External Flows(Flow around Immersed Objects)

@ Agus Saptoro, Deeptangshu Chaudhary, 2011

Australia Malaysia

ChE 323 Transport Phenomena

What are we going to learn…..?

1. Flows: Internal (last week) vs external flow

2. Drag and lift forces

3. Drag and lift coefficient

References:McCabe et al, Unit Operations of Chemical EngineeringCengel et al, Fluid Mechanics: Fundamental and ApplicationsGriskey, transport Phenomena and Unit Operations: A Combined ApproachKing, Introduction to Fluid Flow

Flows

• Internal Flows flow in pipe or annulus flow through valves flow through contraction or expansion etc

• External Flows flow around a solid / flow around immersed

object

• an F-18 flying at a Mach number of 1.4 at 35,000 feet

a T-38 flying at a Mach number of 1.1 at 13,700 feet

http://www.galleryoffluidmechanics.com/shocks/ifs.htm

http://www.greatmichaelphelps.com (retrieved on 14 March 2011)

http://www.procyclingphotos.com (retrieved on 14 March 2011)

Flow Around Objects in Chemical Engineering Problems

• Fixed bed reactor

• Packed absorption column

• Packed distillation column

• Fluidised bed reactor

• etc

Drag, Lift and Forces

Drag

• A body meets some resistances when it is forced to move through a fluid, especially a liquid.

• The forced a flowing fluid exerts on a body in the flow direction is called DRAG

(Cengel et al, Fluid Mechanics: Fundamental and Applications)

Bluff or blunt body

Streamlined body (contoured & sleek)

• Drag is usually is undesirable effects, like friction

• We tend to do minimisation of drag in automobile, aircraft or submarine since reducing drag = reducing fuel consumption

• Pros of drag: ‘a life safer’ in the brakes of automobile

ForcesDynamic

Fd results from the relative motion of the object and the fluid (shear stress)

Static

Fs results from external pressure gradient (Fb) and gravity (Fg).

bgd FFFF

Dynamic Forces

For flow around a submerged object a drag coefficient Cd is defined:

2

2opd

d

uACF

U0 is the velocity of the approaching stream, ρ is the density of the fluid, A is the projected area of the particle, and Cd is the drag coefficient analogous to the friction factor in pipe flow (keep this in mind).

Projected AreaThe projected area used in the Fk is the area “seen” by the fluid.

Spherical Particle

pA4

22 DR

Projected Area

Cylinder

For objects having shapes other than spherical, it is necessary to specify the size, geometry and orientation relative to the direction of flow.

Axis perpendicular to flow Rectangle LDAp

Axis parallel to flow Circle4

2DAp

For flow around a submerged object a drag coefficient Cd is defined:

2

2opd

d

uACF

Drag coefficient

2/

/2o

pdd

u

AFC

The local drag coefficient varies along the surface as a result of the changes in the velocity boundary layer in the flow direction.Average drag coefficient for a surface of length L

dxxCL

CL

dd 0

1

Drag Force• Drag force is the net force exerted by a fluid on a body in the

direction of flow due to the combined effects of wall shear and pressure forces

• Drag caused by frictional effects (due directly to wall shear stress, ) is called skin friction drag / friction drag,

• Drag dues directly to pressure, P, is called pressure drag/form drag (strongly dependence on the form or shape of the body)

pressuredfrictiondd FFF ,, pressuredfrictiondd CCC ,,

dF w

Friction drag depends on the orientation of the body, magnitude of the wall shear stress and viscosity

Friction drag coefficient is analogous to the friction factor in pipe flow. In laminar flow, it is independent of surface roughness and it is a strong function of surface roughness in turbulent flow

The pressure drag is proportional to the frontal area and to the difference between the pressure acting on the front and back of the immersed body

Example: drag coefficient of a car

(Cengel et al)

Drag coefficients of common geometries

• Drag coefficient, in general, depends on Reynold number• Drag coefficient exhibits different behaviour in the low

(creeping), moderate (laminar) and high (turbulent)• At low Re (Re < 1), called creeping flow region, the inertia

effects are negligible and the fluid wraps around the body smoothly

• The drag force acting on a spherical object at low Re

• This Stokes Law is often applicable to dust particles in the air and suspended solid particles in water

Stokes Law

Drag coefficients at low Re

At low Re, the shape of the bodydoes not have a major influenceon the drag coefficient

Drag coefficient for high Re

Example: Application of drag force/coefficient

Parallel Flow over flat plate

The actual of the engineering critical Re for a flat plate A generally accepted value for the critical Re

Flat Plate

Local friction coefficient

Average friction coefficient

Flat plate

Flow over cylinders and spheres

• Flow over cylinders and spheres is frequently encountered in practice, e.g. the tubes in a shell-and-tube HX involve both internal flow through the tubes and external flow over the tubes

6

105

5

102Re

102Re102

102Re

x

xx

x

Laminar

Transition

Fully-Turbulent

Effect of surface roughness

Example

A 2.2 cm OD pipe is to cross a river at a 30 m wide section while being completely immersed in water. The average flow velocity of water is 4 m/s and the water temperature is 15 C. determine the drag force exerted on the pipe by the river

Terminal Velocity(Superficial Velocity)

Static Forces

Static forces exist in the absence of fluid motion. They include the downward force of gravity and the upward force 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

EquilibriumWhen 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.

bgd FFF 0

gVgmu

ACF fppt

fpd 20

2

Terminal Velocity

fdp

pfpt CA

Vgu

2

If the particle has a uniform density , the particle mass is Vprp and

gVu

AC gfpt

fpd 2

02

pfdp

pfpt CA

mgu

2

or

Lift• The components of the pressure and wall shear forces in the

direction normal to the flow tend to move the body in that direction and their sum is called lift

Average lift coefficient for theentire surface

Example: Effect of spin on a tennis ball

Keypoints

1. External flows

2. Drag and lift forces

3. Drag and lift coefficient

Any question?