Transcript 1.Development of Pipe Correlation

The Alaskan pipeline, a significant accomplishment of the engineering profession,
transports oil 1286 km across the state of Alaska. The pipe diameter is 1.2 m, and
the
44 pumps are used to drive the flow. This chapter presents information for
designing
systems involving pipes, pumps, and turbines.
Typical Applications
· For flow in a pipe, find the pressure drop
or head loss.
· For a specified system, find the flow
rate.
· For a specified flow rate and pressure
drop, determine the size of pipe required.
· For a system with a pump, find the pump
specifications (power, head, flow rate).
· For a specified elevation change and flow
rate, find the power that can be produced
by a turbine.
1.Development of Pipe Correlation
1.1 Analytical Approach
Viscous Flow in Pipes
2
1
2
2
P1 v
P2 v

 z1  hp  
 z2  hl , m
 2g
 2g
Σhl
Major Losses
1. Poisueille eq.
2. Darcy’s eq.
Minor Losses
1. Sudden expansion
2. Sudden contraction
3. Fittings and valves
Pressure Drop in Laminar Flow
Assumptions
incompressible
Newtonian
one dimen.
St.st.
Horizontal
pipe
const.
velocity
The forces on it are:
1- pδA in flow direction
2- p’δA on the reverse direction.
3- friction force acting on its outer-surface.
(on the reverse direction of flow)
.
1.1.1 Force Balance on the Element
p
pdA  (p  dl)dA  (2 rdl)  0
l
or
p
 dldA  2 rdl
l
2
since :
dA   r
And assume (p) changes with (l) only 
dp r

. 
dl 2
..................
(1)
p dp
i.e.,

l dl
...................
(2)
dp r
dv
     .
dr
dl 2
dv r dp
  [ ]
dr 2 dl
r dp
dv  [ ]dr
2 dl
...................
(3)
....................
(4)
.....................
(5)
dp r2
 v  [ ]
 constant ............ (6)
dl 4
From boundary conditions:
v0
at
rR
v  vmax
at
r0
constant  vmax
2
2
dp R
[ ]
dl 4
dp R
r 2
 v  [ ] [1  ( ) ]
dl 4
R
.................... (7)
.................... (8)
Eq. (8) /(7) 

v
vmax
r 2
1( )
R
....................... (9)
1.1.2 Relation between vav. & vmax :
v avg
1
  vdA
.......... ........
A
2
r

R
1
r
 2  v max (1  2 )2 rdr
R r  0
R
2v max r  R
r3

 (r  2 ) dr
2
R r 0
R
2v max R 2 R 4
 2 [  2]
R
2 4R
v
 v  max
avg 2
for laminar
flow
............................
(11)
(10)
From (7) in (11)
constant  vmax
vavg
dp R
[ ]
dl 4
dp D2
[ ]
; D  2R  inside
dl 32
32vl
 p 
D2
2
diameter
.................... (7)
of
pipe .............. (12)
Hagen - Poiseuille ...................
(13)
Example:
1.1.3 Use of friction factor (f) for friction
losses determination
drag force(fric tion force)
f 
P.R2
wetted surface unit area
product of density (  ) times 1 v 2
2
2RL
 v2 / 2
shear stress  at the surface

 v2
2
4 fLv 2
N/m 2 Darcy - Weisbach equation,........ (14)
,
Pf 
2D

•For laminar flow sub. ΔPf by Hagen-Poiseuille in
Darcy’s eq. :
32 vL
4 fLv

.......... ......... (15)
2
D
2D
64
4f 
Re
64
f' 
;
4 f  f'  Moody friction factor
Re
16
64
 ffanning 
&
fMoody 
........ (16)
Re
Re
2
•For turbulent flow f = φ( Re, ϵ) ; ϵ is roughness
factor
•f is predicted from Moody diagram or fanning
Flow in viscous sub-layer near
rough and smooth walls
Colebrook equation graphic
representation of Moody Diagram
Example:
Ans. a)
b)
Example:
Example: