steady conduit flow

8/10/2019 Steady Conduit Flow

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3. Closed Conduit Flow

Flow in closed conduits (pipe, if conduit is circular in section, and duct otherwise) differs from

that of open channel flow in the mechanism that derives the flow. In the case of open channel

flow, flow occurs due to the action of gravity. In closed-conduit flow, however, although gravity

is important, the main driving force is the pressure gradient along the flow. The emphasis of this

section will be on pipes.Flow in pipes is an example of internal flow, i.e., the flow is bounded by the walls, in contrast to

external flow where the flow is unbounded. For internal flows, the fluid enters the conduit at one

point and leaves at the other. t the entrance to the conduit there appears what is !nown as

entrance regionwith in which the viscous boundary layer grows and finally at the downstream

end of this region covers the entire cross section. The flow beyond the entrance region is said to

have fully developed. The fully developed flow is characteri"ed by a constant velocity profile (for

a steady flow), a linear drop in pressure with distance, and a constant wall shear stress.

The entrance length isa function of

#eynolds number andis given by relations

below$

Re06.0d

Le

for laminar flow, and

6/1Re4.4

d

Le

for turbulent flow.

%here

vd

Re =

Laminar flow in

pipes

#ecall that flow can be classified into one of two types, laminar or turbulent flow (with a smalltransitional region between these two). The non-dimensional number, the #eynolds number, #e,

is used to determine which type of flow occurs$

&aminar flow$ #e '

Transitional flow$ ' #e ' *

Turbulent flow$ #e + *

Derivation of basic equations of steady laminar flow in pipes


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onsider a case of steady laminar flow in a circular pipe shown below$

ince the flow is steady

velocity distribution remains

the same through out the

length of the pipe. /ence

acceleration of the flow is

"ero. /ence the sum of all

forces for the fluid element

shown should be "ero.

( ) rzpds

d

onlysoffunctionaispandds

dz

s

z sinrsin

ds

dp

rsds

dpsrsinsr

rAandsAWbutsinWsr*Asds

dpppA

22

2

+=

=

=

+=

=++

===

+

(

,

(

,

)(

)(

but for laminar flowdy

dv=

ubstituting this and simplifying one obtains the relationship for velocity as$

)pz(ds

d

4

rRV

22

+

=

Thus the velocity distribution in a circular pipe under laminar flow condition is parabolic, with

maximum value at the center.

)pz(ds

d

4

RV

2

max +=

For a hori"ontal pipe

ds

dp

4

RV

2

max

=

The discharge through the pipe is obtained as

0

*

*

R)pz(

ds

ddr.r)pz(

ds

drR

R

+=

+

=

The average velocity,


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!"

V#2$$$

%

"

V#2

!

$$

ds

d$or)zp(

ds

d

2

V)zp(

ds

d

#2

")pz(

ds

d

&

R

A

V

2

'

2

f

2

'

2

max22'

=

==

=

=+

=+=+==

%is is *no+n as t%e $a,en -oiseuille /ormula for !aminar flo+

This e1uation for head loss due to friction is commonly written as

,

V

"

!

Re%f

2* =

Turbulent Flow

In turbulent flow there is no longer an explicit relationship between mean stress and mean

velocity gradient u3r (because momentum is transferred more by the net effect of random

fluctuations than by viscous forces). /ence, to relate 1uantity of flow to head loss we re1uire an

empirical relation connecting the +all shear stress and the average velocity in the pipe.

For turbulent flow, the boundary shear stress is ta!en as (

(V

o

= and the derivation of thee1uation for the friction head loss proceeds in the same way as in the case of laminar flow.

onsider a segment of an inclined circular pipe conveying a fluid of density and viscosity0,

in 4 5 6"3&

For steady uniform flow, since there is no acceleration, F 5 m a1

(78 7) 9 6" 8 :o7& 5 , where 7 is the wetted perimeter

ubstituting 67 5 (7- 7) and dividing the whole expression by , one gets

6796" 5 :o&3# where # 5 37/ence (67 96")3& 8 ;3


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graphical summary of past experimental results has been presented by moody. This chart,

!nown as the Aoody diagram, is a plot of the friction factor as a function of #eynolds number

and the relative roughness of the pipe wall, i.e. B3> where B is the roughness in consistent units.

n empirical e1uation for the friction coefficient is also given by olebroo! and %hite,

+=

fR

.

".lo,

f e

C,

D@

, , which applies in both smooth and rough turbulent "ones.

Hazen-Williams FormulaThe /a"en-%illiams Formula has been developed specifically for use with water and has been

accepted as the formula used for pipe-flow problems in Eorth merica. It reads

3* for pipes

Local Losses (Minor Losses)In addition to head loss due to friction there are always head losses in pipe lines due to bends,Gunctions, valves etc. uch losses are called Ainor losses. For completeness of analysis these

should be ta!en into account. In practice, in long pipe lines of several !ilometers their effect may

be negligible but for short pipeline the losses may be greater than those for friction.

&ocal losses are usually expressed in terms of the velocity head, i.e.

,

V*% ii

= where !iis the minor loss coefficient

Losses at Sudden Enlargement

onsider the flow in the sudden enlargement, shown in figure below, fluid flows from

section to section . The velocity must reduce and so the pressure increases (as follows

from =ernoulli). t position H turbulent eddies occur which give rise to the local headloss.

pply the momentum e1uation between positions and to give$A- 2A21 :(V2- V)

Eow use the continuity e1uation to remove . (i.e. substitute 1 A2V2)A- 2A21 :A2V2(V2- V)

#earranging gives ( )(,(,( VV

,

V

,

=

*


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2


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D


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0


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Multiple ipe s!stemsIn most practical pipe-flow problems the system constitutes multiple pipes Goined in different

ways. uch complex systems can be one or a combination of the following types

i) pipes in series$ here one pipe ta!es the fluid after

the other so that the same flow rate passes through

out the entire pipe system.

ii) pipes in parallel" in paraIle pipes two or more

pipes branch from a point (node) and reGoin some

distance downstream. /ence at the node the flow is

divided into the pipes whereas the pipes flow underthe same energy difference between the nodes.

iii) #ranching pipes$ such pipes branch off from the main

and may return to it. Typical example is pipes that

convey flow from multiple reservoirs.

iv) ipe networ$s" such a system consists of pipes

interconnected in such a way that the flow ma!es a circuit.

ipes in series

In such a system the same flow passes through all the pipes involved and hence the usual

problems are either$

To determine for a given head /, orTo determine the re1uired head / to maintain a certain flow rate.

The latter problem is relatively simple as the friction coefficients for each pipe can easily be

computed.

For datum through =. the energy e1uation including the loss terms ta!es the form$


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,

V*

,

V

"

!f

,

V*

,

V

"

!f

,

V*

,

V

"

!f

,

V*%

%$

%,

V.

,

V.

etcexa

enBA!

BA!

BA!

BB

B

AA

A

@

@

@

@

@

@

@

,

,

,

,,

,

,

,)(

)(

)(

++++++=

=

+++=++

ince the flow rate is the same through out the pipes, the above e1uation can be reduced to

+++

+

++=

*

@

@

C

@

@

@*

@

(

C

(

(

(*

,

,

C

,

,

,*

,

,

(

(

(

,2

"

*

"

!f

"

*

"

!f

"

*

"

!f

"

*

,

$ etcexen

To determine the flow rate, since #e is not !nown, assume values of the friction coefficient forthe pipes and compute the value of from the e1uation above. %ith this value of compute #e

and based on B3> determineffor each pipe. This iterative procedure is repeated until the assumed

and computed values of the friction coefficient are closer to each other.

ipes in parallel

In such arrangements the flow must satisfy$

i) 5 l 9 9 @ ii) hf(-=)5 hfl5 hf5 hf@

The common types of problems and the recommended procedures are given below.

i) to determine the discharge for a given head difference between and =. ince in such a case,

the head loss is !nown, one can write

,

V

"

!f%

f(

(

,

,

,

,,= and solve for


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#ranching pipes

uch an arrangement ofpipes falls in neither of the

above two (i.e. parallel orseries) categories. The pipes

do not also from a networ!

of complete loops. typical

example is the three-

reservoir problem shown in

the figure.

The problem is often to find

the flow rate (including the

direction) in each pipe. s

the elevation of the /L& at

the Gunction is not !nown,the flow can not be readily

computed. /ence the

procedure for solution starts by assuming a value for this head at the Gunction. The flow rate in

each pipe is then computed for the assumed head at the Gunction. The flow rates computed in sucha way are then chec!ed if they satisfy continuity. If the sum of the discharges in the pipes is less

than "ero (with flow away from the Gunction ta!en negative), then this is means the assumed head

is too high and it is reduced for the next trial. The procedure is repeated until the sum of the flow

rates is very close to "ero.

ipe networ$s

7ipes that are interconnected in such a way that they ma!e loops (or circuits) form a networ!. In

such systems the flow in any of the pipes may come from different circuits and as such it is not

simple to !now the direction of flow by observation. 7ipe networ! problems involve the analysisof existing systems, i.e. the determination of flow rate in each pipe, pressure at Gunctions (or

nodes), the head losses in the pipes and the selection of appropriate material and si"e.

The solution of networ! problems always uses iterative procedures that ma!e use of the following

two facts$

o the flow into a Gunction must e1ual the flow out of the Gunction, i.e. at each node (and for

the entire system) continuity must be satisfied,

o the algebraic sum of the head losses around any circuit must add up to "ero.

=elow is outlined a method (commonly !nown as the /ardy-ross method. after 7rof.

/ardy ross)

o by careful inspection of the networ!, assume a reasonable distribution of flow rate in the

pipes so that continuity is satisfied at each node,o compute the head loss in each pipe. For this either the >arcy-%eisbach e1uation can be

used with the friction coefficient determined from the Aoody diagram, or other methods

as discussed below. The >arcy- %eisbach e1uation can be reformulated as

>at


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(?(g

,r

(

,

(

,

(

(

(

(

=

+=

=

+=

+=

n"

!f+%ere

r"

!f

,AV

"

!f

,% n!

Industrial (commercial) pipe-friction formulas are also used in practice, which are generally of

the form$

m

n

n

n

!

"

!Rr+%ere

r"

!R%

=

==

# is a resistance coefficient, which, in the case of the /a"en-%illiams formula is given as

# 5 l.2DC3,

n 5 .0C, and m 5 *.0D* and depends upon the roughness and is given in the following table.

7ipe material and condition

Jxtremely smooth, straight pipes? asbestos-cement *

steady conduit flow

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