Basic Theory & Formulae Training Module

Cook Legacy CollegeTraining Module T-100:

Basic Theory & Formulae

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Goals of This 15-Minute Module

Review mathematical fundamentals.

Understand common formulae for use in our projects

Apply theory to common problems

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3

Fundamentals of Small Hydro

TH 103:Review and

Quiz

TH 102: Commonly

Applied Formulae

TH 101: Mathematical Fundamentals

4

Math Fundamentals: Area & Volume

FormulaRectangle:Area = Length X WidthA = lwPerimeter = 2 X Lengths + 2 X WidthsP = 2l + 2wParallelogramArea = Base X Heighta = bhTriangleArea = 1/2 of the base X the heighta = 1/2 bhPerimeter = a + b + c(add the length of the three sides)

Note: Mass=Volume & Density

5

Math Fundamentals: Trigonometry

"SOHCAHTOA" is a helpful mnemonic for remembering the definitions of the trigonometric functions sine, cosine, and tangent i.e., sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent.

6

Basic Algebra

PEMDASIt stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

To Solve Equations:1. Combine Like Terms2. Isolate the variable

7

Basic Calculus

Our most common use of calculus is to define the arc or area under a curve.

8

Discharge EquationsThe basic discharge equation is Discharge over an ogee crest (Bureau of Reclamation, U.S. Department of the Interior, 1987)QO=Cwlh1.5, whereQO = Flow discharge over an ogee crestCw=Discharge coefficient from (Water Resources Engineering)h =Head over spillwayl = Length of Coanda weir

Discharge through a Coanda Screen slot (Wahl, 2001)The Coanda discharge equation solves iteratively for the throughput of each progressive slot. It takes the common form of a discharge equation, based on the dimensionless Reynolds, Weber, and Froude numbers adjusted by empirically based constants.∆q=(CCV*CF)*s'*√(2gE)CCV=Coanda Screen discharge coefficient accounting for the effects of velocityCF=Coanda Screen discharge through an orifices'= Length of slot from tail of one wire to lead of next, g=Acceleration due to gravity, E=Specific energy equation based on depth, velocity, and angle of screen surface

9

Deflection Calculation

Radius of Rod 0.25inMoment of Inertia 0.003067959in^4

DUL = Design Uniform Load 0psiE = modulus of elasticity (psi) 28300000psi

W = load (lbs) 0lbsl = length of beam (in) 27iny = deflection (in) 0in

Allowable Deflection of Beam - Industry Standard = Length of beam / 360 0.075

Failure Risk? NO

10

Physics of Compressed Air: Theory

Boyle’s Law: Where:P1=Pressure1; P2=Pressure 2V1=Volume 1; V2=Volume 2T1=Temperature 1; T2=Temperature 2Note: We discount temperature.

Two Implications:1) The volume of air in the tank (at 150 PSI) is smaller than the volume at the screen (at around 10 PSI)

2) Air bubbles injected at depth will “grow” and split as they experience lower pressure while rising through the water.

Understanding an AirBurst™ System requires understanding of some basic physics of compressed air.

11

Process for Sizing Site

Flow-Duration Curve

0.0

10.0

20.0

30.0

40.0

50.0

0 10 20 30 40 50 60 70 80 90 100Percent Time Flow Equalled or Exceeded (%)

Flo

w (

m³/

s)

• Estimate flow duration curve based on: Measurements of flow over time

Size of drainage above site, specific run-off, and shape of flow duration curve

The easiest way to mess up a high head site is to overestimate the flow (as this is the only variable)

12

Process for Sizing Site

Flow-Duration Curve

0.0

10.0

20.0

30.0

40.0

50.0

0 10 20 30 40 50 60 70 80 90 100Percent Time Flow Equalled or Exceeded (%)

Flo

w (

m³/

s)

Picking the TurbineEstimate the installed capacity at various points on the FDCEstimate the cost of equipment cost associated with those. This includes questions such as the size of the equipment and how many nozzles will be used.

Integrate the area under the curve with various scenarios and pick the best payback. Sometimes this can be approximated by simply slicing the FDC into a few scenarios.

For example:The green line has high capacity to “catch” high flow events, but may not run enough to be worth the cost.The blue line will be cheaper, but may leave power on the table. The red line may be the best return.

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