Aero-Hydrodynamic Characteristics

Asst. Dr. Muanmai Apintanapong

In handling and processing : air and water is used as carrier for transport or separation.

Pneumatic separation and conveying has been in use for many years.

Fluid flow or fluid mechanics find increasingly wide applications in handling and processing

Knowledge of physical properties which affect the aero- and hydrodynamic behavior of

agricultural products becomes necessary.

Mechanics of particle motion in fluids

To describe, two properties need:Drag coefficientTerminal velocity

Drag Coefficient

For particle movement in fluids, drag force is a

resistance to its motion. Drag coefficient is a coefficient related to drag

force. Overall resistance of fluids

act to particle can be described in term of drag

force using drag coefficient.

Comparing with fluid flow in pipe principle, drag coefficient is similar to friction coefficient or friction factor (f).

flowofvolumeunitenergykinetic

stressshearf

2

2

2

1

21

)/(

AvfF

V

mv

AFf

For drag coefficient:

flowofvolumeunitenergykinetic

areaperforcedragCD

2

2

2

1

21

)/(

AvCF

V

mv

AFC

D

D

Frictional drag coefficient For flat plate with a laminar boundary layer:

For flat plate with a turbulent boundary layer

5.0

328.1

R

DN

C

58.2log

455.0

R

DN

C

sphereofdiameterplateoflengthD

DvNR

Frictional drag coefficient For flat plate with a transition region:

RRD NN

C1700

)(log

455.058.2

sphereofdiameterplateoflengthD

DvNR

If a plate or circular disk is placed normal to the flow, the total drag will contain negligible frictional drag and does not change with Reynolds number (NR)

Sphere object

At very low Reynolds number (<0.2), Stoke law is applicable. The inertia forces may be neglected and those of viscosity alone

considered.

RD N

C24

Terminal or Settling Velocity

Settling velocity (vt): the terminal velocity at which a particles falls through a fluid.

When a particle is dropped into a column of fluid it immediately accelerates to some velocity and

continues falling through the fluid at that velocity (often termed the terminal settling velocity).

The speed of the terminal settling velocity of a particle depends on properties of both the fluid and the particle:

Properties of the particle include:

The size if the particle (d).

The shape of the particle.

The density of the material making up the particle (p).

FG, the force of gravity acting to make the particle settle downward through the fluid.

FB, the buoyant force which opposes the gravity force, acting upwards.

FD, the “drag force” or “viscous force”, the fluid’s resistance to the particles passage through the fluid; also acting upwards.

Particle Settling Velocity

Put particle in a still fluid… what happens?

Speed at which particle settles depends on:

particle properties: D, ρp, shape

fluid properties: ρf, μ, Re

Fg

Fd

FB

STOKES Settling Velocity

Assumes: spherical particle (diameter = dP)

laminar settling

FG depends on the volume and density (P) of the particle and is given by:

FB is equal to the weight of fluid that is displaced by the particle:

Where f is the density of the fluid.

33

66 PPPPG gdgdF

33

66 PffPB gdgdF

FD is known experimentally to vary with the size of the particle, the viscosity of the fluid and the speed at which the

particle is traveling through the fluid.

Viscosity is a measure of the fluid’s “resistance” to deformation as the particle passes through it.

vdvACF PPfDD 32

1 2

Where (the lower case Greek letter mu) is the fluid’s dynamic viscosity and v is the velocity of the particle; 3d is proportional to the area of the particle’s surface over which

viscous resistance acts.

RD N

C24

From basic equation, F = mg = resultant force:

With v = terminal velocity or vt:

DBG FFFdt

dvmmaF

0dt

dvmFFF DBG

In the case of 0.0001<NR<0.2, terminal velocity can be determined by using CD =24/NR:

18

)()(

3

42 gd

C

gdv fPP

fD

fPPt

In the case of 0.2<NR<500, terminal velocity can be determined by using CD as:

687.015.0124

RR

D NN

C

In the case of 500<NR<200,000, terminal velocity can be determined by using CD as:

44.0DC

Example:

A spherical quartz particle with a diameter of 0.1 mm falling through still, distilled water at 20C

dP = 0.0001m P= 2650kg/m3 f= 998.2kg/m3

= 1.005 10-3 Ns/m2 g = 9.806 m/s2

Under these conditions (i.e., with the values listed above) Stoke’s Law reduces to:

For a 0.0001 m particle: vt = 8.954 m/s or 9 mm/s

18

)(2 gdv fPP

t

25 )10954.8( Pt dv

Stoke’s Law has several limitations:

i) It applies well only to perfect spheres.

