!"#$%&!"'(&$)&*$*+,&
-)%&.&'/&0#1$)2,
*345&6
Note: The two general equations that govern uniform steady fluid
flow in pipes and / or tubings are the continuity equation and the
mechanical energy balance
The Continuity Equation (Simple Mass Balance):
! = r!$!%! = r" $ "%"
#
The Mechanical Energy Balance:
Reynolds Number and Flow Regimes
Reynolds number (Re) is a dimensionless number defined as the
ratio of inertia and viscous forces.
#$ =
!"r
µ
where D is the pipe diameter, u is the average velocity, r is the fluid
density and µ is the fluid viscosity.
Reynolds Number and Flow Regimes
In case of a non-circular cross-section, D must be replaced by the
equivalent diameter, Deq:
æ $+'"" !"#$%&'()*!)+#) ö
0#. = 1+/ = 1 ç
÷
2#%%#3
!,#+&-#%#+
è
ø
In the preceding equation, rH is called the hydraulic radius
Reynolds Number and Flow Regimes
Pipe flow regimes:
Dependence of pipe flow regime on Re
Flow regime
Reynolds
number
Laminar
Re < 2100
Transition
2100 < Re <
4000
Re > 4000
Turbulent
Pressure
Gradient is
proportional to
Q
Variable
Q1.8 ~ Q2
Laminar flow in pipe
Velocity profile
(
-D" ! !
#$%& =
' -%
(µ)
)
Volumetric flow rate (Hagen-Poiseuille equation)
p% ! æ $" - $# ö
&=
ç
÷
'µ è ( ø
p%! æ $" - $# ö
&=
ç
÷
"#'µ è ( ø
• Frictional dissipation
• Maximum velocity (at r = 0)
• Mean velocity
%=
"µ#$
r&
!
%! æ &' ö
( "#$ =
ç÷
)µ è &* ø
$
% = & = & !"#
'
• The Kinetic Energy Term
Kinetic energy per unit mass:
#!
$% = a
!&"
The kinetic energy correction factor (a) accounts for the variation
of local velocity due to the influence of solid boundary. The
mechanical energy balance equation therefore, can be written as
$"
%"!
$!
%!!
&
&
+ a"
+
'" + ( =
+ a!
+
'! + )
r
!&# &#
r
!&# &#
$"
%"!
$!
%!!
&
&
+ a"
+
'" + ( =
+ a!
+
'! + )
r
!&# &#
r
!&# &#
For laminar flow in pipe:
a=!
For turbulent flow in pipe:
a = !"#$ - !"!%
For uniform distribution of velocity:
a =!
Note: u or
! is the average velocity and v is a point velocity
Skin Friction and Form Friction
Skin friction
• If any surface is in contact with a fluid and a relative motion
exists between the surface and the fluid, the transfer of
momentum results in a tangential stress or drag on the surface
that is oriented parallel to the direction of flow. This
phenomenon is called skin friction.
• Skin friction is generated in unseparated boundary layers; for
example, in straight pipes.
Skin Friction and Form Friction
Form friction
• Whenever a fluid changes path to pass around a solid body set
in the flow path, the fluid accelerates and significant frictional
losses consequently occurs because of acceleration and
deceleration of the fluid. This phenomenon is called form drag
or form friction.
• Form friction is an energy dissipation that occurs when boundary
layer separates and form wakes; for example, flow through
valves, fittings, and obstruction such as sudden contraction or
enlargement of cross section.
Friction in Pipes
Friction factor
• Friction factor is a dimensionless wall stress defined as the ratio of
the wall stress to the inertial force per unit area that would result
from the impingement of a stream of density r and velocity u
normally against a wall.
Friction in Pipes
Friction factor
Fanning friction factor:
Darcy friction factor:
$=
t!
# r%"
"
&" =
t!
#
$
% r'
• Evaluation of friction factor
Îö
æ
" #$%#"! = f ç &'( ÷
!ø
è
• where:
Î
roughness of pipe
Î
!
relative roughness of the pipe
Effective surface roughness of some pipe materials
• For laminar flow (Re < 2100)
!"
