(N)CEWB222 Chapter 4 - Specific Energy

7/24/2019 (N)CEWB222 Chapter 4 - Specific Energy

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CEWB222 HYDRAULIC

ENGINEERING

Chapter 4 - Specific Energy

(Alternate Depths)



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The Energy Principles



ENERGY EQUATION

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Lh Z g

V P Z

g

2

2

2

2

2

1

1

1

1

22

VP

Where:Z1 = elevation of streamline above the datum at cross-Section 1.

P1 = Pressure

= Fluid specific weight

P/ = Pressure head

V2

= streamline velocityV2/2g = velocity head

g = local acceleration of gravity

hL= hydraulic losses

= correction factors



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consider the special case where the channel bottom

is horizontal (S0 =0); HL= 0 and 1 = 2 = 1.0

ENERGY EQUATION

g

V y

g

V y

E E

g

V y E

22

2

2

2

2

2

1

1

21

2

22

2

11

1

and Y z

P

X S Y z

P

o



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2

2

2

22

1

2

1

2211

22

A

gA

Q y

gA

Q y

V AV

Apply Continuity Equation:

2

2

2

22

1

2

1

22

y

vyq

gy

q y

gy

q

ENERGY EQUATION

For rectangular channel or wide:



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2 1z

WS

h

v2/2g

vy

hLEGL

HGL

Channel Bottom

Datum

z E z y gA

Q z y

g

V Head

2

22

22Specific

Energy

ENERGY IN OPEN CHANNELS



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Specific Energy, EE = energy (head) measured with respect to

the channel bottom

Multiply through by y2 and arrange to find

y3 – E y2 + q2/2g = 0

What kind of equation is this? How

many roots? Significance of roots?

SPECIFIC ENERGY

2

22

2y2g

v

yE gy

q



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SPECIFIC ENERGY CURVEFor a given specific energy, there are 2

possible depths, low stage Y1 and high stage

Y2

. The low stage is the alternate depth of the

high stage and vice versa.

-The specific energy is a minimum at point C.

-The critical depth of flow , Yc corresponds to

this condition of minimum specific energy.

When the depth of flow is lesser than the

critical depth, Y1 <Yc and the velocity of flow isalso greater than the critical velocity

V1 >Vc

Therefore the flow is supercritical flow

When the depth of flow is greater than the

critical depth,

Y2 > Yc and the velocity of flow is

also less than the critical velocity

V2 <VC

Therefore the flow is subcritical flow

The position of the specific energy curve

changes when discharge changes.



Specific Energy Curve

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Specific Energy Curve



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Specific Energy Diagram

0.0

2.0

4.0

6.0

8.0

10.0

0.0 2.0 4.0 6.0 8.0 10.0

Specific Energy (ft)

Depth(f

t)

Rectangular

channel:

B = 20 ft

Q = 600 ft3/s.Fr = 1

Fr = 2

E = y

Fr = V/(gy)

0.5



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Critical Flow & Depth (Yc)

Critical flow occurs when the specific energy is minimum for a given discharge.The depth for this condition may be determined if we solve for dy/dE

E = y + Q2 /2gA2 and

dE/dy = 0 (set dE/dy equal to zero)

013

2

dy

dA

gA

Q

dy

dE

Where, dA=Tdy

13

2

c gA

cT Q



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g

V

gA

Q

T

A

c

cc

c

2

c

c

2

2

T

A

g

Q

T

A 2

c

3

c

Critical Flow & Depth (Yc)

Or

General

Geometry



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Critical Flow for Rectangular Channel



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Specific Energy Equations



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EXAMPLE 1Determine the critical depth for water flowing at 10 m3 /s

in a trapezoidal channel with bottom width 3 m and sideslopes of 2:1 (H:V).

T=3+4y

y 1

2

y

3 m

1

2

cT

3cA

g

2Q

A=3y+2y2

Solution:

Under critical flow conditions



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EXAMPLE 1

cy43

3)2cy2cy3(

81.9

210

And since Q=10m3

/s and g=9.81m/s2

, then under criticalflow conditions

Solving for yc by trial and error yields the critical depth

yc=0.855m



EXAMPLE 2

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EXAMPLE 2

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Thank You

(N)CEWB222 Chapter 4 - Specific Energy

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