Encyclopedia of Environmental Science and Engineering, Volume I and II

WATER FLOW 1279

Bends The effect of the presence of bends is to induce

secondary flow currents which are responsible for the addi-

tional energy dissipation:

hK

V

bb

2

2 g

. (25)

The bend loss coefficient, K b , depends on the ratio of the

bend radius, r, to the pipe diameter, d, as well as the bend

angel. For a 90
 bend and r / d ratio varying from 1 to 12,

values of K b range from 0.20 to 0.07.

Gates and Gate Valves The gate and gate valve loss can

be expressed as:

hK

V

gg

2

2 g

(26)

The value of the loss coefficient, K g , for gates depends on

a variety of factors. The value of K g for the case having the

bottom and sides of the jet suppressed ranges from 0.5 to 1.0.

for typical values of K g for gate valves see Table 7.

Energy-discharge Relation

In pressure conduit flow, the water is transmitted through a

closed boundary conveying structure without a free surface.

Figure 3 illustrates graphically the various forms of energy

losses which could take place within the conduit. The follow-

ing energy relation can be written:

hh h hlfent tc (28)

in which h end  entrance loss, h tc  transition loss, h f  skin

friction loss. If H denotes the total head required to produce

the discharge and h v represents the existing velocity head,

H  h l  h v. (29)

Writing Eq. (29) in terms of the velocity heads and their

respective loss coefficients,

HC

V

K

V

K

VV

f

LV

D

K

l

v

 



2

2

2

2

2

2

2

gggg

g

ent

1

2

tc

2

2

1

2

22

2

−

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢

⎢

VV 22

2 g

⎤

⎦

⎥,

(30)

where K v  combined velocity head and exit loss coefficient.

By the continuity equation:

AV^11 A V2 2 (31)

and

VA

A

12 22 V

1

2

2

22 gg

.^

Equation (30) could be expressed as,

HC

V

V

K

A

A

K

A

A

fL

gD

K

l

v





2

2

2

2

2

1

2

2

2

1

2

2

2

2

1

2

g

g ent tc

⎛

⎝⎜

⎞

⎠⎟

−

⎛

⎝

⎜

⎞

⎠

⎟

⎡⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

(32)

in which

CK

A

A

K

A

A

fL

D

lt ent^2 cKv

1

2

2

2

1

2

2

1

2

⎛

⎝⎜

⎞

⎠⎟

−

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

g ⎥

(33)

V

C

H

l

(^212)
1
 / 2 g (34)
QAV
A
C
gH
CA H


22
2
1
12
2
2
2
/
g ,
(35)
TABLE 7
K g for gate values
Fully open 0.2
3
(^4)
open 1.3
1
(^2)
open 5.5
1
(^4)
open 24.0
Exit Loss In general the entire velocity head is lost at
exit and the exit loss coefficient, K e is unity in the equation:
hK
V
ee
2
2 g

. (27)

hent

htc

TEL

Transition

H

hl

–V^2 /2g

hf

hv

p/y

FIGURE 3 Energy relations.

C023_003_r03.indd 1279C023_003_r03.indd 1279 11/18/2005 11:12:13 AM11/18/2005 11:12:13 AM