t expect you to know how to do the integration, but I do expect you to unde
• Solve equation by integrating:
C
Q C Q C
Q Q
t
Q Q
V
C t
Q Q
V
s s w w
s w
s w
o
s w =
+
+
− −
 +

















+ −
 +











1 exp exp
(section 1)
where Co = initial concentration in the lake (concentration at t = 0).
• What does this equation tell you?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.] exp (x) – this is “e” on your calculator
• Steady-state solution
dC
dt
=





0
• When concentration no longer changing with time.
• So:
•
•
•
• Rewrite equation:
[QsCs + QwCw] t = [Qs + Qw] C t
IN OUT
Solve for C:
C
Q C Q C
Q Q
s s w w
s w
=
+
+
C. Include Waste Decay
•
•
• , then you need to include it in your model.
• Decay is often represented as a “first-order” process. – 1
st order –
means
dC
dt
= −KC
K = rate
• 1
st
-order decay can often be a good approximation.
• Include 1st
-order decay in the completely-mixed model:
V ( )
dC
dt
= QsCs + QwCw − Qs + Qw C − VKC
– add decay to “mass out” part of the equation  -VKCt, where KC is the decay
rate we
• Solve equation for C by integrating:
C
Q C Q C
Q Q VK
t
Q Q
V
K C t
Q Q
V
K
s s w w
s w
s w
o
s w =
+
+ +
− −
+
+


















+ −
+
+












1 exp exp
• Also want the steady-state result
dC
dt
=





0
[QsCs + QwCw] t = [Qs + Qw +KV] C t
IN OUT
Solve for C:
C
Q C Q C
Q Q KV
s s w w
s w
=
+
+ +
Example #1
A lake has a volume of 1000 m3
. Flow into the lake from a stream is 100 m3
/day. A factory
currently discharges 10 m3
/day into the lake. The factory wants to double its discharge
concentration of toluene from 10 mg/m3
to 20 mg/m3
. The stream has 1 mg/m3
toluene from
upstream industry. Toluene is found to decay at a rate of 0.01/day. How will doubling the
factory discharge concentration affect the toluene levels in the lake?
Look
at
steady-state solutions to compare:
• This is a very simple model – important to understand assumptions.
V=1000 m3
Factory
10 m3
/d
Stream
100 m3
/d out
E. Model Assumptions
1.
2.
3.
4.
5.
6. are constant.
F. Multi-cell modeling
• The completely-mixed model may give reasonable results, but you may want to
•
•
•
•
• EXAMPLE
1
st:
Completely Mixed Model (1 “Cell”)
Break up: Completely Mixed Model (2 “Cells”)
Cell #1 Cell #2
Qin
Cin
Qout
Cout
Qin
Cin
Q1
C1
Qout
Cout
G. Example #2
It is believed (based upon analysis of lots of data and modeling) that the lake in Example 1 can
be better represented by dividing the lake into 2 cells, the first cell 250 m3
and the second cell
750 m3
. How will this affect the results of the analysis?
We’ll
do
the
“before” picture.
Factory
10 m3
/d
Stream V=750 m3
out 100 m3
/d
V=250 m3
Lake Characteristics
•
•
•
•
• processes that affect mixing.
III. Physical Processes to Consider
A. Hydrodynamic Mixing
B. Wind-blown Mixing – Wind blowing across a lake can create
circulation “cells” that cause mixing.
lakes may have one mixing cell that mixes all the way to the bottom [DIAGRAM]
y since they are caused by shear from the upper cells. – Show shear with books on
table – similar with fluids like water! [DIAGRAM]
Wind
Wind
Slower
mixing as
you go
deeper
B. Temperature Effects
1. Density of Water – Need to look at density of water. Everyone
sa
Example: at
2. Resistance to Mixing – How hard it is to mix water depends on the
density difference. So, for example, if you had two bowls of water:
Water Density Changes with Temperature
992
993
994
995
996
997
998
999
1000
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Temperature (°C)
Density (mg/mL)
ml
mg  = 0.3
ml
mg  = 0.008
Less dense
More dense
Ttop = 31C
Tbottom = 30C
Ttop = 5C
Tbottom = 4C
Harder to Mix Easier to Mix
3. Temperature Effects on Lakes → Stratification
• Summer – Get stratification due to hot air temperatures that heat the
top part of the lake
• – picture shows idealistic stratification.
Epilimnion –tem
perature.
Thermocline – –
) 5-20 ft thick. Very difficult to mix.
Hypolimnion -–heat up. Little to no mixing (no wind).
[ADD DEPTH TO DIAGRAM]
How does it get this way? Air starts getting warmer and sun’s rays
stronger – warms surface of lake. Also, incoming runoff water and
stream water is warmer. – Warmer water decreases in density, making it
more difficult for wind to mix it with lower depths. – So water near the
surface stays near the surface and continues getting warmer throughout
the summer. As it gets warmer at the surface, it gets even harder to mix
– Feedback Loop.
• How does this affect contaminants?
The residence time is shorter.
Residence time = time an average water molecule spends in the water
body.
Cross-section
Epilimnion
Hypolimnion
Thermocline
hypolimnion
thermocline
epilimnion
Stream in LAKE Stream out
Fall – Get “Fall turnover”
[ADD DEPTH TO DIAGRAM]
0 5 10 15 20 25
Temperature (°C)
• Winter – 2 cases.
[ADD DEPTH TO DIAGRAM]
0 5 10 15 20 25
Temperature (°C)
• Spring
•
•
•
• [ADD DEPTH TO DIAGRAM]

0 5 10 15 20 25
Temperature (°C)
Another
ice
How do you know if stratification is significant?
Densimetric Froude number
N
v
gd
DF
o
=


where
NDF = densimetric Froude number (-) – nondimensional (make sure units
cancel)
v = average flow-through velocity (m/s), Q/wd
Q = flow rate into and out of lake (m3
/s)
w = average width (m)
d = average depth (m)
 = change in water density over depth d (kg/m3
)
ml
mg 1
m
kg 1
3
=
o = reference density (kg/m3
). For water, use 1000 kg/m3
g = gravitational acceleration (9.81 m/s2
)
What it tells you (these are some general guidelines):
NDF > 1 then fully mixed
1 > NDF > 0.1 then weak stratification
NDF < 0.1 then strong stratification
Example 3
A lake is 200 m long, 20 m wide, and 10 m deep. A stream flows into and out of
the lake at 1000 m3
/day. The temperature at the top is 20C (
w 3 m
kg
 = 998.3
) and
at the bottom is 10C (
w 3 m
kg
 = 999.73
). Is the lake likely to be stratified?
Example 4
You have determined that the lake in Example 1 is strongly stratified in the
Summer. The epilimnion is 8 feet deep, so the volume of water in the
epilimnion is 300 m3
. How does this information change your analysis of
Example 1?
V=300 m3 now – smaller volume that contaminants affect.
At