Chapter 5. Incompressible Flow in Pipes and Channels · 2008-10-21 · Incompressible Flow in Pipes and Channels * Relations between maximum velocity ... Dilute polymer solutions

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

Chapter 5. Incompressible Flow in Pipes and Channels

Shear Stress and Skin Friction in Pipes (전단응력및표면마찰)

* Shear-stress distribution

For fully developed flow,

∑ ←=∴==

0&

FVV abab ββ

from Eq. (4.51):

0=−+−∑ = gwbbaa FFSpSpF

( )aabb VVmF ββ −∑ = &

From Eq. (4.52),

sF−=


0)2()(22 =−+−∑ = τπππ rdLdpprprF

02=+

rdLdp τ

--- Eq. (5.1)

로나누면dLr2π

단면적방향으로의압력은일정)r과는무관 (관의

ww rr == at& ττ

02=+∴

w

wrdL

dp τ--- Eq. (5.2)

Eq. (5.2)에서 Eq. (5.1)을빼면,

or rrw

w⎟⎟⎠

⎞⎜⎜⎝

⎛=

ττrrw

w ττ=


* Relation between skin friction & wall shear

펌프에의한일이없고마찰을고려할경우의 Bernoulli 방정식은

fbb

bbaa

aa hVgZpVgZp

+++=++22

22 αρ

αρ

일반적으로 pa > pb이므로 로표시할

--- Eq. (4.71)

수있고 fully developed flow인

수평관을대상으로하며마찰은유체와관벽사이의 skin friction hfs만존재하므로

ppp ab ∆−=

sababab ppZZVV ∆=∆=== &,,, αα (압력강하는표면마찰에의한것이므로)

이경우 Bernoulli 식은

fss

fssaa hphppp

=∆

+∆−

=ρρρ 즉, --- Eq. (5.4)

From Eq. (5.2), 02=+

∆−

w

wsrL

p τ

LD

Lr

h w

w

wfs

τρ

τρ

42==∴

를소거하면sp∆


* Friction factor (마찰계수), f

여기서정의하는마찰계수 f 는 Fanning friction factor

또다른마찰계수로 Blasius or Darcy friction factor가있는데이는 4f 에해당

222

2/ VVf ww

ρτ

ρτ

=≡ --- Eq. (5.6)

wall shear stressdensity × velocity head

(단위면적당전단력)(단위부피당운동에너지)즉,

skin friction hfs와 friction factor f와의관계:

242 2V

DLfpL

rh s

w

wfs =

∆==

ρτ

ρ--- Eq. (5.7)

22 VLDpf s

ρ∆

=∴DVf

Lps

22 ρ=

∆ --- Eqs. (5.8)-(5.9)or


* Flow in noncircular channels

In evaluating the diameter in noncircular channels, an equivalent diameter (등가지름)

Deq is used.

rH : hydraulic radius (수력학적반지름)Heq rD 4=

pH L

Sr ≡ S : cross-sectional area of channelLp : wetted perimeter

44/2 D

DDrH ==π

π4

4/4/ 22io

oi

ioH

DDDDDDr −

=+−

=ππππ

Deq=Do-Di

44

2 bb

brH ==

Deq=bDeq=D

1) Circular tube: 2) Annular pipes: 3) Square duct:

단면이원형이아닌관의경우 Reynolds number Re또는 friction factor f등의계산시에

D대신 Deq혹은 r 대신 2rH를대입하여계산가능함을의미.


