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DVC Calculus Field TripCarollo Engineers

November 15th, 2010

Dan FrostKaty Rogers

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Field Trip Outline

• Introduction of speakers• Introduction to Carollo• Office tour• Calculus at Carollo and lunch• Questions

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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- Algae to Biofuels

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Katy Rogers

• BS Civil Eng. from UC Davis in 2007• MS Env. Eng. from Stanford in 2008• Carollo Engineers since 2008• Experience Highlights:

- City of Modesto Tertiary Wastewater Treatment Plant- City of Modesto Engineer’s Report- City of Turlock THM Study- City of San Francisco CIP

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Introduction to Carollo Engineers

• 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.

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Carollo Statistics• Founded 1933• 630+ employees• 300+ professional engineers• Multi disciplined

- Sanitary/environmental- Structural- Mechanical- Electrical- Instrumentation- Civil- Chemical

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Company Organization

CEO

Partners

Associates

EngineersDraftersGraphics

Document ProcessingBusiness Development

Human Resources

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Carollo currently maintains 32 offices in 12 states

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What types of projects do we work on?

• Water treatment• Wastewater treatment• Infrastructure • Research and Development• Water Reuse• Construction Management• Integrated Water Resources

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Key drivers for water and wastewater projects include:

• Growth• Regulations• Aging Infrastructure• Management/Public Policy

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Project Examples….

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Orange County Sanitation Districts Orange County, CA360 mgd (combined)

Reclamation Plant No. 2Huntington Beach, CA

Wastewater Treatment Plant No. 1Fountain Valley, CA

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x/13 City of Stockton, CA

Westside Sewer Interceptor41,000 LF/72 inch pipe

City of Phoenix, AZSouthern Avenue Interceptor160,000 LF/48-90 inch pipes

GCDCorpOv903.ppt13

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Clark County Water Reclamation District, NV130-mgd of biological phosphorus removal

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x/15 Napa Sanitation District, CA10-mgd DynaSand filter for reuse

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Stockton WWTP Facility Aerial

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What do we produce for our clients?

• Proposals and Statements of Qualifications• Preliminary studies and reports• Master plans• Design plans and specifications

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Office Tour

• Library and Central Files• Engineering groups

- Civil/Process- Structural- Electrical and instrumentation- Mechanical

• Graphics • Business development• Drafting• ISG/Tech Support

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Calculus at Carollo

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Calculus at Carollo

• Do we use it?• How do we use it?

- Hydraulic calculations- Volume calculations- Structural analysis

• Questions?

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Do we use calculus at Carollo?

• Short Answer: Yes• Long Answer:

- We use simple calculus-derived equations- We also use software to solve several equations

simultaneously

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Algebraic equations derived from calculus:Calculus Algebra Used For:

Area under a curve Calculating rain volume, excavation volume, etc

Conservation of Mass Mass and flow balances (calculating flows and masses when multiple streams are involved)

Conservation of Energy Hydraulic design (structures, pipes, pumps, valves, etc), Hydraulic Profiles

Rate Equation Treatment design (reaction time)

b

a

dx)]x(g)x(f[A x)YY(21A 21

XkX)SK

S(VXQ ds

moo

WWewo XQX)QQ(

21111111 QQVApVAp 0

V V V V

dM u dV dV u dV u dVdt t t

02

1

2

1

2

1 gdzVdVdp

kCdtdCR

o

C

C

Ro

o

CCClnk

b2bb

a2aa

ghv21P

ghv21P

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• Start with Newton’s Second Law….

Derivation of the Bernoulli Equation

Bernoulli’s Equation

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How we use the Bernoulli Equation at Carollo:• 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

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Calculating flow volumes from hydrographs• Hydrograph - flow vs. time (e.g. storm, river)• Volume of water over time = Area under curve

Time

Flow

Time

Flow

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• Estimate volume of water from a rainfall event- Size culverts, pipelines- Wastewater storage

• Actual data not defined by simple equations- Apply approximation methods

• Trapezoidal Rule: A = Δx(yo/2 + y1 + y2 + yn/2)• Average End Area Method: A = ½ (y1+y2) * Δx

Calculating flow volumes from hydrographs

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Hydrograph Example

• Receive raw data from clientTreatment Plant Flow during Rainfall Event Typical Treatment Plant Flow

Time Flow Time Flowt Q t Q0 5 0 51 51 1 5.52 89 2 63 119 3 6.54 141 4 75 155 5 7.56 161 6 87 159 7 8.58 149 8 99 131 9 9.5

10 105 10 1011 71 11 10.512 29 12 11

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Hydrograph Example

• Receive raw data from clientTreatment Plant Flow during Rainfall Event Typical Treatment Plant Flow

Time Flow f(x) = f(t) = Q = -2t2+ 50t + 5 Time Flow g(x) = g(t) = Q = 0.5t+5t Q t Q0 5 0 51 51 1 5.52 89 2 63 119 3 6.54 141 4 75 155 5 7.56 161 6 87 159 7 8.58 149 8 99 131 9 9.5

10 105 10 1011 71 11 10.512 29 12 11

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Hydrograph Example

• Graph raw data

0 1 2 3 4 5 6 7 8 9 10 11 120

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120

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180Flow from Rainfall = f(x)Base Flow = g(x)

Time (Hours)

Flow

(ga

l/hr)

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Hydrograph Example

• Volume due to Rainfall (I&I) = Area btw Curves- Integrate equation for curves

0 1 2 3 4 5 6 7 8 9 10 11 120

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180Flow from Rainfall = f(x)Base Flow = g(x)

Time (Hours)

Flow

(ga

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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- 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

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Hydrograph Example

• Volume due to Rainfall (I&I) = Area btw Curves- Approximation (Avg. End Area Method)

0 1 2 3 4 5 6 7 8 9 10 11 120

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120

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180Flow from Rainfall = f(x)Base Flow = g(x)

Time (Hours)

Flow

(ga

l/hr)

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Hydrograph Example

• Approximation: 1,252 Gallons

Time FlowAvg

Flow Duration Volume

t QQ

(gal/hr) t (Hrs) gal0 51 5.5 5.25 1 5.252 6 5.75 1 5.753 6.5 6.25 1 6.254 7 6.75 1 6.755 7.5 7.25 1 7.256 8 7.75 1 7.757 8.5 8.25 1 8.258 9 8.75 1 8.759 9.5 9.25 1 9.25

10 10 9.75 1 9.7511 10.5 10.25 1 10.2512 11 10.75 1 10.75

Sum = 96

Time FlowAvg

Flow Duration Volume

t QQ

(gal/hr) t (Hrs) gal0 51 51 28 1 282 89 70 1 703 119 104 1 1044 141 130 1 1305 155 148 1 1486 161 158 1 1587 159 160 1 1608 149 154 1 1549 131 140 1 140

10 105 118 1 11811 71 88 1 8812 29 50 1 50

Sum = 1,348

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Structural Analysis

• Structural engineers use calculus to determine the maximum stress of structural elements under different loads.

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Structural Analysis

• Shear at a point is the sum of all vertical forces acting on an object.

• Moment at a point is the total bending moment acting on an object.

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Structural Analysis

• Step 1: Determine all forces using plane static equilibrium equations.

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Structural Analysis• 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.

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Structural Analysis• 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.

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Structural Analysis• In summary….

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Another example….

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More examples….

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Software used to analyze complex systems• Hydraulix• BioTran/BioWin• EnerCalc• STAAD

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Questions?

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End of Presentation