chapter 4 energy equation printing.doc

Energy Equation and its Application 1/60
Chapter “4”
When the student finishes studying this chapter, he should be able to:
 Understand the use and limitations of the Bernoulli’s equation,
and applying it to solve a variety of the fluid flow problems,
 Calculate various flow properties using the energy and hydraulic
grade lines,
 Work with the energy equation expressed in terms of heads, and
use it to determine power output and pumping power
requirements,
 Understanding the different velocity and flow rate measurement
techniques and learn their advantages and disadvantages.

Energy Equation and its Applications
Fig. (1)- An Elemental Cylindrical Stream tube.
Figure (1) shows a small "element" stream tube of cylindrical section.
Taking this to be a control volume at an instant time, the forces acting
in the direction of flow along the stream tube are:
 Pressure Forces
 Upstream end,
 Downstream end,
 Circumference, and
 Weight Force
= A
.
P

=  A
dP
P 

=Zero (pressure forces p cancel) *
=   

 cos
g
ds
.
A
cos
g
m 




* The force due to the surrounding fluid acting normal to the sides
of the element, perpendicular to its axis (the fluid is at rest). The
resultant of these forces in any direction must be zero)
Where A is the cross-sectional area of the element at right angles to
the streamlines, and the weight component in the direction of motion
  dz
A
g
ds
dz
ds
A
g 
 



(since dz
cos
ds 
 )
The forces tending to accelerate the fluid mass are the pressure forces
on the two ends of the element
  A
dP
A
dP
P
A
P 



The mass of the element is s
A 
 . , and its acceleration for steady flow
can be expressed as  
ds
dV
V . Thus applying a
m
F  , we get
 
ds
dV
V
.
A
ds
dz
.
A
g
A
dP 


 

Dividing by ds
.
A
 , and in the limit
0
s
d
dV
V
s
d
z
d
g
s
d
p
d
1




……………………………… (1)
This equation is commonly referred to as the one-dimensional Euler's
Equation. It applies to both compressible and incompressible flow,

since the variation of  over the element length s
 is small. Dividing
by g , Eq. (1) can also be expressed as
0
g
2
V
d
z
d
g
p
d 2












……………………………… (2)
For case of incompressible fluid ( or  = constant), Eq. (2) can be
integrated to give
H
head
total
constant
g
2
V
z
g
p
d 2




 
  …….………… (3)
This is Bernoulli's equation for steady flow of a frictionless
incompressible fluid along a streamline. Each term in the above
equation ordinarily expressed in L-feet or meters-, represents
 Newton meters of energy per Newton of fluid flowing in SI units,
and
 Foot pounds of energy per pounds of fluid flowing in BG units)
If there is no flow,
constant
z
g
p



This equation shows that for an incompressible fluid at rest, at any
point in the fluid the summation of the pressure head g
/
p  plus the
elevation z is equal to the sum of these two quantities at any other
points.

Bernoulli's Equation
 (Steady flow along stream line)
General
.
const
Z
g
2
V
dP 2



 
 Incompressible flow ( .
const

 )
.
const
Z
g
2
V
P 2




 The Bernoulli equation states that the sum of the kinetic, potential
and flow energies of a fluid particle is constant along a stream line
during steady flow
Flow energy

/
P 2
V
2
1
 Z
g
 = Constant

Kinetic energy Potential energy
Static, Dynamic, Compressible Flow
 The kinetic and potential energies of the fluid can be
converted to flow energy (and vice versa) during flow,
causing the pressure to change. Multiplying the Bernoulli's
equation by the density  ,
.
const
Z
g
V
2
1
P 2


 
 (along a stream line)
Each term in the equation has pressure units, and thus each
term represents some kind of pressure.
 It is often convenient to represent the level of mechanical
energy graphically using heights to facilitate visualization of
the various terms of the Bernoulli equation. This is done by
dividing each term of the Bernoulli's equation by g to give:
Pressureheady
g
P
 g
2
V 2
 Z
 = H
Velocity head Elevation heady Total head


g
P

Is the Pressure head; it represents the height of a fluid
column that produces the static pressure P.

