1. PHILADELPHIA UNIVERSITY
Department of Civil Engineering
Hydraulics
(670441)
CHAPTER 3
Water Flow in Pipes
Instructor:
Eng. Abdallah Odeibat
Civil Engineer, Structures , M.Sc.
1
2. 3.1 DESCRIPTION OF PIPE FLOW
In hydraulics, the term pressure pipe flow refers to full
water flow in closed conduits of circular cross sections
under a certain pressure gradient. For a given
discharge (Q), pipe flow at any location can be
described by the pipe cross section, the pipe elevation,
the pressure, and the flow velocity in the pipe.
The elevation (h) of a particular section in the pipe is
usually measured with respect to a horizontal
reference datum such as mean sea level (MSL). The
pressure in a pipe generally varies from one point to
another, but a mean value is normally used at a given
cross section. In most engineering computations, the
section mean velocity (V) is defined as the discharge
(Q) divided by the cross-sectional area (A). 2
3. 3.2 THE REYNOLDS NUMBER
Near the end of the nineteenth century, British
engineer Osborne Reynolds performed a very carefully
prepared pipe flow experiment.
Figure shows the schematics of a typical setup for the
Reynolds experiment. A long, straight, glass pipe of
small bore was installed in a large tank with glass
sides. A control valve (C) was installed at the outlet
end of the glass pipe to regulate the outflow. A small
bottle (B) filled with colored water and a regulating
valve at the bottle's neck were used to introduce a fine
stream of colored water into the entrance of the glass
pipe when the flow was initiated. 3
5. Water in the large tank was allowed to settle very
quietly in a room for several hours so that water in
every part of the tank became totally stationary. Valve
C was then partially opened to allow a very slow flow
in the pipe. At this time, the colored water appeared as
a straight line extending to the downstream end,
indicating laminar flow in the pipe. The valve was
opened up slowly to allow the pipe flow rate to increase
gradually until a certain velocity was reached; at that
time, the thread of color suddenly broke up and mixed
with the surrounding water, which showed that the
pipe flow became turbulent at this point.
5
6. Reynolds found that the transition from laminar to
turbulent flow in a pipe actually depends not only on
the velocity but also on the pipe diameter and the
viscosity of the fluid. Furthermore, he postulated that
the onset of turbulence was related to a particular
index number. This dimensionless ratio is commonly
known as the Reynolds number (NR) and it can be
expressed as
When expressing Reynolds number for pipe flow, D is the
pipe diameter, V is the mean velocity, and v is the
kinematic viscosity of the fluid, defined by the ratio of
absolute viscosity (µ) and the fluid density (ρ).
6
7. It has been found and verified by many carefully
prepared experiments that for flows in circular pipes
the critical Reynolds number is approximately 2,000.
At this point, the laminar pipe flow changes to a
turbulent one.
The transition from laminar to turbulent flow does not
happen at exactly NR = 2,000 but varies from
approximately 2,000 to 4,000 based on differences in
experimental conditions.
This range of Reynolds number between laminar and
turbulent flow is commonly known as the critical zone.
7
8. Under ordinary circumstances, water loses energy as it
flows through a pipe. A major portion of the energy loss
is caused by
1. friction against the pipe walls and
2. viscous dissipation occurring throughout the flow.
Wall friction on a moving column of water depends on
the roughness of the wall material (e) and the velocity
gradient [(dV/dr)] at the wall. For the same flow rate, it
is evident in next Figure that turbulent flow has a
higher wall velocity gradient than that of laminar flow;
hence, a higher friction loss may be expected as the
Reynolds number increases.
8
9. At the same time, momentum transfer of water
molecules between layers is intensified as the flow
becomes more turbulent, which indicates an increasing
rate of viscous dissipation in the flows. As a
consequence, the rate of energy loss in pipe flow varies
as a function of the Reynolds number and the
roughness of the pipe wall.
9
12. 3.4 ENERGY IN PIPE FLOW
Water flowing in pipes may contain energy in various
forms. The major portion of the energy is contained in
three basic forms:
1. Kinetic Energy
2. Potential Energy, and
3. Pressure Energy.
12
13. The three forms of energy may be demonstrated by
examining the flow in a general section of pipe, as
shown. This section of pipe flow can represent the
concept of a stream tube that is a cylindrical passage
with its surface everywhere parallel to the flow
velocity; therefore, the flow cannot cross its surface.
13
14. Bernoulli equation:
Therefore, the algebraic sum of the velocity head,
the pressure head, and the elevation head
accounts for nearly all the energy contained in a unit
weight of water flowing through a particular section of
pipe. In reality, however, a certain amount of energy
loss occurs when the water mass flows from one
section to another. Accounting for this loss in
engineering applications is discussed next.
14
16. Figure depicts schematically the heads at two locations
along a pipeline. At section 1, the upstream section, the
three heads are V2/2g, P1/ˠ, and h1. (Note that the
energy per unit weight of water results in a length or
height dimensionally.)
