10.3-Pipe Head Loss, fluid mech
[ Pobierz całość w formacie PDF ]
Pipe Head Loss
10.3 Pipe Head Loss
This section presents the DarcyWeisbach equation, which is used for calculating head loss in a straight run of
pipe. This equation is one of the most useful equations in fluid mechanics.
Combined (Total) Head Loss
Pipe head loss is one type of head loss; the other type is called component head loss. All head loss is classified
using these two categories:
(10.4)
Component head loss
is associated with flow through devices such as valves, bends, and tees.
Pipe head loss
is
associated with fully developed flow in conduits, and it is caused by shear stresses that act on the flowing fluid.
Note that pipe head loss is sometimes called major head loss, and component head loss is sometimes called
minor head loss. Pipe head loss is predicted with the DarcyWeisbach equation.
Derivation of the Darcy-Weisbach Equation
To derive the DarcyWeisbach equation, start with the situation shown in Fig. 10.4. Assume fully developed and
steady flow in a round tube of constant diameter
D
. Situate a cylindrical control volume of diameter
D
and
length
L
inside the pipe. Define a coordinate system with an axial coordinate in the streamwise direction (
s
direction) and a radial coordinate in the
r
direction.
Figure 10.4
Initial situation for the derivation of the Darcy-Weisbach equation.
Apply the momentum equation (6.5) to the control volume shown in Fig. 10.4.
(10.5)
1 of 3
1/15/2009 12:59 AM
Pipe Head Loss
Select the streamwise direction and analyze each of the three terms in Eq. (10.5). The net efflux of momentum is
zero because the velocity distribution at section 2 is identical to the velocity distribution at section 1. The
momentum accumulation term is also zero because the flow is steady. Thus, Eq. (10.5) simplifies to Σ
F
= 0.
Forces are shown in Fig. 10.5. Summing of forces in the streamwise direction gives
(10.6)
Figure 10.5
Force diagram.
Figure 10.4b shows that sin α = (
z
/
L
). Equation (10.6) becomes
(10.7)
Next, apply the energy equation 7.29 to the control volume shown in Fig. 10.4. Recognize that
h
p
=
h
t
= 0,
V
1
=
V
2
, and α
1
= α
2
. Thus, the energy equation reduces to
(10.8)
Combine Eqs. (10.7) and (10.8) and replace
L
by
L
. Also, introduce a new symbol
h
f
to represent head loss in
pipe.
(10.9)
Rearrange the right side of Eq. (10.9).
(10.10)
Define a new πgroup called the
friction factor f
that gives the ratio of wall shear stress (τ
o
) to kinetic pressure
(ρ
V
2
/2):
(10.11)
In the technical literature, the friction factor is identified by several different labels that are synonymous: friction
factor, Darcy friction factor, DarcyWeisbach friction factor, and the resistance coefficient. There is also another
coefficient called the Fanning friction factor, often used by chemical engineers, which is related to the Darcy
Weisbach friction factor by a factor of 4.
2 of 3
1/15/2009 12:59 AM
Pipe Head Loss
This text uses only the DarcyWeisbach friction factor. Combining Eqs. (10.10) and (10.11) gives the Darcey
Weisbach equation:
(10.12)
To use the DarcyWeisbach equation, the flow should be fully developed and steady. The DarcyWeisbach
equation is used for either laminar flow or turbulent flow and for either round pipes or nonround conduits such
as a rectangular duct.
The DarcyWeisbach equation shows that head loss depends on the friction factor, the pipelengthtodiameter
ratio, and the mean velocity squared. The key to using the DarcyWeisbach equation is calculating a value of the
friction factor
f
. This topic is addressed in the next sections of this text.
Copyright ¨ 2009 John Wiley & Sons, Inc. All rights reserved.
3 of 3
1/15/2009 12:59 AM
[ Pobierz całość w formacie PDF ]