g
2
V 2
Is the Velocity head; it represents the elevation needed
for a fluid to reach the velocity V during frictionless free
fall.
Z Is the Elevation head; It represents the potential head of
the fluid.
H Is the total head.
 Bernoulli's equation can be expressed in terms of heads as:-
The sum of pressure, velocity, and elevation heads along a stream
line is constant along a stream line is constant during steady flow
when the compressibility and frictional effects are negligible.
Hydraulic Grad Line (HGL)
is a line that represents the sum of the static pressure and the
elevation heads,
Z
g
P


Energy Grad Line (EGL)

is a line that represents the total head of the fluid ,
Z
g
V
g
P


2
2

The difference between the heights of (HGL) and (EGL) is equal
to the dynamic head,
g
2
V 2
About the "EGL" and the "HGL"
 For stationary bodies such as reservoirs or lacks, the EGL and
HGL coincide with the free surface of the liquid.
(The elevation of the free surface Z in such cases represents
both the EGL and HGL since the velocity is zero and the state
pressure "gage" is zero).

 The EGL is always a distance g
2
V 2
above the HGL. These two
lines approach each other as the velocity decreases, and they
diverge as the velocity increases.
In an idealized Bernoulli's type, EGL is horizontal and its height
remains constant.
For open channel flow, the HGL coincides with the free surface of
the liquid, and the EGL is a distance g
2
V 2
above the free
surface.
At a point exit, the pressure head is zero (atmospheric
pressure) and thus the HGL coincides with the pipe outlet.
The energy loss due to friction effects causes the EGL and
HGL to slope down- ward in the direction of flow. The slope is
a measure of the head loss in the pipe. A component that
generates significant frictional effects (valve) cause a sudden
drop in both EGL and HGL at that location.
A steep jump in EGL and HGL whenever mechanical energy
is added the fluid by a pump, and a steep drop occurs
whenever mechanical energy is removed from the fluid by a
turbine.

 The pressure (gage) of a fluid is zero at locations where the
HGL intersects the fluid. The pressure in a flow section that
lies above the HGL is negative, and the pressure in a section
that lies below the HGL is positive. An accurate drawing of a
piping system and the HGL can be used to determine the
regions where the pressure in the pipe is negative (below the
atmospheric pressure). This enables us to avoid situations in
which the pressure drops below the vapor pressure of the
liquid (which cause cavitations).
 When the flow is irrotational, the Bernoulli's equation becomes
applicable between any two points along the flow (not just on
the same streamline),
2
2
1
2
2
2
















 Z
g
V
g
P
Z
g
V
g
P


Streamlines

Change in EGL & HGL due to change in diameter of the pipe

Change in EGL & HGL due to flow through a nozzle.

Rise in EGL&HGL due to pump

Drop in EGL and HGL due to turbine.

EGL and HGL in a straight pipe

Sub atmospheric pressure when pipe is above HGL
Applications
I- Orifices
Orifices are regularly shaped, submerged openings through which flow
is propelled by differences in energy between the upstream and
downstream sides of the opening. The stream of flow expelled from the
orifice is called the jet. When the jet exits the orifice, adverse velocity
components cause it to contract to a point after which the flow area

remains relatively constant and the flow lines become parallel (see Fig.
2). This point is called the vena contracta.
Fig. (2) - Cross-sectional View of Typical Orifice Flow.
1- Small orifice
Fig. (2) shows a jet of water issuing from a large tank through a small
orifice. At a small distance from the tank, the streamlines are straight
and parallel, and the pressure is atmospheric. Applying Bernoulli's
equation between points  and  gives
g
2
V
Z
g
P
g
2
V
Z
g
P 2
2
2
2
1
1



















…………….……….. (4)
Variables and assumptions:
 The reference datum is at the centerline/centroid of the
orifice, 1
z and 2
z are equivalent