The algebraic sum of these three heads gives the point
a above the energy datum. The distance measured
between points a and b represents the total head, or
the total energy contained in each unit weight of water
that passes through section 1.
16
17. During the journey between the upstream and
downstream sections, a certain amount of hydraulic
energy is lost because of friction (i.e., primarily
converted to heat). The remaining energy in each unit
weight of water at section 2 is represented by the
distance between points a' and b' in Figure. Once
again, this is the total head and is the sum of the
velocity head, the pressure head, and the elevation
head.
17
18. The elevation difference between points a' and a"
represents the head loss (hi), between sections 1 and
2. The energy relationship between the two sections
can be written in the following form
This relationship is known as the energy equation, but
occasionally it is mistakenly called Bernoulli's
equation (which does not account for losses or
assumes they are negligible). For a horizontal pipe of
uniform size, it can be shown that the head loss
results in a pressure drop in the Pipe because the
velocity heads and the elevation heads are equal.
18
19. Figure displays a few other noteworthy hydraulic
engineering concepts. For example, a line may be drawn
through all of the points that represent total energy along
the pipe. This is called the energy grade line (EGL). The
slope of the EGL represents the rate at which energy is
being lost along the pipe. A distance V2/2g below the EGL
is the hydraulic grade line (HGL).
19
23. 3.5 LOSS OF HEAD FROM PIPE FRICTION
Energy loss resulting from friction in a pipeline is
commonly termed the friction head loss (hf). This is the
loss of head caused by pipe wall friction and the viscous
dissipation in flowing water.
Friction loss is sometimes referred to as the major loss
because of its magnitude, and all other losses are
referred to as minor losses.
Several studies have been performed during the past
century on laws that govern the loss of head by pipe
friction. It has been learned from these studies that
resistance to flow in a pipe is 23
24. 1. independent of the pressure under which the water
flows,
2. linearly proportional to the pipe length (L),
3. inversely proportional to some power of the pipe
diameter (D),
4. proportional to some power of the mean velocity (V),
and
5. related to the roughness of the pipe, if the flow is
turbulent. 24
25. The most popular pipe flow equation was derived by
Henri Darcy (1803 to 1858), Julius Weisbach (1806 to
1871), and others about the middle of the nineteenth
century. The equation takes the following form:
This equation is commonly known as the Darcy-
Weisbach equation. It is conveniently expressed in
terms of the velocity head in the pipe. Moreover, it is
dimensionally uniform since in engineering practice
the friction factor (f) is treated as a dimensionless
numerical factor; hf and V2/2g are both in units of
length.
25
26. 3.5.1 FRICTION FACTOR FOR LAMINAR FLOW
This indicates a direct relationship between the friction
factor ( f ) and the Reynolds number (NR) for laminar
pipe flow. It is independent of the surface roughness of
the pipe.
26
27. 3.5.1 FRICTION FACTOR FOR TURBULENT FLOW
When the Reynolds number approaches a higher value—
say, NR >> 2,000—the flow in the pipe becomes practically
turbulent and the value of f then becomes less dependent
on the Reynolds number but more dependent on the
relative roughness (e/D) of the pipe.
The quantity e is a measure of the average roughness
height of the pipe wall irregularities, and is the pipe
diameter. The roughness height of commercial pipes is
commonly described by providing a value of e for the pipe
material. It means that the selected pipe has the same
value of f at high Reynolds numbers as would be obtained
if a smooth pipe were coated with sand grains of a uniform
size e. The roughness height for certain common
commercial pipe materials is provided next.
27
34. 3.6 EMPIRICAL EQUATIONS FOR FRICTION
HEAD LOSS
One of the best examples of empirical equations is the
Hazen-Williams equation, which was developed for
water flow in larger pipes (D≥ 5 cm, approximately 2
in.) within a moderate range of water velocity (V≤
3m/sec, approximately 10 ft/sec).
This equation has been used extensively for the
designing of water-supply systems in the Untied
States. The Hazen-Williams equation, originally
developed for the British measurement system, has
been written in the form
34
35. where S is the slope of the energy grade line, or the
head loss per unit length of the pipe (S = hf /L), and Rh
is the hydraulic radius, defined as the water cross-
sectional area (A) divided by the wetted perimeter
(P). For a circular pipe, with A = D2/4 and P = D,
the hydraulic radius is
The Hazen-Williams coefficient, CHW, is not a function
of the flow conditions (i.e., Reynolds number). Its
values range from 140 for very smooth, straight pipes
down to 90 or even 80 for old, unlined tuberculated
pipes. Generally, the value of 100 is taken for average
conditions. The values of CHW for commonly used
water-carrying conduits are listed in Table 35
37. Note that the coefficient in the Hazen-Williams
equation shown in previous Equation, 1.318, has units
of ft0.37/sec. Therefore, Equation is applicable only for
the British units in which the velocity is measured in
feet per second and the hydraulic radius (Rh) is
measured in feet. Because 1.318 ft0.37/sec = 0.849
m0.37/sec, the Hazen-Williams equation in SI units may
be written in the following form:
where the velocity is measured in meters per second
and Rh is measured in meters.
