12.2 Flow in Pipes Revisited
d.l
1.0
OJ)
ta
..c: 0.9
u
cJJ
:.a 0.8
d.l
;;.~
--a
S 0.6
S ;:s
u 0.5
C;
I::
parabolic flow
/
.9 0.4
~/~
'"
I::
d.l
S 0.3
/ '"
:.a
~
I
0.2
~
I::
,
0
Z
'"
0.1
/
..,
,
,
,I
\
379
1.0 11
ta
0.9 "5
cJJ
:.a
0.8 :>.
rs
0.7 ~
S
0.6 ' *
0.5 ~
o
. til
0.4 ~
S
OJ :.a
§
0.2 Z
0.1
0.0 -1G....,.."""'j=-,--..,--..,--,-,---,----,--,~,__,.__,.__,.___,_,__,___,_,__+_ 0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Non-dimensional distance from pipe axis
Fig. 12.2: Non-dimensional cumulative discharge for flow through pipe
For the parabolic velocity profile (Eq. 12.1) in the tube, the shear stress, T,
on the tube wall becomes:
or using Eq. (2.104) for average velocity u we get:
After some manipulation we can represent t.he shear stress, T, as:
in which C f is the skin friction coefficient:
where Reynolds number, Re, is defined as:
'liD
Re=-.
l/
(12.5)
(12.6)
(12.7)
( 12.8)
(12.9)
d.l
1.0
OJ)
ta
..c: 0.9
u
cJJ
:.a 0.8
d.l
;;.~
--a
S 0.6
S ;:s
u 0.5
C;
I::
parabolic flow
/
.9 0.4
~/~
'"
I::
d.l
S 0.3
/ '"
:.a
~
I
0.2
~
I::
,
0
Z
'"
0.1
/
..,
,
,
,I
\
379
1.0 11
ta
0.9 "5
cJJ
:.a
0.8 :>.
rs
0.7 ~
S
0.6 ' *
0.5 ~
o
. til
0.4 ~
S
OJ :.a
§
0.2 Z
0.1
0.0 -1G....,.."""'j=-,--..,--..,--,-,---,----,--,~,__,.__,.__,.___,_,__,___,_,__+_ 0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Non-dimensional distance from pipe axis
Fig. 12.2: Non-dimensional cumulative discharge for flow through pipe
For the parabolic velocity profile (Eq. 12.1) in the tube, the shear stress, T,
on the tube wall becomes:
or using Eq. (2.104) for average velocity u we get:
After some manipulation we can represent t.he shear stress, T, as:
in which C f is the skin friction coefficient:
where Reynolds number, Re, is defined as:
'liD
Re=-.
l/
(12.5)
(12.6)
(12.7)
( 12.8)
(12.9)