 The reference pressure is the atmospheric pressure,
0
Patm  , point  occurs at the vena contracta and the
existing stream pressure is 0.
 A hydrostatic condition prevails there H
g
/
p1 
 .
 The reservoir is sufficiently large and point  is sufficiently
far from the orifice for the velocity 1
V to be negligible.
Consequently, g
2
/
V
H 2
2

i.e. gH
2
V
V 2
1 
 …………….…….. (5)
Fig. (3) - Discharge through a Small Orifice.
This formula, which was developed by Torricelli in 1943, states that the
discharge velocity equals the speed that a frictionless particle would
attain if it fell freely from point  to point . For more accurate results,
one may account for friction by introducing an experimentally
determined velocity coefficient V
C . The coefficient of velocity depends on

the size and shape of the opening as well as the distance from the free
surface to the opening H . For well designed, rounded opening, the
value of V
C is usually between 0,98 to 1.0.
The discharge may be calculated by applying the continuity equation
gh
2
A
C
A
V
Q c
v
c
th 
 ………………………..…….. (6)
Where c
A is the area of the jet at vena contracta.
The area of the jet, however, is smaller than that of the orifice, due to
the convergence of the streamlines, as it shown in Fig.(4). The
contraction of the jet is called the vena contracta. Experiments have
shown that the jet area c
A and the orifice area A are called by:
A
C
A c
c
Where c
C is the coefficient of contraction, and is normally = 0.62 for
sharp-edged outlet.

A. Sharp-edged Orifice
c
C = 0.62, v
C =0.98
B. Well- rounded entrance
c
C = 1.00, v
C =0.98
C. Sharp-edged entrance
c
C = 1.00, v
C =0.86
Fig. (4) Flow through Various Types of Orifices.
Hence, the actual discharge
gH
2
A
C
C
Q c
v
th  ………………………..…….. (7)
or gH
2
A
C
Q d
th  ………….. (8)
Where d
C is the coefficient of discharge = c
V C
.
C
Note:
 The actual velocity attained at the vena
contracta is slightly less, and a coefficient of
velocity is defined as the ratio of the actual
(mean) velocity to the theoretical. i.e. actual
mean velocity gH
2
Cv
 .

 The coefficient of contraction c
C is defined as
the ratio of the area of the vena contracta to
the area of the orifice itself.
 Because of the two effects of friction and
contraction, the discharge from the orifice is
less than the theoretical value and the
coefficient of discharge d
C is defined as the
ratio of the actual discharge to the theoretical
discharge.
discharge
l
Theoretica
discharge
Actual
Cd 
velocity
l
Theoretica
orifice
of
Area
velocity
Actual
contract
vena
of
Area
Cd



v
c
d C
C
C 


2- Large Orifice
If the orifice is large (head of liquid is less than 5 times the height of the
orifice), then the small orifice equation will no longer be accurate. This
is a result of the significant of h (and therefore,V ) across the orifice. In
such cases, the flow rate (i.e. discharge) has to be determined by
integrating the discharges through small elements of the area as
follows:
For any element strip of thickness dh, across a rectangular orifice,
h
g
2
dh
b
dQ  …………………………….…………….. (9)
Thus the total discharge is:
dh
h
g
2
b
Q
1
2
H
H
5
.
0

 …………………………….…………….. (10)
 
2
/
3
1
2
/
3
2 H
H
g
2
B
3
2
Q 

………………….…………….. (11)

Fig. (5) - Discharge Through a large Orifice.
If 1
H tends to zero, the upper edge of the orifice no longer influences
the flow. This is corresponding to a thin plate weir. Consequently, Eq.
(11) forms the basic equation of thin plate weir.
3- Submerged Orifice
A submerged orifice is one that discharges a jet into fluid of the same
kind. The orifice illustrated in Fig. (6), for example, discharges liquid
into more of the same liquid. A vena contracta again forms, and the
pressure there corresponds to the head 2
H . Application of Bernoulli's
equation between points  and  gives:
g
2
V
Z
g
P
g
2
V
Z
g
P 2
2
2
2
1
1



















or    
g
2
V
0
H
0
0
H
2
2
2
1 



 ……………………...(12)
and so the theoretical velocity )
H
H
(
g
2
V 2
1
2 
 .
In other words, Torricelli's formula is still applicable provided that H
refers to the difference of head across the orifice. Except for very small

orifices, the coefficients for a submerged orifice are little different from
those for a producing a free jet. The kinetic energy of a submerged jet
is usually dissipated in turbulence in the receiving fluid.
Fig. (6) - Discharge Through a Submerged Orifice.
4- Underflow Gates
A variety of underflow gate structures is available for flow rate control.
Three types are illustrated in Fig. (7).
The flow under a gate is said to be free outflow when the fluid issues
as a jet of supercritical flow with a free surface open to the atmosphere
as shown in Fig. (7). In such cases it is customary to write this flow rate
as the product of the distance, a, between the channel bottom and the