37
39. Another popular empirical equation is the Manning
equation, which was originally developed in metric
units. The Manning equation has been used
extensively for open-channel designs (discussed in
detail in Chapter 6). It is also quite commonly used for
pipe flows. The Manning equation may be expressed
in the following form:
where the velocity is measured in meters per second
and the hydraulic radius is measured in meters. The n
is Manning's coefficient of roughness, specifically
known to hydraulic engineers as Manning's n.
39
40. In British units, the Manning equation is written as
where Rh is measured in feet and the velocity is
measured in units of feet per second. The coefficient in
Equation serves as a unit conversion factor because
1m1/3/sec = 1.486 ft1/3/sec. Next Table contains typical
values of n for water flow in common pipe materials.
40
43. 3.7 FRICTION HEAD LOSS—DISCHARGE
RELATIONSHIPS
Many engineering problems involve determination of
the friction head loss in a pipe given the discharge.
Therefore, expressions relating the friction head loss to
discharge are convenient.
Noting that A = D2/4 and g = 32.2 ft/sec2 for the
British unit system, we can rearrange the Darcy-
Weisbach equation, as
For high Reynolds number, this equation can be
rewritten as
43
45. 3.8 LOSS OF HEAD IN PIPE CONTRACTIONS
A sudden contraction in a pipe usually causes a
marked drop in pressure in the pipe because of both
the increase in velocity and the loss of energy to
turbulence. The phenomenon of a sudden contraction
is schematically represented in Figure
45
46. The loss of head in a sudden contraction may be
represented in terms of velocity head in the smaller
pipe as
46
47. Head loss from pipe contraction may be greatly
reduced by introducing a gradual pipe transition
known as confusor as shown in Figure.
The head loss in this case may be expressed as
47
49. The loss of head at the entrance of a pipe from a large
reservoir is a special case of loss of head resulting from
contraction. Because the water cross-sectional area in
the reservoir is very large compared with that of the
pipe, a ratio of contraction of zero may be taken.
For a square-edged entrance, where the entrance of the
pipe is flush with the reservoir wall as shown in Figure
3.12 (a), the Kc values shown for D2/D1 = 0.0 in Table
3.5 are used.
The general equation for an entrance head loss is also
expressed in terms of the velocity head of the pipe:
49
51. 3.9 LOSS OF HEAD IN PIPE EXPANSIONS
The behavior of the energy grade line and the
hydraulic grade line in the vicinity of a sudden pipe
expansion is schematically depicted in Figure.
51
52. The loss of head from a sudden expansion in a pipe can
be derived from the momentum considerations. The
magnitude of the head loss may be expressed as
The head loss resulting from pipe expansions may be
greatly reduced by introducing a gradual pipe
transition known as a diffusor.
The head loss in this pipe transition case may be
expressed as 52
53. The values of K’E vary with the diffusor angle ( ):
A submerged pipe discharging into a large reservoir is
a special case of head loss from expansion. The flow
velocity (V) in the pipe is discharged from the end of a
pipe into a reservoir that is so large that the velocity
within it is negligible. 53
54. From general case Equation we see that the entire
velocity head of the pipe flow is dissipated and that the
exit (discharge) head loss is
where the exit (discharge) loss coefficient Kd = 1.0
54
55. 3.10 LOSS OF HEAD IN PIPE BENDS
Pipe flow around a bend experiences an increase of
pressure along the outer wall and a decrease of
pressure along the inner wall.
55
56. In hydraulic design the loss of head due to a bend, in
excess of that which would occur in a straight pipe of
equal length, may be expressed in terms of the velocity
head as
56
57. 3.11 LOSS OF HEAD IN PIPES VALVES
Valves are installed in pipelines to control flow by
imposing high head losses. Depending on how the
particular valve is designed, a certain amount of
energy loss usually takes place even when the valve is
fully open. As with other losses in pipes, the head loss
through valves may also be expressed in terms of
velocity head in the pipe:
The values of Kv vary with the type and design of the
valves. When designing hydraulic systems, it is
necessary to determine the head losses through any
valves that are present. 57
64. 3.12 METHOD OF EQUIVALENT PIPES
The method of equivalent pipes is used to
facilitate the analysis of pipe systems containing
several pipes in series or in parallel. An
equivalent pipe is a hypothetical pipe that
produces the same head loss as two or more pipes
in series or parallel for the same discharge. The
expressions presented for equivalent pipes
account for the losses from friction only.
64
65. 3.12.1 PIPES IN SERIES
Suppose we employ the Darcy-Weisbach equation to
solve the series pipe problem. In terms of the
discharge (Q), Equation becomes
65
