> 4000 turbulent and water/t… · equation” by well estimating of “ fricti factr , f “ ;...

Lect.No.2 Asst.Prof.Dr. Jaafar S. Maatooq

1st Semester Flow Dynamics in Closed Conduit (Pipe Flow) 1 of 14

The flow in closed conduit ( flow in pipe ) is differ from this occur in open

channel where the flow in pipe is at a pressure (does not have a free surface ) .

The flow in pipe can be demonstrated such as :-

- Laminar flow ,

- Transitional flow ,

- Turbulent flow .

To distinction between the above features , the well known “ Reynold,s

Number” can be used , according to experiments that given by “ Osborn

Reynold in 19th

century “ .

1-Reynold’s Experiment

In 1883, Osborne Reynolds demonstrated that there are two distinctly

different types of flow by injecting a very thin stream of colored fluid having

the same density of water into a large transparent tube through which water is

flowing. And from the feature of streaming this dye fluid , Reynold give a

number can be considered as a boundary between flow faces , this number is a

function of , flow velocity , fluid density , pipe diameter , and fluid viscosity ,

where ;

R= f (V , ρ , υ (or μ ) , D ) …………………….. (1)

and then , R=VDρ

μ or R=

VD

υ ;

R= Reynolds No.,

μ = dynamic viscosity ,

υ = kinematic viscosity .

See Figure(1) , below for Reynold”s experiments ;



Fig.(1) : Experiments shows the flow state as demonstrated by Reynolds

Observations (dye) Reynolds

Number, Re

Flow

Classification



Transitional/

Turbulent

> 4000

Turbulent



2-Viscous (Real) Flow in Conduits ,Head Loss in Pipes from

Friction ( Major Losses)

The head loss between two points in a circular pipe carrying a fluid under

pressure can be found by ; hf=Δp/γ

Where: ∆p = p1 − p2 , and can be measured by using piezometer tubes.

The velocity of the flow can be found by using a Pitot tube. The reading of

the Pitot tube is the total head = pressure head + velocity head



The total “ friction head loss “ (hL) , can be calculated using “ Darcy

Equation” by well estimating of “ friction factor , f “ ; where :-

Also the “ friction head loss“ (hL) , can be calculated by using Hazen William

Equation , where ;



3-Head Loss versus Discharge



The friction factor of “Darcy Equation” can be estimated , using “ Moody

Diagram”

as shown in Fig.(2) , below ;

Fig.(2): Friction Factor estimation as presented by Moody



4-Method to Determine Darcy-Weisbach friction factor ( f )

PIPE FLOWS

Laminar (R < 2,000) Turbulent (R > 4,000)

f = 64/R

Smooth Transitional Wholly Rough

(δv > e) (0.071e ≤ δv ≤ e) (δv < 0.071e)

Turbulent (Smooth):

Prandtle ………..

√

√

for R > 4000 ….. (2)

Blasisus ………..

for 3000 < R < 100000 … (3)

Turbulent ( Transitional) :

Colebrook ……..

√ -

√ ……………… (4)

Turbulent ( Wholly Rough ):

Von- Karamen …

√

………………………. (5)



5-Direct Calculation of Flow Velocity

Combining the “Darcy” and “Colebrook” equations yield’s the explicit

equation for average flow velocity in pipe :-

- √

√ ………………….. (6)

Where S=hf / L and ( ν ) is a kinematic viscosity

When using Eq.4 (Colebrook equation) and due to the implicit form of this

formula for “f” , it can be use the following formula to find a friction factor

which presented by “Moody” :-

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

Eq.7 can be used just with :-

R ranged between 4000 – 10000000

e/D up to 0.01

and from the above limitations the accuracy of “f” resulting is within (+/- 5% ) .



6-Types of Water flow Problems

In design and analysis of pipe systems that involve the use of the “Moody

Diagram” or “Colebrook formula” , it is usually found a three types of

problems in practice . In all these problems the fluid type and roughness of

pipe must be specified . The classification of the three problem can be shown

in the following ;

Problem Type Given Find

1 L , D , Q hL

2 L , D , hL Q

3 L , hL , Q D

The solution of the above problems can be cleared as the following steps

1- The solution of problems of the first type is by using directly the “Moody

Chart” .

2- The solution of problems of the second type obtained by:-

*Assume fully turbulent flow region (high Reynold’s number) , for a given

roughness of pipe .

*From this assumption find “friction factor” .

*By using Darcy formula the flow rate can be obtained .

*The friction factor can be corrected using Moody diagram or Colebrook

equation and the above process is repeated until the solution converges .

3- The solution of problems of the third type will be :-

*Start calculation by assume a pipe diameter .

*The head loss is calculated by this assumption is then compared to the

given head loss .

*The calculation repeated with another pipe diameter until solution obtained.



Swamee and Jain in 1976 suggested the following explicit relations to avoid

iteration . The results from this relation are within 2% with the results

obtained by using Moody chart ;

- ….. ……. (8)

It is valid just for :-

10-6

< e/D < 10-2

3000 < R < 3x108

-

…… (9)

It valid for R > 2000

….(10)

It is valid just for 10-6

< e/D < 10-2

& 5000 < R < 3x108

7-Simplified Equations to Calculate Head Losses in Commercial Pipes

The new empirical equations used for head losses calculation in most

commercial pipes that may be used in practice were submitted by Ibrahim

Can by using in direct solution of head losses without need to use Colebrook

Equation or Moody diagram . These proposed formulas are listed in Table

below :-



Empirical Equation Pipe Type

=0.000934

1.818

D4.821 PVC ( e=0.0015mm )

=0.00103

1.882

D4.963 Steel (e=0.05mm )

=0.00112

1.929

D5.08 Asphalted Cast Iron (e=0.12mm)

=0.00141

1.974

D5.205 Concrete (e=0.5mm)

Note that the above equations used with SI-units where ; Q in (m3/s) , L in

(m) , D in (m) and hL in (m) . The maximum error from the above formulas

compared with measured in practice not exceed +/- 2% .

> 4000 turbulent and water/t… · equation” by well estimating of “ fricti factr , f “ ;...

Documents