bottom of the gate (gate opening) times the convenient reference
velocity   2
1
1
y
g
2 . This is
1
d y
g
2
a
C
q 
Where q is the flow rate per unit width. The discharge coefficient, d
C , is
a function of the contraction, a
/
y
C 2
c  and the depth ratio a
/
y1 .
Typical values of the discharge coefficient for flow outflow (or free
discharge) from a vertical sluice gate are on the order of 0.55 to 0.60
as indicated by the top line in Fig. (8).
As indicated in Fig. (7b), in certain situations the jet of water issuing
from under the gate is overlaid by a mass of water that is quite
turbulent. Typical value of d
C for these drowned outflow cases are
indicated as the series of lower curves in Fig. (8)

(a) Vertical Gate, free flow (b) Drowned Outflow Gate, submerged flow
(b) Radial Gate (c) Drum Gate
Fig. (7)-Three Variations of Underflow Gates.
Fig. (8) - Typical Discharge Coefficient for Underflow Gate.

Obstruction Flow Meters: Nozzle and Orifice Meters
Consider incompressible steady flow of a fluid in a horizontal pipe
diameter D that is constricted to a flow area of diameter d , as shown
in Fig. (9). Applying the continuity equation and the Bernoulli's equation
between a location before the constriction (point) and the location
where constriction occurs (point ) can be written as
 
D
d
Constriction
Fig. (9) – Flow through a Constriction in a Pipe.
Continuity Equation 2
2
1
1 V
A
V
A
Q 

or
    2
2
2
1
2
1 V
D
d
V
A
A
V 


 ……………………….. (13)
Bernoulli's equation ( 2
1 z
z  ):

2
2
1
2
g
2
V
g
P
g
2
V
g
P





















……………………….. (14)
Combining Eqs. (1) and (2) and solving for 2
V gives
  )
1
(
g
P
P
g
2
)
D
/
d
(
1
g
P
P
g
2
V 4
2
1
4
2
1
2











 









 
 ……………………….. (15)
where D
d

 is the diameter ratio.
The velocity in Eq. (15) is obtained by assuming no losses, and thus it
is the maximum velocity that can occur at the constriction side. Once
2
V is known, the flow rate can be determined from
2
2 V
A
Q 
In reality, some pressure losses due to
 Frictional effects are inevitable, and thus the velocity will be
less,
 Also, the fluid stream will continue to contract past the
obstruction, and the vena contracta area is less than the flow
area of the obstruction.
Both losses can be accounted for by incorporating a correction factor
called the discharge coefficient d
C whose value (which is less than 1)
is determined experimentally. Then the flow rate for obstruction flow
meter can be expressed








 


g
P
P
g
2
)
1
(
A
C
Q 2
1
4
o
d
.
act


……………………….. (16)
where:
4
/
d
A
A 2
2
o 

 : is the cross sectional area of the hole, and
D
/
d

 : is the ratio of the whole diameter to pipe
diameter.
The value of d
C depends on both and the Reynolds
number 
/
D
V
R 1
e  . Charts and curve-fit correlation for d
C are
available for various types of obstructions.
If the obstruction flow meter equipped with a manometer across it
  h
g
P
P
P L
i
2
1 
 




or
  h
1
G
.
S
h
1
g
P
P
g
P
L
L
i
L
2
1
L



















Substituting into Eq. (4) gives
  H
g
2
)
1
(
A
C
1
G
.
S
h
g
2
)
1
(
A
C
Q
4
o
d
L
4
o
d
.
act

 




…….. (17)

This analysis shows that the flow rate through a pipe can be
determined by constricting the flow and measuring the decrease
in pressure due to the increase in velocity at the constriction site.
Noting that the pressure drop between two points along the flow
can be measured easily by a manometer or a differential pressure
transducer.
The most common types of obstruction meters are: the Venturi-meters
(Fig.10 a), the orifice meters (Fig. 10 b), and the nozzle meters (Fig. 10
c).
Net
head
loss
Cost
(a)
Venturi
meter
Small Large

(b)
Orifice
meter
Large Small
(c)
Nozzle
meter
Medium Medium
Fig. (10)-Typical Devices for Measuring Flow-rate in Pipes.
1- Venturi Meter
The Venturi meter, invented by an American engineer (1842-1930)
and named in honor of Giovanni Venturi (1746-1822), an Italian
physicist who first tested conical expansions and contraction. The
Venturi meter is the most accurate flow meter in the group, but it is also
the most expensive. Its gradual contraction and expansion prevent flow
separation and it suffers only frictional losses on the inner wall
surfaces. The head loss for Venturi meters due to friction is very low

(only about 10%). Thus they should be preferred for applications that
cannot allow large pressure drops. Owing to its streamlined design, the
discharge coefficient of Venturi meters is very high for most flows. In
the absence of specific data, we can take 
d
C 0.98.
2- Nozzle Meter
In nozzle meters, the plate is replaced by a nozzle, and thus the flow in
the nozzle is streamlined. As the result, the vena contracta is
practically eliminated and the head loss is small. However, flow nozzle
meters are more expensive than orifice meters. The value of d
C can
be taken to be 0.96 for flow nozzle.
3- Orifice Meter
The orifice meter has the simplest design and it occupies minimal
space as it consists of a plate with a hole in the middle. Some orifice
meters are sharp-edged, while others are beveled or rounded. The
sudden change in flow area in orifice meters causes considerable head
loss or permanent pressure loss. The value of d
C can be taken to be
0.61 for orifices.

Fig. (11) – Schematic for the Orifice meter.
Note:
 H is the difference in piezometric head of the fluid in the meter, not
the difference of levels of the manometer liquid.
 d
C varies somewhat with the rate of flow, the viscosity of the fluid
and the surface roughness, a value of about 98
.
0 is usual with
fluids of low viscosity.
 To ensure that the pressure measured at each section is the
average, connections to the manometer are made via a number of
holes into an annular ring,
 A common ratio of diameters D
/
d = ½. Thus 1
2 A
/
A = ¼ and
1
2 V
/
V = 4. Although a smaller throat area gives a greater and

more accurately measured difference of pressure, the subsequent
dissipation of energy in the diverging part is greater. Moreover, the
pressure at the throat may become low or enough for dissolved
gases to be liberated from the liquid, or even for vaporization to
occur.
A weir is an obstruction or dam built across open channel over
which liquid flow, often through a specially shaped opening. Weirs
are typically installed in open channels such as streams to
determine discharge (flow rate). The basic principle is that the
discharge over a weir is directly related to the water depth (H)
upstream from the weir (see Fig.12a). Weirs are classified to the
shape of this opening. The flow over the lowest point of structure
surface or edge over which water flows is called crest, whereas
the stream of water that exits over the weir is called the nappe.
Depending on the weir design, flow may contract as it exit over
the top of the weir, and, as with orifices, the point of maximum
contraction is called vena contracta (see Fig.12a ).
The contraction can be contracted or suppressed by designing
the weir such that its shape conforms to the shape of the channel.
This type of weir is called suppressed weir (B = b) in rectangular
III- Weirs

channel. The nappe then contracts in the vertical direction only
and not horizontally. With a contracted weir, the crest and nappe
vary from channel to such degree that a significant contraction of
flow area does not occur (see Fig.12c ).
In addition to suppressed and contracted weir types, weirs are
also distinguished as either sharp-crested or broad-crested. A
sharp-crested weir has a sharp upstream edge formed so that the
nappe flows clear of the crest. Broad-crested weirs have crests
that extend horizontally in the direction of flow far enough to
support the nappe and fully develop hydrostatic pressures for at
least a short distance.
Weirs can also be distinguished by their shapes. The most
common types of weirs are the rectangular weir, the triangular "V-
notch" weir, and the trapezoidal (or Cipolletti) weir (see Fig 13).

