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 the44 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

m hzg

vPhz

gvP

lp ,22 2

222

1

211

Pressure Drop in Laminar Flow

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

2

2

02

r Ad : since

rdl dldAlp

or

(1) .................. )rdl (dA)dllp

p(pdA

And assume (p) changes with (l) only

.

dldp

lp

.,e.i

(2) ................... r

.dldp

2

(5) ..................... dr]dldp

[r

dv

(4) .................... ]dldp

[r

drdv

(3) ................... r

.dldp

drdv

2

2

2

(6) ............ constantr

]dldp

[v

4

2

From boundary conditions: 

0r at vv

Rr at 0v

max

(7) .................... R

]dldp

[vconstant max

4

2

(8) .................... ])Rr

(1[4R

]dldp

[v 22

(9) ....................... )Rr

(1v

v 2

max

Eq. (8) /(7)

1.1.2 Relation between vav. & vmax :

]42

[2

)(2

2)1(1

1

2

42

2max

2

3

2max

02

2

max2

RRR

Rv

dr Rr

rRv

rdr Rr

vR

(10) .................. vdAA

v

Rr

0r

Rr

r

avg

(11) ............................ flow laminar for 2

maxv avgv

From (7) in (11)

(13) ................... Poiseuille - Hagen D

vl32p 2

(7) .................... R

]dldp

[vconstant max

4

2

(12) .............. pipe of diameter inside 2R D ; 32D

]dldp

[v2

avg

Example:

1.1.3 Use of friction factor (f) for friction losses determination

(14) ,........ N/m , D2Lvf4

P

2v

surface theat stressshear

2/v RL2

R.P

v21 times )( density of product

areaunit surface wettedforce) tionforce(fric drag

f

22

f

2

2

2

2

equation Weisbach -Darcy

(16) ........ Re64

f & Re16

f

factor friction Moody'ff4 ; Re64

'f

Re64

f4

(15) ................... D2Lvf4

DvL32

Moodyfanning

2

2

•For laminar flow sub. ΔPf by Hagen-Poiseuille in Darcy’s eq. :

•For turbulent flow f = φ( Re, ϵ) ; ϵ is roughness factor•f is predicted from Moody diagram or fanning diagram

Flow in viscous sub-layer near rough and smooth walls

Colebrook equation graphic representation of Moody DiagramExample:

Ans. a)

b)

Example:

Example: