DVC Calculus Field Trip
Carollo Engineers
Dan Frost
Katy Rogers
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November 15th, 2010
Field Trip Outline
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• Introduction of speakers
• Introduction to Carollo
• Office tour
• Calculus at Carollo and lunch
• Questions
Dan Frost
• BS/MS Env. Eng. from Cal Poly in 2008
• Carollo Engineers since 2008
• Experience Highlights:
- Digester Rehab/FOG Facilities – Fresno & DSRSD
- Napa Wastewater Master Plan
- City of San Leandro WPCP Rehabilitation Project
- City of Stockton CIP and Energy Management Plan
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- Algae to Biofuels
Katy Rogers
• BS Civil Eng. from UC Davis in 2007
• MS Env. Eng. from Stanford in 2008
• Carollo Engineers since 2008
• Experience Highlights:
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-
City of Modesto Tertiary Wastewater Treatment Plant
City of Modesto Engineer’s Report
City of Turlock THM Study
City of San Francisco CIP
Introduction to Carollo Engineers
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• We are the largest firm in the United States
dedicated solely to water and wastewater
treatment.
• Carollo Engineers provides planning, design, and
construction management services for municipal
clients.
Carollo Statistics
•
•
•
•
Founded 1933
630+ employees
300+ professional engineers
Multi disciplined
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-
Sanitary/environmental
Structural
Mechanical
Electrical
Instrumentation
Civil
Chemical
Company Organization
CEO
Partners
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Associates
Engineers
Drafters
Graphics
Document Processing
Business Development
Human Resources
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Carollo currently maintains
32 offices in 12 states
What types of projects do we work on?
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• Water treatment
• Wastewater treatment
• Infrastructure
• Research and Development
• Water Reuse
• Construction Management
• Integrated Water Resources
Key drivers for water and wastewater
projects include:
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• Growth
• Regulations
• Aging Infrastructure
• Management/Public Policy
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Project Examples….
Wastewater Treatment
Plant No. 1
Fountain Valley, CA
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Orange County Sanitation
Districts Orange County, CA
360 mgd (combined)
Reclamation Plant No. 2
Huntington Beach, CA
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City of Phoenix, AZ
Southern Avenue Interceptor
160,000 LF/48-90 inch pipes
City of Stockton, CA
Westside Sewer Interceptor
41,000 LF/72 inch pipe
GCDCorpOv903.ppt13
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Clark County Water Reclamation District, NV
130-mgd of biological phosphorus removal
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Napa Sanitation District, CA
10-mgd DynaSand filter for reuse
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Stockton WWTP Facility
Aerial
What do we produce for our clients?
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• Proposals and Statements of Qualifications
• Preliminary studies and reports
• Master plans
• Design plans and specifications
Office Tour
• Library and Central Files
• Engineering groups
- Civil/Process
- Structural
- Electrical and instrumentation
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- Mechanical
• Graphics
• Business development
• Drafting
• ISG/Tech Support
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Calculus at Carollo
Calculus at Carollo
• Do we use it?
• How do we use it?
- Hydraulic calculations
- Volume calculations
- Structural analysis
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• Questions?
Do we use calculus at Carollo?
• Short Answer: Yes
• Long Answer:
- We use simple calculus-derived equations
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- We also use software to solve several equations
simultaneously
Algebraic equations derived from calculus:
Calculus
Algebra
Area under a curve
b

A  [ f ( x )  g( x )]dx
a
Conservation of Mass
dM

 

    u  dV  
dV     u  dV       u   dV  0
V t
V
V t
dt V


A
1
( Y1  Y2 )  x
2
p1A 1V1  p1A1V1  Q1  Q2
Qo Xo  V(
 mS
)X  k d X
Ks  S
Used For:
Calculating rain volume,
excavation volume, etc
Mass and flow balances
(calculating flows and
masses when multiple
streams are involved)
 (Q o  Q w ) X e  Q W X W
Conservation of Energy

2
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1
dp

2
2
  VdV   gdz  0
1
1
1 2
v a  gha
2
1
 Pb  v b2  ghb
2
Pa 
Rate Equation
CR

Co
dC
 kC
dt
k   ln
Co
Co  CR
Hydraulic design
(structures, pipes,
pumps, valves, etc),
Hydraulic Profiles
Treatment design
(reaction time)
Derivation of the Bernoulli Equation
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• Start with Newton’s Second Law….
Bernoulli’s Equation
How we use the Bernoulli Equation at
Carollo:
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• To determine pressure, velocity, and elevation at
points within a hydraulic system
• To size pipes, valves, pumps, and turbines
• To determine headloss through a pipe due to
friction and connections
• To develop hydraulic profiles
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Applications of the Bernoulli Equation
Determining flow from a tank or reservoir
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Applications of the Bernoulli Equation
Sizing and designing culverts
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Applications of the Bernoulli Equation
Sizing pumps and pipelines
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Applications of the Bernoulli Equation
Determining the flow, pressure, and
headloss at points within a parallel pipe
system
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Hydraulic Profile for the City of Modesto
WWTP
Calculating flow volumes from
hydrographs
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Flow
Flow
• Hydrograph - flow vs. time (e.g. storm, river)
• Volume of water over time = Area under curve
Time
Time
Calculating flow volumes from
hydrographs
• Estimate volume of water from a rainfall event
- Size culverts, pipelines
- Wastewater storage
• Actual data not defined by simple equations
- Apply approximation methods
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• Trapezoidal Rule:
• Average End Area Method:
A = Δx(yo/2 + y1 + y2 + yn/2)
A = ½ (y1+y2) * Δx
Hydrograph Example
• Receive raw data from client
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Treatment Plant Flow during Rainfall Event
Time
t
0
1
2
3
4
5
6
7
8
9
10
11
12
Flow
Q
5
51
89
119
141
155
161
159
149
131
105
71
29
Typical Treatment Plant Flow
Time Flow
t
Q
0
5
1
5.5
2
6
3
6.5
4
7
5
7.5
6
8
7
8.5
8
9
9
9.5
10
10
11 10.5
12
11
Hydrograph Example
• Receive raw data from client
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Treatment Plant Flow during Rainfall Event
Time
t
0
1
2
3
4
5
6
7
8
9
10
11
12
Flow f(x) = f(t) = Q = -2t2+ 50t + 5
Q
5
51
89
119
141
155
161
159
149
131
105
71
29
Typical Treatment Plant Flow
Time Flow g(x) = g(t) = Q = 0.5t+5
t
Q
0
5
1
5.5
2
6
3
6.5
4
7
5
7.5
6
8
7
8.5
8
9
9
9.5
10
10
11 10.5
12
11
Hydrograph Example
• Graph raw data
180
Flow from Rainfall = f(x)
160
Base Flow = g(x)
Flow (gal/hr)
140
120
100
80
60
40
20
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0
0
1
2
3
4
5
6
7
Time (Hours)
8
9
10
11
12
Hydrograph Example
• Volume due to Rainfall (I&I) = Area btw Curves
- Integrate equation for curves
180
Flow from Rainfall = f(x)
160
Base Flow = g(x)
Flow (gal/hr)
140
120
100
80
60
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40
20
0
0
1
2
3
4
5
6
7
Time (Hours)
8
9
10
11
12
Hydrograph Example
• Integration of f(t):
- Q = Flow During Rain Event = f(t) = -(2t)2 + 50t + 5
-  f(t) =  (-(2t)2 + 50t + 5)dt = -4/3t3 + 25t2 +5t + C
- At t = 0, Q = 5 => C = 5
- From t = 0 to t = 12 => [-4/3(12)3 + 25(12)2 +5(12) + 5]
- [-4/3(0)3 + 25(0)2 +5(0) + 5]
= 1,356 gallons
• Integration of g(t):
- Q = Typical Flow = g(t) = 0.5t+5
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-  g(t) =  (0.5t+5)dt = 1/4t2 + 5t + C
- At t = 0, Q = 5 => C = 5
- From t = 0 to t = 12 => [1/4(12)2 + 5(12) + 5] – 5 = 96 gallons
• Total Flow Due to Rainfall = 1,356 – 96 = 1,260 gallons
Hydrograph Example
• Volume due to Rainfall (I&I) = Area btw Curves
- Approximation (Avg. End Area Method)
180
Flow from Rainfall = f(x)
160
Base Flow = g(x)
140
Flow (gal/hr)
120
100
80
60
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40
20
0
0
1
2
3
4
5
6
7
Time (Hours)
8
9
10
11
12
Hydrograph Example
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• Approximation: 1,252 Gallons
Avg
Time Flow Flow Duration Volume
Q
t
Q (gal/hr) t (Hrs)
gal
0
5
1 51
28
1
28
2 89
70
1
70
3 119 104
1
104
4 141 130
1
130
5 155 148
1
148
6 161 158
1
158
7 159 160
1
160
8 149 154
1
154
9 131 140
1
140
10 105 118
1
118
11 71
88
1
88
12 29
50
1
50
Sum = 1,348
Avg
Time Flow Flow Duration Volume
Q
t
Q (gal/hr) t (Hrs)
gal
0
5
1 5.5 5.25
1
5.25
2
6
5.75
1
5.75
3 6.5 6.25
1
6.25
4
7
6.75
1
6.75
5 7.5 7.25
1
7.25
6
8
7.75
1
7.75
7 8.5 8.25
1
8.25
8
9
8.75
1
8.75
9 9.5 9.25
1
9.25
10 10 9.75
1
9.75
11 10.5 10.25
1
10.25
12 11 10.75
1
10.75
Sum =
96
Structural Analysis
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• Structural engineers use calculus to determine the
maximum stress of structural elements under different
loads.
Structural Analysis
• Shear at a point is the sum of all vertical forces acting on
an object.
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• Moment at a point is the total bending moment acting on
an object.
Structural Analysis
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• Step 1: Determine all forces using plane static equilibrium
equations.
Structural Analysis
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• Step 2: Calculate shear at a point by integrating the load
function w(x) or the area under the load diagram up to
that point.
Structural Analysis
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• Step 3: Calculate moment at a point by integrating the
shear function V(x) or the area under the shear diagram
up to that point.
Structural Analysis
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• In summary….
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Another example….
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More examples….
Software used to analyze complex
systems
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• Hydraulix
• BioTran/BioWin
• EnerCalc
• STAAD
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Questions?
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End of Presentation