Maths
A2 Maths
Tunnelworks A2 Maths Student sheet 1Student sheet
TUNNELWORKS
Modelling sewer and CSO flow
A2
MATHS
STUDENT
SHEET
The Thames Tideway Tunnel will control discharges from 34 combined sewer outfalls (CSOs)
that currently discharge untreated sewage directly into the River Thames after it rains. By
creating a model of how water flows though each sewer and CSO engineers can model the
total
capacitysewer
in the and
tunnel.
Modelling
CSO flow
The Thames Tideway Tunnel will control discharges from 34 combined sewer outfalls (CSOs) that
currently
sewage
directly
into the River Thames after it rains. By creating a
A basic discharge
model foruntreated
water flow
through
a CSO.
model of how water flows though each sewer and CSO engineers can model the total capacity in
the
Thistunnel.
basic curve shows how water flows through an example sewer when:
• The rainfall has an intensity of 2mm / hour.
A basic model for water flow through a CSO.
• The rainfall lasts for 1 hour.
This basic curve shows how water flows through an example sewer when:
Using a real equation
The rainfall has an intensity of 2mm / hour.
The rainfall lasts for 1 hour.
3
2
f(x) = 0.5x – 2x + 2x
Flow rate
•
•
Max. flow capacity
of the sewer = 0.6m 3/s
diagram
0
1
2
Time (hours)
Under these conditions the maximum flow into the sewer is at its threshold flow rate of 0.6m3/s.
Underthan
these
the maximum
flow
into
the
sewer is
ataitsCSO
threshold
flow rate ofinto the
More
thisconditions
and at present,
any excess
flow
will
overflow
into
which discharges
0.6m3/s.
More
than
this
and
at
present,
any
excess
flow
will
overflow
into
a
CSO which
River Thames.
discharges into the River Thames. Eg:
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1
Task 1
A2 Maths
Transformations allow this curve to represent other rainfall conditions.
Student sheet
Sketch the transformations to this curve that would show the following conditions:
Task 1
• Intensity 9mm/hour, which creates a maximum flow of 0.98m3/s
Transformations
allow thiswhich
curvecreates
to represent
other rainfall
conditions.
• Intensity 14mm/hour,
a maximum
flow of 1.20
m3/s
100J
Sketch the transformations to this curve that would show the following conditions:
• Assume
Intensitythe
9mm/hour,
which
creates
a maximum
flow of 0.98m3/s
rainfall still
lasts
1 hour
in each case.
• Intensity 14mm/hour, which creates a maximum flow of 1.20 m3/s
Assume
the sketch,
rainfall still
lastsany
1 hour
each
case.
For each
shade
areainthat
represents
the excess flow from the sewer, into the CSO
ForTideway
each sketch,
shade
any areafor
that
represents the excess flow from the sewer, into the CSO
Tunnel
will capture
treatment.
that would currently discharge into the River Thames, and which once constructed the Thames
Tideway Tunnel will capture for treatment.
that would currently discharge into the River Thames, and which once constructed the Thames
CSO
Sewer
Level in sewer
at threshold
flow rate
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Discharge into
River Thames
2
Task 2
The basic curve above (and the conditions it represents) can be represented by the
function:
A2 Maths
f(x) = 0.5x3 – 2x2 + 2x
Student sheet
Task 2
Modify this function so the transformed curve represents the following conditions:
The basic curve above (and the conditions it represents) can be represented by the
function: f(x) = 0.5x3 – 2x2 + 2x
• Intensity 1mm/hour and maximum flow of 0.50m 3 /s
• Intensity 6mm/hour and maximum flow of 0.82m 3 /s
Modify this function so the transformed curve represents the following conditions:
• Intensity 1mm/hour and maximum flow of 0.50m /s
• Intensity 6mm/hour and maximum flow of 0.82m3/s
(Hint:
Assume
the rainfall
still
lasts 1 hour.
Use
the new
3
/s maximum flow rates to calculate the
• Intensity
16mm/hour
and
maximum
flow of
1.22m
3
• Intensity 16mm/hour and maximum flow of 1.22m
/s
3
scale factor.):
(Hint: Assume the rainfall still lasts 1 hr. Use the new maximum flow rates to calculate the scale factor.):
Now modify the function so it represents 2mm / hour rainfall that lasts:
Now modify the function so it represents 2mm / hour rainfall that lasts:
• 0.5 hours, with the flow through the sewer lasting 1 hour in total.
hours,with
with the
the flow
flow through
•• 33hours,
throughthe
thesewer
sewerlasting
lasting44hours
hoursinintotal.
total.
• 0.5 hours, with the flow through the sewer lasting 1 hour in total.
(Hint: Use
scale
factor.)
(Hint:
Use the
the new
new total
totalflow
flowtimes
timestotocalculate
calculatethe
the
scale
factor.)
Task 3 3
Task
Making the model more realistic
In reality the sewer always has a minimum of 0.4m3/s
wastewater flowing through it,
In reality the sewer always has a minimum of 0.4m3/s wastewater flowing through it, even
The sewer
has
a minimum
flow rate of 0.4 m3/s even when there’s no rain.
even
when
there
is
no
rain.
when there is no rain.
So for a ‘basic’ rain event the curve is now:
Flow rate
Max. flow capacity
of the sewer = 0.6m 3/s
Min. sewer flow rate
= 0.4m 3/s
0
1
Time (hours)
2
Modify
the function
functionf(x)
f(x)so
soit itnow
nowrepresents
represents
this
more
realistic
situation
as shown
in graph
the graph
Modify the
this
more
realistic
situation
as shown
in the
3
above. The
conditions
should
stillstill
be 0.6m
/s. What
two two
above.
The maximum
maximumflow
flowunder
underbasic
basicrainfall
rainfall
conditions
should
be 0.6m3/s.
What
transformations do
transformations
doyou
youneed
needtotoinclude?
include?
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3