che354 immersed

7/29/2019 ChE354 Immersed

1/23

Flow Around Immersed Objects

Incompressible Flow


2/23

Goals

Describe forces that act on a particle in a fluid.

Define and quantify the drag coefficient for

spherical and non-spherical objects in a flow

field.

Define Stokes and Newtons Laws for flowaround spheres.


3/23

Flow Around Objects

There are many processes that involve flow through aporous medium such as a suspension of particles:

Packed Bed Chemical Reactor

Food Industry

Oil Reservoirs


4/23

Forces

Dynamic

Fk results from the relative motion

of the object and the fluid (shear

stress)

Static

Fs results from external pressure

gradient (Fp) and gravity (Fg).

pgk FFFFMM 12


5/23

Dynamic Forces

A drag force (friction) arises in situations in which

moving fluids are in contact with a solid surface (recall

pipe flow.

Fwis the kinetic force exerted on the solid wall or

particle, A is the wetted surface area, K is the kinetic

energy per unit volume, and fis the friction factor.

K

AFf w

2

2

V

w

headvelocitydensity

shearwallf


6/23

Dynamic Forces

For flow around a submerged object a drag coefficientCd is defined and the equation becomes:

2

2

0uAC

F dk

U0 is the approach velocity (far from the object), is

the density of the fluid, A is the projected area of the

particle, and Cd is the drag coefficient analogous tothe friction factor in pipe flow (keep this in mind).


7/23

Projected Area

The projected area used in the Fkis the area seen bythe fluid.

Spherical Particle

A 4

2

2 DR


8/23

Projected Area

Cylinder

For objects having shapes other than spherical, it is

necessary to specify the size, geometry and orientationrelative to the direction of flow.

Axis perpendicular to flow Rectangle LDA

Axis parallel to flow Circle

4

2D

A


9/23

Drag Coefficient

The drag coefficient, like the friction factor in pipes

depends on the Reynolds number

0DuRe

D is particle diameter or a characteristic length and

and are fluid properties.


10/23

Drag Coefficient

For slow flow around a sphere and Re


11/23

Drag CoefficientReCRe d 2410 44.01000 dCRe


12/23

Why Different Regions?As the flow rate increases wake drag becomes an important

factor. The streamline pattern becomes mixed at the rear ofthe particle and at very high Reynolds numbers completely

separate in the wake. This causes a greater pressure at the

front of the particle and thus an extra force term due to

pressure difference.


13/23

Static Forces

Static forces exist in the absence of fluid motion. They

include the downward force of gravity and the upwardforce of buoyancy that results from the gravity induced

pressure gradient in the z-direction

gVgmF pppg 1P

ghPP f 12

gV

ghAPPAF

fp

fb

12

bF

gF


14/23

Total Force

The gravity and buoyancy forces on an object immersedin liquids do not generally balance each other and the

object will be in motion.

bgkt FFFF

bF

gF

What is the direction of Fk?

It is always opposite to the direction ofparticle motion


15/23

Equilibrium

When a particle whose density is greater than that of the

fluid begins to fall in response to the force imbalance, it

begins to accelerate (F=ma). As the velocity increases

the viscous drag force also increases until all forces are in

balance. At this point the particle reaches terminal

velocity.

bgk FFF 0

gVgmuACF fppt

fpd 20

2


16/23

Terminal Velocity

If the particle has a uniform density , the particle

mass is Vpp and

gVuAC gfptfpd 2

0

2

fd

pfp

t

C

Dgu

3

4

Use: Falling ball viscometer to measure viscosity

fdp

pfp

t

C

mgu

3

2


17/23

Settling VelocityThe settling (terminal) velocity of small particles is

often low enough that the Reynolds number is lessthan unity (Cd = 24/Re).

18

2gD

ufpp

t

1Re

Between 1000


18/23

Criterion for Settling Regime

Reynolds number is a poor criteria for determining theproper regime for settling. We can derive a value K that

depends solely on the physical parameters

312

fpfp

g

DK

K < 2.6 Stokes Law

Newtons LawK > 68.9


19/23

Example

A cylindrical bridge pier 1 meter in diameter is submerged

to a depth of 10m in a river at 20C. Water is flowing pastat a velocity of 1.2 m/s. Calculate the force in Newtons on

the pier.

smkgx

mkg

water

water

3

3

10005.1

2.998

smu 2.10


20/23

Example

2

2

0uACF dk

6

3

3

0 10192.1

10005.1

12.12.998

smkg

msmmkgDuRe

Fig. 7.3 gives Cd 0.35

Projected Area = DL = 10 m2

Ns

m

m

kgmFk 515,22.12.99810

2

35.02

22

3

2


21/23

ExampleEstimate the terminal velocity of limestone particles

(Dp = 0.15 mm, = 2800 kg/m3) in water @ 20C.


22/23

Example

9.3

10005.1

2.99828002.99898.9

105.1

31

2

3

3324

sm

kgm

kg

m

kg

s

m

mK

Guess Re = 4 Cd = 9.0

s

m

m

kg

mm

kg

s

m

ut 02.0

2.9980.93

105.12.99828008.94

3

4

32

310005.1

02.02.998105.13

34

smkg

smmkgmRe


23/23

Example

Guess Re = 2 Cd

= 15

U0 = 0.015 m/s Re = 2.3

Guess Re = 2.5 Cd = 12

U0 = 0.017 m/s Re = 2.6

che354 immersed

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