The drag force (3dvt) is derived experimentally only for spheres.Non-spherical particles will experience a different distribution of viscous drag.

ii) It applies only to still water.

Settling through turbulent waters will alter the rate at which a particle settles; upward-directed turbulence will decrease vt whereas downward-directed turbulence will increase vt.

Coarser particles, with larger settling velocities, experience different forms of drag forces.

iii) It applies to particles 0.1 mm or finer.

Stoke’s Law overestimates the settling velocity of quartz density particles

larger than 0.1 mm.

When settling velocity is low (d<0.1mm) flow around the particle as it falls smoothly

follows the form of the sphere.

Drag forces (FD) are only due to the viscosity of the fluid.

When settling velocity is high (d>0.1mm) flow separates

from the sphere and a wake of eddies develops in its lee.

Pressure forces acting on the sphere vary.

Negative pressure in the lee retards the passage of the particle, adding a new resisting force.

Stoke’s Law neglects resistance due to pressure.

iv) Settling velocity is temperature dependant because fluid viscosity and density vary with temperature.

Temp. vt

C Ns/m2 Kg/m3 mm/s

0 1.792 10-3 999.9 5

100 2.84 10-4 958.4 30

Grain size is sometimes described as a linear dimension based on Stoke’s Law:

Stoke’s Diameter (dS): the diameter of a sphere with a Stoke’s settling velocity equal to that of the particle.

18

2sPf

t

gdv

gv

dPf

tp

18

Set ds = dP and solve for dP.

Measurement of terminal velocity

a) Direct measurementb) Estimating settling velocity based

on particle dimensions.

Settling velocity can be measured using settling tubes: a transparent tube filled with still water.

In a very simple settling tube:

A particle is allowed to fall from the top of a column of fluid, starting at time t1.

The particle accelerates to its terminal velocity and falls over a vertical distance, L, arriving there at a later time, t2.

The settling velocity can be determined:

a) Direct measurement

2 1

L

t t

vt

A variety of settling tubes have been devised with different means of determining the rate at which particles fall. Some apply to individual particles while others use bulk samples.

Important considerations for settling tube design include:

i) Tube length: the tube must be long enough so that the length over which the particle initially accelerates is small compared to the total length over which the terminal velocity is measured.

Otherwise, settling velocity will be underestimated.

ii) Tube diameter: the diameter of the tube must be at least 5 times the diameter of the largest particle that will be passed

through the tube.If the tube is too narrow the particle will be slowed as it settles

by the walls of the tube (due to viscous resistance along the wall).

iii) In the case of tubes designed to measure bulk samples, sample size must be small enough so that the sample doesn’t settle as a mass of sediment rather than as discrete particles.

Large samples also cause the risk of developing turbulence in the column of fluid which will affect the measured settling

velocity.

b) Estimating settling velocity based on particle dimensions.

Settling velocity can be calculated using a wide variety of formulae that have been developed theoretically and/or experimentally.

Stoke’s Law of Settling is a very simple formula to calculate the settling velocity of a sphere of known density, passing through a still fluid.

Stoke’s Law is based on a simple balance of forces that act on a particle as it falls through a fluid.

Terminal velocity determination

There are two methods for determination of terminal velocity.Terminal velocity from drag coefficient-

Reynold’s number relationship.Terminal velocity from time-distance

relationship.

Terminal velocity from drag coefficient-Reynold’s number relationship

Both CD and NR include a velocity term, calculation of vt from CD and NR

relationship require a trial-and-error solution. To eliminate trial-and-error

solution, case of vt or dp is unknown, the term CDNR

2 or CD/NR are first calculated and plot against NR.

For spherical particles:

P

fPf

fPPfRD

W

dgNC

2

2

32

8

3

4

323

4

tf

fP

R

D

v

gN

C

CDNR2 is first calculated and plot

against NR

CD.NR2

NR

CD/NR is first calculated and plot against NR

CD/NR

NR

Terminal velocity from time-distance relationship

In free fall of an object in still air, the net force on the object is the difference in the force of

gravity (mg) and the resultant frictional or drag force (kv2). This net force is also equal

to m(dv/dt) After integration this equation and determine

velocity in term of displacement (distance) with time (vt =s/t)

tv

g

g

vs

t

t coshln2

Distance (s)

Time (t)

In such cases, from linear portion

Slope = s/t =vt

t

t

v

g

g

vslope coshln

2

or calculate vt from equation: