Chapter 5. Incompressible Flow in Pipes and Channels · 2008-10-21 · Incompressible Flow in Pipes...

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−=

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

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=

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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∆

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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.

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

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)

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

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

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

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)

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

- 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

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

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 ↑

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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'=∴

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* Friction factor chart

↑Dk /

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

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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.

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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)

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

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

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

* 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

Unit Operations Chapter 5. Incompressible Flow in Pipes and Channels

∫−=∫− 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