#=
$%
"#
$! =
%&
fM = fD
f = 0.0085
• Evaluation of Friction Factor
• The friction factor chart is a log-log plot of f versus Re at different
values of relative roughness. In addition, friction factor can be
estimated using empirical equations as follows:
• Colebrook-White equation
Î ö !"#$%
æ
& = !"%'%() ç *"#+, ÷ +
- ø ./ &
è
!"#$%&
ü'(&)*+,#+"(&)-,#"*&,.-*#/&#.$&#0*102$(#&*$3+"(&+4&1-4$5&"(&#.$&6"2$1*""78
9.+#$&$:0-#+"(;
ü<*+,#+"(&)-,#"*&)"*&-&=+5$&*-(3$&")&>$&+4&3+?$(&+(&<+30*$&@8A&")&B$**CD4&
6EF&E-(51""7/&G!"&$5+#+"(
• Explicit form Colebrook-White equation (approximate)
ì
Î !"#$% æ
Î #&"% ö ö ü
æ
' = í -#"()(*+ ç ,"!-. *+ ç ,"!-. +
÷ ý
/
01
/ 01 ø ÷ø þ
è
è
î
-!
• Blasius equation (for hydraulically smooth surface; 4000 < Re <
105)
!"!%&
'=
()!"#$
• Churchill equation (for smooth and rough tubes; Re > 4000)
!"#
é
$
Î æ % ö ù
= -&'() ê!"*% + ç
÷ ú
+ è ,- ø úû
.
êë
• Evaluation of Frictional Dissipation (F) in MEB
!
• Skin friction (FS)
$%&'
(" =
!)#*
• Form friction (FF)
• Form friction can be evaluated in terms of loss coefficient, K,
which is defined as the number of velocity heads lost due to
fluids passing through valves, fittings or any obstructions.
Alternately, form friction can also be estimated in terms of the
equivalent length of a pipe that has the same effect (i.e.
pressure drop due to friction) as the valve, fitting, or obstruction
under in the system considered.
• Evaluation of form friction by LOSS COEFFICIENT (K) METHOD
$!
"" = å %
!&#
• Form friction due to valves and fittings
K values for some fittings and valves are available in Perry’s CHE
Handbook. A sample selection is shown below
Type of fitting or valve
Loss coefficient, Kf
45o ell, standard
0.35
45o ell, long radius
0.20
Gate valve
¼ open
24.0
½ open
4.50
¾ open
0.90
fully open
0.17
• Form friction due to sudden changes in cross section
• Loss coefficient (K) for sudden enlargement and sudden
contraction of cross section
Velocity (u) to be
used in equation
K
Sudden
enlargement of
cross section
Sudden
contraction of
cross section
æ
$ ö
%# = ç "- " ÷
$! ø
è
!
æ $ ö
% " = &'( ç # - ! ÷
è $# ø
Upstream
velocity (u1)
Downstream
velocity (u2)
H-I&J055$(&,"(#*-,#+"(&")&,*"44&4$,#+"(K&H1I&
4055$(&$(2-*3$L$(#&")&,*"44&4$,#+"(
• Total frictional dissipation (or total friction loss)
" = "! + å ""
!
é %
ù &
' = ê ($ + ) " + ) # + å ) $ ú
ë *
û !+#
• Evaluation of form friction by EQUIVALENT LENGTH (Le) METHOD
%&"' #(!
"=
!)$*
$! = $" + å $#
• where Le is the equivalent length of valve or fitting, LS is the total
length of straight pipes, and LT is the total equivalent length of
pipes and fittings.
• For a specific pipe fitting or valve, there is a corresponding Le/D.
A sample selection is shown in the following table below
Type of fitting
Le/D
Angle valve (open)
160
Gate valve (open)
6.5
Square 90o elbow
70
Sudden contraction, 4:1
15
Sudden contraction, 2:1
11
Sudden enlargement, 1:4
30
Sudden enlargement, 1:2
20
• The relation ship between K and Le is given by the following
equation:
"!
# = $%
&