이 동 현 상(Transport phenomena)

2009 년숭실대학교 환경화학공학과

3.1. FLOW PAST IMMERSED OBJECTS AND PACKED AND FLUIDIXED BEDS

3.1A. Definition of Drag coefficient for Flow Past Immersed Objects

skin drag (wall drag):

form drag:

stagnation point:

Chap. 3 Principles of Momentum Transfer and Applications

Stream line

For flow past immersed objects drag coefficient CD: the ratio of the total drag force per unit area to v0

2/2

Ap: p.123for a sphere, Ap=Dp

2/4for a cylinder, Ap= LDp

total drag force FD

Reynolds number NRe

2/

/20v

AFC pD

D

pDD Av

CF 2

20

Dp L

0

Re

vDN p

not a pipe diameter!

3.1B. Flow Past Sphere, Long Cylinder, and Disk

in the laminar region for NRe < 1.0(Stokes’ law)

03 vDF pD

Re0

24

/

24

NvDC

pD

Ex 3.1-1

3.5. NON-NEWTONIAN FLUIDS3.5A. Types of Non-Newtonian Fluids

For Newtonian fluids

and =const.

a plot of vs. (-dv/dr): straight line (linear through the origin)

The slope is .

For Non-Newtonian fluids a plot of vs. (-dv/dr): not linear through the origin

- time-independent fluids: major- time-dependent fluids- viscoelastic fluids

dr

dv

3.5B. Time-Independent Fluids

1. Pseudoplastic fluids- majority of non-Newtonian fluids

K: n:

2. Dilatant fluids

3. Bingham plastic fluids- a plot of vs. (-dv/dr): linear but not through the origin- yield stress 0: to initiate flow

n

dr

dvK

(n < 1) Ostwald-de Waele equation

n

dr

dvK

(n > 1)

* n = 1 Newtonian fluid

dr

dv 0

3.5H. Velocity Profiles for Non-Newtonian Fluids

1. Pseudoplastic fluids and dilatant fluids

nn

nnn

Lx R

rR

KL

pp

n

nv

/)1(

0

/)1(0

/1

0 121

2. Bingham plastic fluids

R

rR

R

rR

L

ppv L

x 114

0

2

20

3.6. DIFFERENTIAL EQUATION OF CONTINUITY3.6A. Introduction

- Now, we use a differential element for a control volume. differential balance

- equation of continuity: differential equation for the conservation of mass

3.6B. Types of Time Derivatives and Vector Notation

1. Partial time derivative of

2. Total time derivative of

3. Substantial time derivative of

t

dt

dz

zdt

dy

ydt

dx

xtdt

d

)(

vtz

vy

vx

vtDt

Dzyx

6. Differential operations

gradient

divergence

Laplacian

others

zk

yj

xi

ˆˆˆ

z

v

y

v

x

vv zyx

)(

2

2

2

2

2

22

zyx

3.6C. Differential Equation of Continuitya mass balance through a stationary element volume x y z

(rate of mass acc.) = (rate of mass in) – (rate of mass out)

zyvzyvdirectionxinoutmassofrateinmassofratexxxxx

)()()()(

tzyxonaccumulatimassofrate

)(

z

vv

y

vv

x

vv

tzzzzzyyxyy

xxxxx

)()()()()()(

)()()()(

vz

v

y

v

x

v

tzyx

(Equation of Continuity)

Incompressible fluids constant density

or

zv

yv

xv

z

v

y

v

x

v

t zyxzyx

z

v

y

v

x

v

zv

yv

xv

tzyx

zyx (Equation of Continuity)

(Equation of Continuity))( vz

v

y

v

x

v

Dt

D zyx

0)(

z

v

y

v

x

vv zyx

0Dt

D

3.7. DIFFERENTIAL EQUATIONS OF MOMENTUM TRANSFER OR MOTION

3.7A. Derivation of Equations of Momentum Transfer- equation of motion: differential equation for the conservation of momentum

p. 189-190 참조

.accmomentum

ofrate

systemonacting

forcesofsum

outmomentum

ofrate

inmomentum

ofrate

x

pg

zyyz

vv

y

vv

x

vv

t

vx

zxyxxxxz

xy

xx

x

(Equation of motion)gpDt

vD

)(

(x component)

x

pg

zyyDt

Dvx

zxyxxxx

(x component)

=

3.7B. Equations of Motion for Newtonian Fluids with Varying Density and Viscosity- for Newtonian fluids in rectangular coordinates

)(3

22 v

x

vxxx

)(3

22 v

y

vyyy

)(3

22 v

z

vzzz

x

v

y

v yxyxxy

y

v

z

vzy

zyyz

z

v

x

v xzxzzx

x

pg

z

v

x

v

z

x

v

y

v

yv

x

v

xDt

Dv

xxz

yxxx

)(3

22

(x component of equation of motionfor varying density and viscosity)

3.7B. Equations of Motion for Newtonian Fluids with Constant Density and Viscosity- for Newtonian fluids in rectangular coordinates

constant and

x

pg

z

v

x

v

z

x

v

y

v

yv

x

v

xDt

Dv

xxz

yxxx

)(3

22

x

pg

z

v

y

v

x

v

Dt

Dvx

xxxx

2

2

2

2

2

2

0)(

z

v

y

v

x

vv zyx

(x component of Navier-Stokes equation)

x

pg

z

v

y

v

x

v

z

vv

y

vv

x

vv

t

vx

xxxxz

xy

xx

x

2

2

2

2

2

2=

(continuity equation)

3.8 USE OF DIFFERENTIAL EQUATIONS OF CONTINUITY AND MOTION3.8A. Introduction

The purpose and uses of the differential equations of motion and continuity- to apply these equations to any viscous-flow problem

The strategy to solve a given specific problem

(1) Simplification- to discard the terms that are zero or near zero in the equation of conti

nuity and motion

(2) Integration with boundary conditions and/or initial conditions- boundary conditions: at wall (ex: no slip v=0)

at center (ex: symmetry dv/dx=0)- initial conditions: at the beginning

*steady-state 0t

3.8B. Differential Equations of Continuity and Motion for Flow between Parallel Plates

Ex 3.8-1

3.8C. Differential Equations of Continuity and Motion in Stationary and Rotating Cylinders

Ex 3.8-3 (=pp. 83-85)Ex 3.8-4

3.10. BOUNDARY-LAYER FLOW AND TURBULENCE3.10A. Boundary Layer Flow

- boundary layer:

for laminar flow

x:v:

Chap. 3 Principles of Momentum Transfer and Applications

xvN xRe,

NRe < 2105 : laminarNRe > 3106 : turbulent

3.10. BOUNDARY-LAYER FLOW AND TURBULENCE3.10B. Boundary-Layer Separation and Formation of Wakes

- wake: