LECTURE 1: Review of pipe flow: Darcy-Weisbach,
Manning, Hazen-Williams equations, Moody
diagram
1.1. Important Definitions
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.
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).
Pressure (P) in the pipe varies from one point to another, but a mean value is normally used
at a given cross section.
Mean velocity (V) is defined as the discharge (Q) divided by the cross-sectional area (A)
1.2. 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 dissipiation in
flowing water. It is also called major loss.
The most popular pipe flow equation was derived by Henry Darcy (1803 to 1858), Julius
Weiscbach (1806 to 1871), and the others about the middle of the nineteenth century. The
equation takes the following form and is commonly known as the Darcy-Weisbach
Equation.
hf LV
D
(1.1)
When Reynolds Number (NR) is less than 2000 flow in the pipe is lminar and friction factor is
calculated with the following formula;
f
1
N
(1.2)
When Reynolds Number (NR) is greater or equal to 2000, the flow in the pipe becomes
practically turbulent and the value of friction factor (f) then becomes less dependent on the
Reynolds Number but more dependent on the relative rougness (e/D) of the pipe. The
roughness height for certain common commercial pipe materials is provided in Table 1.1.
Table 1.1 Roughness Heights ,e, for Certain Common Materials
Friction factor can be found in three ways:
1. Graphical solution: Moody Diagram
2. Implicit equations : Colebrook-White Equation
3. Explicit equations: : Swamee-Jain Equation
2
Figure 1.1. Friction factors for flow in pipes: the Moody Diagram. Source: From Hwang et al., Fundamentals of Hydraulic Engineering Systems,
2010
3
Example 1.1(Use of Moody Diagram to find frition factor): A commercial steel pipe, 1.5 m
in diameter, carries a 3.5 m3/s of water at 200C. Determine the friction factor and the flow
regime (i.e. laminar-critical; turbulent-transitional zone; turbulent-smooth pipe; or turbulentrough pipe)
Solution:
To determine the friction factor, relative roughness and the Reynolds number should be
calculated.
For commercial steel pipe, roughness heigth (e) = 0.045 mm
Relative roughness (e/D) = 0.045 mm/1500 mm = 0.00003
V= Q/A=(3.5 m3/s)/[(π/4)(1.5 m)2]=1.98 m/s
NR=DV/ν=[(1.5 m)(1.98 m/s)]/(1.00x10-06 m2/s)=2.97x106
From Moody Diagram, f=0.011
The flow is turbulent-transitional zone.
f=0.011
e/D= 0.00003
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Colebrook-White Equation:
√
log D
.
.
NR √
!
(1.3)
Swamee-Jain Equation :
'
(.
,/. 2.03
)*+ ( /.0 1 6.7 8
45
(1.4)
1.3. Emprical Equations for Friction Head Loss
1.3.1 Hazen-Williams equation:
It was developed for water flow in larger pipes (D≥5 cm, approximately 2 in.) within a
moderate range of water velocity (V≤3 m/s, approximately 10 ft/s). Hazen-Williams equation,
originally developed fort he British measuerement system, has been writtten in the form
1.318:;< =>(. ? (.
(1.5)
S= slope of the energy grade line, or the head loss per unit length of the pipe (S=hf/L)
Rh = the hydraulic radius, defined as the water cross sectional area (A) divided by wetted
perimeter (P). For a circular pipe, with A=πD2/4 and P=πD, the hydraulic radius is
@
=> A πD
/
πD
B
(1.6)
CHW= Hazen-Williams coefficient. The values of CHW for commonly used water-carrying
conduits are given in Table 1.2.
The Hazen-Williams equation in SI units is written in the form of
0.849:;< =>(. ? (.
Velocity in m/s and Rh is in meters.
5
(1.7)
Table 1.2. Hazen-Williams coefficient for different type of pipes
1.3.2. Manning’s Equation
Manning equation has been used extensively open channel designs. It is also quite commonly
used for pipe flows. The Manning equation may be expressed in the following form:
/
F => ?/
(1.8)
n= Manning’s coefficient of roughness. Typical values of n for water flow in common pipe
materials is given in Table 1.3
In British units, the Manning equation is written as
Where V is units of ft/s.
6
.G
F
/
=> ? /
(1.9)
Table 1.3. Manning Rougness Coefficient for pipe flows
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1.4. Friction Head Loss-Discharge Relationships
Head loss equations can be written in the form of dicharge and a resistance coefficient (R) in the
following form.
hf RQJ
(1.10)
X is dimensionless and the dimension of R depends on the friction equation and the unit
system chosen.
1.4.1. Darcy-Weisbach Equation:
hf L(Q/A)
D
LQ
DA
L
DA
Q
(1.11)
R
x=2
1.4.2 Hazen Williams Equation
V B. C. R S (. . S (.
R S Hydraulic radius,
\/] ^_^
`_aa_b c_de_a_
h
S = Slope of EGL, !
L
B = 1.318 in British Systems
B = 0.85 in SI Unit
C = Hazen – Williams coefficient (for concrete C = 130)
A. V Q B. C. A. Q
D (.
R
S
(.
. Q TR
L
h U
L
CW.X2 .D3.X0
.
.GG
BW.X2
RHazen-W
h R H^h_ijW . Q.G
x = 1.85
8
Z .Q.G
1.4.3. Manning’s Equation
V
i
. R S / . S/
Manning’s
constant
Slope of EGL =
ST
L
Hydraulic
Radius
B = 1 in SI System
B = 1.49 in British System
Q A.
h U
B
i
. R S / . S/
(.l L.i
B
.
D2.//
Z . Q
Rmanning
x = 2by using a) Hazen Willimas b) Manning’ s
Example 1.2: Calculate the friction head loss
equation and c) Darcy-Weisbach equation for a commercial steel pipe (new) with 1.4 m
diameter and a flowrate of 3.3 m3/s at 100C water.
a) Hazen Williams Equation
L
.GG
R U W.X2 3.X0 . W.X2 Z
C
.D
B
RU
C = 140
B = 0.85
(((
(W.X2 .(.)3.X0
.
.GG
((.G)W.X2
Z
R = 0.22
h (0.22)(3.3 m /s).G
h 2.0 m
b) Manning’s Equation
RU
(.l L.i
.
B
D2.//
Z
n = from Table 1.3
B = 1 in SI System
RU
(.l (((( e).((.(()
.
(.)2.//
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Z = 0.17
Table 1.2
h 0.17(3.3)
h 1.85 m
c) Darcy Weisbach Equation
R
V = Q/A =
R_ V.D
υ
=
.L
DA
. e/ /]
π(.)
/
(.)(. e)
.(r(st
= 2.14 m/s
= 2.29 r 10j
e = Roughness Height (mm) Table 1.1 e = 0.045 mm
e/D =
((.(/((() e
. e
= 3.2 r 10j
f= 0.013from Moody Diagram
R
.L
DA
=
((.().(((( e)
(l.)(.)(π(.)
/)
h 0.199Q
h 0.199(3.3)
h 2.17 m
10
= 0.199
Example 1.3. : The pressure at point A in the suction pipe is a vacuum of 10 min mercury Q
= 3.0 ft /sec. Total head at point A and B with respect to a datum at the base of the reservoir.
γ`^a_ 62.3 Ib/ft Pressure Head =
P
(s.g) mercury = 13.6
V
, Velocity Head
ρ.
Total Head for point A= zA +
Total Head for point B= zB +
zA zB Velocity for suction pipe;
VA Velocity for discharge pipe
+
+
, Elevation Head
z
PA
ρ.
PB
ρ.
a/ /]_\
π((/)
/
5.5 ft/sec
a/ /]_\
VB 8.59 ft/sec
π(G/)
/
(.r.)(j(/)
(.)
Total Head for point A=
+
+ 25 ft= 14.14 ft
.(.)
.(.)
Total Head for point B = 0 +
(G.)
.(.)
+ (25 15 40 ft)=81.15 ft
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