Laminar Flow in Pipes and Channels

* Laminar flow of Newtonian fluids

원형단면을갖는흐름을대상, 속도분포는 centerline에대해대칭

u depends only on r

rrdr

du

w

wµ

τµτ

−=−=

drduµτ −=

Eq. (5.3) 적용r

zcenterline

wall

적분하면 (경계조건: )wrru == at0

( )22

2rr

ru w

w

w −=µ

τ∫∫ −= rr

u

w

ww

rdrr

du0 µτ --- Eq. (5.15)

2

max1 ⎟⎟

⎠

⎞⎜⎜⎝

⎛−=∴

wrr

uu)0at(

2max == rru wwµ

τ --- Eq. (5.17)


Average velocity

--- Eq. (4.11)∫= udSS

V 1

Eq. (5.15) 대입한후적분

rdrdS π2=

( )µ

τµ

τ40

223

wwrw

w

w rrdrrrr

V w =∫ −= --- Eq. (5.18)

이식을 와비교하면, µτ2max

wwru = 5.0max

=u

V --- Eq. (5.19)

In Laminar flow,

Kinetic energy correction factor,

Momentum correction factor,

0.2=α Eq. (4.70)에 (5.15)와 (5.18)을대입해계산

34

=β Eq. (4.50)에 (5.15)와 (5.18)을대입해계산


Hagen-Poiseuille equation

--- Eq. (5.20)µL

DpV s32

2∆=

Eq. (5.7)과 Eq. (5.18)을이용하여 대신보다실제적인 로변환하면,wτ sp∆

232

DVLpsµ

=∆or

VDq4

2π=여기서 이므로 q 와 측정으로부터점도계산가능:sp∆

qLDps

128

4∆=πµ : Hagen-Poiseuille equation

)/(4 DLp ws τ=∆또한 Eq. (5.7)에서 이므로,

DV

wµτ 8

= --- Eq. (5.21)

Eq. (5.21)을 Eq. (5.7)에대입하면 f와 Re 사이의관계가유도됨:

Re1616

==ρµ

VDf --- Eq. (5.22)


* Laminar flow of non-Newtonian liquids

- Power law fluids

반지름 r에따른 velocity profile:

Fig. 5.4. Velocity profiles in the laminar flowof Newtonian and non-Newtonianliquids.

n

drduK−=τ

nrr

Kru

nnw

n

w

w/11

/11/11/1

+−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

++τ


- Bingham model

o

oo

drdu

drduK

ττ

ττττ

<=

>−=−

at0

at

반지름 r에따른 velocity profile:

( ) ⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−= 01

21 ττ

w

ww r

rrrK

u

Fig. 5.5. (a) Velocity profile and(b) Shear diagramfor Bingham plastic flow

- Some non-Newtonian mixtures at high shear violate the zero-velocity (no-slip) b. c.ex) multiphase fluids (suspensions, fiber-filled polymers) “slip” at the wall


Turbulent Flow in Pipes and Channels

viscous sublayer (점성하층):viscous shear ↑, eddy ×

buffer layer (완충층) or transition layer (전이층):viscous shear & eddy 공존

turbulent core (난류중심부):viscous shear ↓, eddy diffusion ↑

C.L.

wall

viscous sublayer buffer layer

turbulent core

Velocity profile for turbulent flow:

much flatter than that for laminar flow

Eddies in the turbulent core: large but low intensity

in the buffer layer: small but high intensity

Most of the kinetic-energy content of the eddies lies in the buffer zone.

C.L.

wall

laminar flowturbulent flow

Re ↑


* Velocity distribution for turbulent flow

In terms of dimensionless parameters

ρτµµ

ρ

ρτ

w

w

yyuy

uuu

fVu

=≡

≡

=≡

+

+

*

*

*2

: friction velocity

: velocity quotient (무차원)

: distance (무차원) y : distance from tube wall

Re based on u* & y)( yrrw +=∴

* Universal velocity distribution equations

i) viscous sublayer:

ii) buffer layer:

iii) turbulent core:

++ = yu05.3ln00.5 −= ++ yu

5.5ln5.2 += ++ yu

intersection으로부터

for viscous sublayer

for buffer zone

for turbulent core

5<+y

305 << +y

30>+yRe > 10,000 이상에서적용가능


* Relations between maximum velocity

& average velocity V

When laminar flow changes to turbulent,

the ratio changes rapidly

from 0.5 to about 0.7,

& increases gradually to 0.87 when Re=106.

max/uV

maxu

For laminar flow, is exactly 0.5.max/uVfrom Eq. (5.19)

* Effect of roughness

Types of roughness

Rough pipe larger friction factor

k : roughness parameter

k/D : relative roughness

For laminar flow, roughness has no effect

on f unless k is so large.

Dkf /&Reofnft'=∴


* Friction factor chart

↑Dk /

Friction factor plot for circular pipes (log-log plot)


* Friction factor for smooth tube* Drag reduction

Dilute polymer solutions in waterdrag reduction in turbulent flow

Application: fire hose (a few ppm of PEO in water can double the capacity of a fire hose)

632.0

62.0

103Re000,3forRe

125.0014.0

10Re000,50forRe046.0

×<<+=

<<= −

f

f

(wide range)

* Non-Newtonian fluids

↑ppm↑n


* Friction loss from sudden expansion

Ke can be calculated theoretically from the momentum balance equation (4.51) and the Bernoulli equation (4.71).

2

2a

efeVKh =

: average velocity of smaller or upstream section)Ke : expansion loss coefficient

aV

2

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

b

ae S

SK for turbulent flow ( )1&1 ≅≅ βα

3/4&2 == βαLaminar flow인경우에는 을사용하면 Ke를구할수있다.


* Friction loss from sudden contraction

cross section of minimum area

2

2b

cfcVKh = : average velocity of smaller or downstream section)

Kc : contraction loss coefficientbV

Kc < 0.1 for laminar flow hfc is negligible.

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

a

bc S

SK 14.0 for turbulent flow (empirical equation)


* Friction loss from fittings

2

2a

fffVKh = : average velocity in pipe leading to fitting

Kf : fitting loss coefficientaV

Table 5.1 Loss coefficients for standard pipe fittings

24

2VKKKDLfh fecf ⎟

⎠⎞

⎜⎝⎛ +++=

skin friction loss coeff.fitting loss coeff.

expansion loss coeff.contraction loss coeff.

Bernoulli equation without pump: ( ) fbaba hZZgpp

=−+−ρ

* Total friction

대입

Ex. 5.2) Homework


05.0≈cK

To minimize expansion loss, the angle between the diverging walls of a conical expander must be less than 7o.

For angles > 35o The loss through this expander can become greater than that through a sudden expansion.

separation point (분리점)

Separation of boundary layer in diverging channel

* Minimizing expansion and contraction losses

. Contraction loss can be nearly eliminated by reducing the cross section gradually.

In this case, separation & vena contracta do not occur.

. Expansion loss can also be minimized by enlarging the cross section gradually


* Flow through parallel plates (Prob. 5.1 & 5.3과연관)

In laminar flow between infinite parallel plates,

yx

W

L

bpa pb

2yτ

τ

upper plate

lower plate

flow

212

bLVpp ba

µ=− 임을보이고 를구하시오.maxmax /,/ uVuu

LWyWpyWp ba τ222 =−(풀이) Force balance: from Eq. (4.52)

yLpp ba τ

=−

dyduµτ −=

대입


∫−=∫− uy

bba dudyy

Lpp

02/)(

µ

적분하면,

( ) 2

max 22⎟⎠⎞

⎜⎝⎛−

=∴b

Lppu ba

µ

b.c. 대입(umax at y=0)

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛−

= 22

22)( yb

Lppu ba

µ

( )L

bpp

dyuWbW

dSuS

V

ba

a

µ12

11

2

2/0

−=

∫∫ ==

212

bLVpp ba

µ=−∴

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=∴

2

max )2/(1

by

uu

32

max=∴

uV

Related problems:

(Probs.) 5.4, 5.8, 5.10, 5.12, 5.13, 5.17, 5.20 and 5.21

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Chapter 5. Incompressible Flow in Pipes and Channels · 2008-10-21 · Incompressible Flow in Pipes and Channels * Relations between maximum velocity ... Dilute polymer solutions