(a)
(b)
(c)

Fig.(12)- Rectangular Sharp-crested weir with end contraction.
a) Sharp-crestedWeirGeometry, b) Suppressed weir (B = b) and c) Contracted weir

(b) Rectangular Weir (c) Triangular Weir (d) Cipolletti Weir
Fig. (13)- Sharp-crested Weir Plate Geometry

Weirs are simple and inexpensive to build and install. Common
materials of construction include metal, fiberglass and wood. However,
they represent a significant loss of head, and are not suitable for
measuring flows with solids that may cling to the weir or accumulate
upstream of it.
The Weir Equation
To determine the basic discharge equation for the weir, we will
consider a horizontal strip of width b and V is the velocity at a depth
h below the free surface, as shown in Fig. (14).
Fig. (14) Definition Sketch for a Sharp-crested Weir.
The head on the weir H is defined as the vertical distance between the
weir crest and the water surface taken far enough upstream of the weir
to avoid local free-surface curvature. Applying Bernoulli's equation

between a point upstream of the weir  and  a point in the plane of
the weir gives:
2
2
2
2
1
2
1
1
z
g
2
V
g
p
z
g
2
V
g
p







……….……….. (17)
Variables and assumptions:
 the reference datum is the crest of the weir, 1
z and 2
z are
equivalent.
 the reference pressure is the atmospheric pressure,
 neglecting streamline curvature and assuming negligible
velocity of approach upstream of the weir ( 0
V1  ),
 the existing stream pressure at point  is 0, and H
g
p1


.
After applying the known variables and assumptions and solving for
2
V , Eq. (17) becomes:
h
g
2
V
V 2 
 …………………………….…………….. (18)
Then  
dh
b
gh
2
dQ  , and integrating from the free surface, 0

h ,
to the weir crest, H
h  gives the total theoretical discharge equation
as:
dh
h
b
g
Qth 
 2
1
2 …………………………….…………….. (19)
This will be different for every differently shaped weir or notch. To
make further use of this equation we need an expression relating the
width of flow across the weir to the depth below the free surface.

1- Rectangular Weir
For a rectangular weir the width does not change with depth so there is
no relationship between b and h . Substituting B
b  in Eq. (19) gives
dh
h
B
g
2
Q 2
1
th 

2
3
th h
g
2
B
3
2
Q  …………………….…….…………….. (20)
In the case of actual flow over a weir, the streamlines converge
downstream of the plane of the weir, and viscous effects are not
entirely absent. Consequently, a discharge coefficient d
C must be
applied to the basic expression on the right hand-side of Eq. (c) to
bring the theory in line with the actual discharge. Thus we have
2
3
2
3
2
h
g
B
C
Q d
act  …………….…………….. (21)
Fig. (15) – Definition Sketch for the Rectangular Weir.

For low-viscosity liquids, the coefficient of discharge d
C is primarily a
function of the relative height H/P. An empirically determined equation
for d
C adapted from Kindsvater and Carter ( ) is
P
H
05
.
0
4
.
0
Cd 
 ……………….……….…………….. (22)
Eq. (22) is valid up to an w
P
H value of 10 as long as the weir is well
ventilated so that atmospheric pressure prevails on both the top and
the bottom of the weir nape.
Note:
Since the values of H is small at low flows, an
accurate determination of d
C becomes
impracticable. It will be seen after that these
considerations lead to alternative weir shapes.
Rectangular Weir with End Contraction
Of weir section does not span the entire width of the channel.
Therefore, their will a contraction of the contraction of the flow section
just downstream of the weir so that the effective length of the weir will

be somewhat less than L. Experiments by Francis indicated that under
the conditions depicted in Fig. () the effective reduction in length is
approximately equal to H
2
.
0 when H
/
L  3. Thus, the formula for a
contracted weir (one with flow contraction due to end walls) is given as:
  2
3
d H
H
n
1
.
0
L
g
2
C
Q 
 …………………………………….. (23)
where n is the number of contraction, normally 2 but sometimes 1..
Franzini page 431
Robenson 211
Fig. (16) – Limiting Properties of Standard Contraction Weir.

2- Triangular ( V-notch ) Weir
A definition sketch for triangular weir is shown in Fig. (17). The primary
advantage of the triangular weir is that it has a higher degree of
accuracy over a much wider range of flow than does the rectangular
weir, because the average width of the flow section increases as the
head increases.
The basic discharge equation for the triangular weir is derived in the
same manner as that for the rectangular weir, expecting of course, the
fact that b is not a constant but a function h . If the angle of the "V "
is  , then the width of the strip is
  







2
tan
h
H
B

…………….…………….. (24)
Fig. (17) - Definition Sketch for the triangular Weir.
Therefore, the discharge is

 dh
h
H
h
2
tan
g
2
2
Q
H
0
2
/
1
th  


Integrating over the total head on the weir, we have
2
/
5
H
0
2
/
5
2
/
3
th H
2
tan
g
2
15
8
h
5
2
h
H
3
2
2
tan
g
2
2
Q











However, a coefficient of discharge must still be used with the
basic equation. Hence we have
2
5
2
2
15
8 /
act H
tan
g
Q

 …………….. (25)
Experimental results with water flow over weirs with 
 60
 and
cm
2
H  indicate that d
C has a value of 0.58. Hence the
discharge equation for the triangular weir with these limitations is
2
5
2
2
179
0 H
tan
g
.
Qact

 …………….. (26)

3- Cipolletti Weir
In order to avoid correcting for end contractions a Cipolletti weir is
often used. It has a trapezoidal shape with side slopes of four

vertical on one horizontal (4V:1H). The additional area adds
approximately enough to the effective width of the stream to
offset the lateral contraction.
B
/2 H
1H : 4V
H
H/4

Fig. (18) - Cipolletti Weir
3- Board-crested Weir
A broad crested weir is a structure in an open channel that has a
horizontal crest above which the fluid pressure may be
considered hydrostatic. It is usually built of concrete. One of its
advantages is that it is rugged and can and can stand up well
under field conditions. A typical configuration is shown in Fig.(19).
The operation of a broad-crested weir is based on the fact that
nearly uniform critical flow is achieved in the short reach above
the weir block. If

( 08
.
0
L
/
H w  , viscous effects are important, and the flow is sub
critical over the weir. However, if 5
.
0
L
/
H w  the streamlines are
horizontal). If the kinetic energy of the upstream flow is negligible,
then
g
2
V 2
1  1
y and the specific energy is 1
1
2
1
1 y
y
)
g
2
/
V
(
E 

 .
Observations show that as the flow passes over the weir block, it
accelerates and reaches critical conditions c
2 y
y  and
0
.
1
F )
2
(
r  . The flow does not accelerate to supercritical
conditions ( 0
.
1
F )
2
(
r  ).
The Bernoulli equation can be applied between point  upstream
of the weir and point  over the weir where the flow is critical to
obtain:
g
2
V
P
y
g
2
V
P
H
2
C
w
cr
2
1
w 




or, if the upstream velocity head is negligible
g
2
V
g
2
)
V
V
(
y
H
2
cr
2
1
2
cr
cr 



Young pag. 442

Fig. (19) - Broad Crested Weir Geometry.
However, because cr
cr
2 y
g
2
V
V 
 , we find that cr
2
cr y
g
V  so
that we obtain
2
y
y
H cr
cr 

or H
3
2
ycr 
Thus, the flow rate is
2
/
3
cr
cr
cr
cr
cr
2
2 y
g
b
y
g
y
b
V
y
b
V
y
b
Q 



It is possible to derive a straightforward expression for th
Q in terms of
H
2
/
3
2
/
3
.
th H
b
g
3
2
Q 






and the actual discharge, act
Q , is given by the weir equation
2
3
2
3
2
3
544
0
3
2 /
d
/
d
/
.
act H
g
b
C
.
H
g
b
C
Q 







……………..
(27)
Where typical values of d
C , the broad-crested weir coefficient,
can be obtained from the following equation,

2
1
1
65
0
/
w
eir
d broadw
)
H/P
(
.
C


…………….. (28)
IV - Stagnation Tube
Consider a curved tube such as that shown in Fig. (20). Note how
some stream lines move to the right and some to the left. But
one, in the center, goes to the tip end stops. It stops because at
this point the velocity is zero- the fluid does not move at this one
point. This point is known as the stagnation point. When
Bernoulli's equation is applied between points  and , note
that 2
1 z
z  , it reduces to
2
2
1
2
2
1
2
1














 V
P
V
P 
 ……………………… (29)
Also note that the velocity at point  is zero (a stagnation point).
Hence, Eq. (29) reduces to
 
