CRITICAL FLOW

BERNOULLI’S EQUATION

g

vz

p

g

vz

p

22

2

22

2

2

11

1

SPECIFIC ENERGY

g

vdEs

2

2

Specific EnergyIn a channel with constant discharge, Q

2211 VAVAQ

2

2

2gA

QdE

g

VdE

2

2

Consider rectangular channel (Area= b.d) and Q= q.b

2

2

2gd

qdE

q is the discharge per unit width of channel

Area

b

d

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

E

d

Specific Energy: Sluice Gate

2

2

2gd

qdE

1

2

21 EE

sluice gate

1d

2d

EGL

d1 and d2 are ___________ depths (same specific energy)

q = 5.5 m2/sd2 = 0.45 mV2 = 12.2 m/s

d1 = 8.07 m

alternateGiven downstream depth and discharge, find upstream depth.

0

1

2

3

4

0 1 2 3 4

E

dSpecific Energy:

Raise the Sluice Gate

2

2

2gd

qdE

1 2

E1 E2

sluice gate

1d

2d

EGL

as sluice gate is raised, d1 approaches d2

E is minimized for a given discharge or

Q is maximized for a given energy

CRITICAL FLOW

Minimum E

2

2

2gA

QdE

P

A

T

dd

d dA

TdddA but

0

1

2

3

4

0 1 2 3 4

E

y

dc

Consider a channel with an irregular cross-section

dd

dA

gA

Q

dd

dE3

2

10

3

2

1gA

TQ 1Fr

Critical Flow exists

Maximum Q

2

2

2gA

QdE

P

A

T

dd

d dA

Consider a channel with an irregular cross-section

3

2

1gA

TQ 1Fr

Critical Flow exists

)(2 dEgAQ

0)(2

)(2

dEg

gAdEg

dd

dA

dd

dQ

but and AQdEg /)(2 TdddA /

General Equation

3

2

1gA

TQ

CRITICAL FLOW

Sample:

A flow of 280 L/s of water is carried by a

90˚-V notch flume with n = 0.011.

Find the critical depth.

dc = 0.44 m

CRITICAL FLOW

For rectangular sections

cc gdV 3

2

g

qd c

3/4

22

c

c

cR

nVS

g

Vd c

c

2

g

Vd cc

22

2

2

cc

ddE Edc

3

2

velocity head = 1/2 (dc)

g

VdE

2

2

CRITICAL FLOWFor rectangular sections

cc gdV 3

2

g

qd c 3/4

22

c

c

cR

nVS

Sample:

A channel having a rectangular section and

base width of 3 m carries a flow of 5.70

m3/s. Find the critical depth, the critical

velocity and the critical slope if n = 0.011.