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### Open Channel Hydraulic

### This is all about RG744 Fall semester 2013

Hydrology and Water Resources RG

Fluid mechanics

- Weight
- Mass Density
- Specific weight
- Specific gravity
- Hydrostatics
- Continuity equation
- Types of flow
- Energy and Energy Head
- Bernoulli’s Equation
- Flow through open channel

Properties of a Fluid

- Weight

W = mg (kN, lb)

- m = mass of fluid (kg, slugs) g = acceleration due to gravity 9.81 m2/sec, 32.2 ft2/sec
- Mass Density
- mass of the fluid per unit volume at a standard temperature and pressure

r = m/V (kg/m3, slugs/ft3)

- V = volume of fluid (m3, ft3)
- In the case of water, neglect the variation in mass density and consider it at a temperature of 4oC and at atmospheric pressure; then r= 1,000 kg/m3

Properties of a Fluid

- Specific Weight
- gravitational force per unit volume
- Units: kN/m3, lb/ft3
- In SI units, the specific weight of water at a standard reference temperature of 4oC and atmospheric pressure is 9.81 kN/m3

g = W/V

- Specific Gravity
- ratio of the specific weight of a given liquid to the specific weight of pure water at a standard reference temperature
- Units????

Sg (fluid) = gfluid/ gwater

- Specific Gravity of water = ?

Problem?

- A reservoir of glycerin has a mass of 1,200 kg and a volume of 0.925 m3. Calculate
- Weight of the glycerin
- Mass density of glycerin
- Specific weight of glycerin
- Specific gravity of glycerin

g = 9.81 ft/sec2, gw = 9800 N/m3.

Terminology

- Open channel flow – any flow path with a free surface (open to atmosphere)
- Can be classified as
- Prismatic channel
- With constant x-section and a constant bed slope
- Non-prismatic
- Varies in both the x-sectional shape and bed slope between any two selected points along the channel length
- Atmospheric pressure acts continuously, constantly and at every location on water surface therefore is neglected

Open Channel Hydraulics

- Variables of open channel flow analysis
- Open channel flow classification based on various criteria
- Time
- Depth
- Space
- Regime (subcritical or supercritical)

Depth of Flow

- Elevation difference between water surface and deepest part of the channel

Channel Slope

- Difference in the channel invert elevation between two locations divided by the distance between them
- In prismatic channel the slope is often constant over a significant channel distance

Hydraulic depth & hydraulic radius

- Hydraulic depth: average depth across the channel

Discharge & Velocity

- Discharge or flow rate: amount of water moving in a channel or stream system
- Velocity: speed at which water moves in an open channel

V = Q/A

V= average channel velocity, Q= discharge, A = x-sec area

- Water movement adds kinetic energy to the system
- Channel velocity is not constant at any location
- Varies both horizontally and vertically for any given channel cross-section
- Velocity near the channel banks is less than the velocity in the center of the channel

Flow Classification

- Uniform vs. non-uniform
- Steady vs. unsteady flow
- One-dimensional vs. multidimensional flows
- Gradually varied vs. rapidly varied
- Subcritical vs. supercritical

Types of Flow

- Uniform Flow
- in which the flow velocity and depth do not change from point to point along any of the streamlines otherwise it is called non-uniform or varied flow
- Laminar Flow
- in which each liquid particle has a definite path and the paths of individual particles do not cross each other
- Turbulent Flow
- if each particle does not have a definite path and the paths of individual particles also cross each other, the flow is called turbulent

Types of Flow

- Steady Flow
- in which the depth and velocity at a point are constant with respect to time
- Unsteady Flow
- if Q is not constant
- One-dimensional Flow
- flow, whose streamlines may be represented by straight lines as opposed to curved lines

Subcritical & Supercritical Flow

- Classification is based on ratio of inertial to gravitational forces at a stream location – Froude number
- If Fr > 1 – flow is ‘supercritical’ and inertial forces dominate, associated with steeper slopes (high velocity and shallow depth)
- If Fr < 1 – flow is ‘subcritical’ – gravitational forces dominate usually calm and tranquil –small slope usually in natural channels - (low velocity and high depth)
- For Fr = 1 both depth and flow are call ‘critical’

Energy

- What is energy?
- Ability to do work?
- Moving fluids possess energy by virtue of its
- Velocity
- Position
- Pressure

Energy and Head

- 3 kinds of energies that can be stored in a waterbody
- Potential: due to elevation/position ‘Z’ (elevation above a fixed datum)

PE = WZ= mgZ

- Kinetic: due to velocity/motion

KE = mv2 = (W/g) v2

- Pressure: amount of work done in moving the fluid element a distance equals to the segment’s length ‘d’

Force F = PA

Work done (Pressure energy) = Fxd = PAd = P(Ad) = P(Volume) = PW/ g

Total Energy

- Total Energy = Potential + Kinetic + Pressure

TE =WZ + (W/g)v2 + PW/ g

- Energy may be expressed as ‘Head’
- divide by ‘W’ throughout
- Represents total energy per unit weight of the fluid

Energy Head

- Total Head

H = Z + v2/g+ P/ g

- Z = Elevation Head (units of length)
- v2/g = Velocity Head (units of length)
- P/ g = Pressure Head (units of length)

Example?

- Given:
- Water in a 6 in diameter pipe with a velocity of 8 ft/s
- Fluid pressure is 4 lb/in2
- Elevation of the center of the pipe above datum is 10 ft
- Required?
- What is total energy head?

Bernoulli’s Equation – conservation of energy

- During a steady flow of a frictionless incompressible fluid, the total energy (total head) remains constant along the flow path

Z + v2/g+ P/ g = constant

Z1 + v12/g+ P1/ g = Z2 + v22/g+ P2/ g

Continuity equation

- Based on the conservation of mass
- Assumption: flowing fluids have constant mass density (incompressible liquid)
- States that the quantity of liquid passing per time unit is the same at all sections

Q1 = Q2 = Q3= ….

- OR A1V1 = A2V2= A3V3= ….
- Q = flow discharge [m3/s];
- V = average velocity of the liquid [m/s];
- A = area of the cross-section [m2];
- and 1, 2, 3 = the number of sections 1-3

GOOD LUCK ;-)

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