1
2
2
1
2
1
P
P
V 


By the equations of hydrostatics, (where the streamlines are
straight and parallel), d
g
P1 
 and  
d
h
g
P2 
  . Therefore,
   
 
d
d
h
g
V
2
/
1 2
1 

 

Which reduces to:

h
g
2
V1 
Thus it is seen that a very simple device such as this curved tube
can be used to measure the velocity of flow.
Fig (20) a- Stagnation Tube b- Pitot Tube
V- Pitot tube
The Pitot tube, named after the French hydraulic engineer who
invented it, is based on the same principle as the stagnation tube,
but it is much more versatile than the stagnation tube. The "Pitot
tube" has a pressure tap at the upstream end of the tube for
sensing the stagnation pressure. There are also ports located

several tube diameters downstream of the front end of the tube
for sensing the static pressure in the fluid where the velocity is
essentially the same as the approach velocity. When Bernoulli's
equation is applied between points  and  in Fig. (20), we
have
g
2
V
z
g
P
g
2
V
z
g
P 2
2
2
2
1
1



















But 1
V = 0, so solving that equation for 2
V gives

























 2
1
1
2 z
g
P
z
g
P
g
2
V


………………….. (30)
Here V
V2  and h
z
g
P



. Hence, we obtain
)
h
h
(
g
2
V 2
1 
 …………………………….…………….. (31)
Where V is the velocity of the stream and 1
h and 2
h are the
piezometric heads at points  and  respectively.
By connecting a pressure gage or manometer between taps that
lead to points 1 and 2 we can easily measure the flow velocity
with the Pitot tube. A major advantage or the Pitot tube is that it
can be used to measure velocity in a pressure pipe; a simple
stagnation tube not convenient to use
in such a situation. In gas flow measurement, where a single
differential pressure gage is connected across the taps, Eq. (31)
simplifies to g
/
P
V 

 , where P
 is the pressure difference
across the taps.

Note:
 The Pitot - static tube is used mainly in
laboratory channels, wind tunnels, pipes and in
pressure conduits of small dimensions.
 It is suitable only for high velocity
measurements (greater than 105 m/s, since
low velocities give a head difference too small
to measure accurately

Fig. (21) - Details and Installation of Pitot tube.
V - Jet Trajectory
A free liquid jet in air will describe a trajectory, or path under the
action of gravity, with a vertical velocity component, which is
continually changing. The trajectory is a streamline, and if air
friction is neglected, Bernoulli's equation may be applied to it, with
all pressure terms zero. Thus, the sum of the elevation and
velocity head must be the same for all points of the curve.
Z
Vo (z)
Vo ( x )= V0 cos 
Vo
2
( x ) / 2g
Vo
2 / 2g
Vo ( x )
P
EGL
V 2 / 2g
Jet axis
Vx = Vo ( x )=
constant
& Vz = Vo ( z ) – g t
Jet Trajectory

Fig. (22) - Jet Trajectory.

If o
V is the initial velocity of the jet as it leaves the nozzle
(Fig.22), the EGL will be a horizontal line at distance g
2
/
V2
o
above the nozzle.
Applying Newton's equations of uniformly accelerated motion to a
particle of the liquid passing from the nozzle to point P, whose
coordinates are z
,
x in time t gives:
t
V
X o
X
 ……………………… (32)
and )
t
g
(
2
1
t
V
Z 2
Zo

 ……………………… (33)
Substituting the value of t in Eq. (33) gives:
2
2
X
X
Z
X
.
V
2
g
X
V
V
Z
o
o
o


 ……………………… (34)
Setting 0
dX
/
dZ  , max
Z occurs when
g
V
V
X o
o Z
X 
 ……………………… (35)
Substituting the value of X into Eq. (34) gives:
g
V
Z o
Z
max
2
2
 ……………………… (36)

Note:
Eq. (34) is that of an inverted parabola having its vertex at
g
V
V
X O
O
O
Z
X 
 and
g
2
V
Z
2
ZO
O 

Representation of the EGL&HGL for Frictionless Flow in a Duct
EGL&HGL for Flow from a Tank

Orifice-meter Construction th
d Q
.
C
Q 

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