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# Chapter 4 Particle Dynamics

By Earl Gomez,2014-05-07 12:08
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Chapter 4 Particle Dynamics

Chapter Four: Single Particle Dynamics

Stokess Law and Corrections

Aerodynamic Diameter

Particle Relaxation Time and Stop Distance

High Reynolds Number

A. Stokess Law

Def.: Drag coefficient

Drag coefficient = (Drag Force/Projected Area)/(Dynamic Pressure)

Based on experimental results as shown above:

?2424?? for Re < 1, or the Stokess law and where Re is the CDRe?dVpp

Reynolds number, and are the density and viscosity of air, d is ??p

the particle diameter, and V is the particle velocity relative to air. pC?0.44 for Re > 1000, or the Newtons law; and in the transition region D

240.687 ??(10.15Re)CDRe

For a spherical particle under Stokes’s law, which is generally valid for the aerosol

in the ambient atmosphere, the drag force is:

F?3??dV, for Re < 1 or the Stokess law Dpp

Assumptions for the above relationships:

1. rigid spherical particle

2. Stokess law or inertial force is much smaller than viscous force

3. continuum fluid

4. free flow without wall effects

5. the density of air is constant or low Mach number flow

Corrections for Stokes law:

1. Slip correction for non-continuum flow:

3??dVpp and ?FDCc

? for 0.1 ?m ?1?2.52dp?Ccd

???d??????12.341.05exp0.39C for all particle sizes C?????d????

where Cc is the Cunningham slip correction factor.

2. near wall correction for non-symmetrical flow pattern:

d/23??dV9pppk?1? and , where h is the distance ?FkDh16Cc

between the center of particle and the surface

3. dynamic shape factor: correction for non-spherical particle

3??dVpp where is the shape factor as shown below ?Fk??DCc

B. Aerodynamic Diameter

At terminal settling velocity, the drag force is equal to the gravitational force

3??dV????dg3()ppppV?Vk?? and , thus psC6c

2??(?)dgpCpc, for Re < 1.0 ?Vs18?k?

2dg?Cppc(????)?1If and , V?psk?18?

Effect of Pressure on Terminal Setting Velocity of Standard Density Spheres at 293K

Particle V at the Indicated Pressure(m/s) TSDiameter

P=0.1 atm P=1.0 atm P=10 atm (?m)

-8-9-100.001 6.9?10 6.9?10 6.9?10

-7-8-90.01 6.9?10 7.0?10 8.7?10

-6-7-70.1 7.0?10 8.8?10 3.5?10

-5-5-51 8.8?10 3.5?10 3.1?10

10 0.0035 0.0031 0.0029

100 0.29 0.25 0.17 Definition: Mobility B is the ratio of the terminal velocity of a particle to the steady

V1state force producing that velocity, thus, for large particles. B??F3??dDp

C. Particle Relaxation Time and Stop Distance

The particle velocity in a still air when released from rest can be derived as (based

nd law): on Newton’s 2

dVpV(t?0)?0m?mg?F?mg?3??dV with pppDpppdt

22ddg???t/?pppppThus, where = particle relaxation time ?V?(1?e)?pp18?18?Thus, particle relaxation time is the characteristic time for particle to transit from

one state to another state.

Relaxation Time for Standard Density Particles at Standard Conditions

Particle Diameter Relaxation Time

(s) (?m)

-90.01 7.0?10

-80.1 9.0?10

-61.0 3.5?10

-410.0 3.1?10

-2100 3.1?10

Because the particle relaxation time is generally very small, the particle dynamics

are generally assumed to be in equilibrium. That is, the transition period is

generally not taken into account for small particles.

The horizontal distance particle traveled when injected horizontally into a still air

is

2dxdxpp??????3??3?? mFdVdpDppp2dtdt

x(t?0)?0with is the particle position, Thus p

2?d?t/??t/?pppp x?V(1?e)??V(1?e)ppoppo18?

Vwhere is the initial particle velocity. po

S??V Therefore, the maximum traveling distance or the stop distance S is ppoNote that the above equation is only valid for Stokes flow. Mercer (1973)

derived the following approximate equation within 3% difference for initial

Reynolds number up to 1500 as

d1/3?????Rep1/30??Re6arctanS ????0?6??g??

Stopping Distance, Initial Reynolds Number, and Time to Travel 95 Percent of

the Stopping Distance for Standard Density Spheres with an Initial Velocity of

10 meters per Second

Stopping Time to Travel aDistance ,

Particle V=10 m/s 95% of Stopping 0aDiameter Distance

Re (mm) (s) (?m) 0

-5-80.01 0.0066 7.0?10 2.0?10

-4-70.1 0.066 9.0?10 2.7?10

-51.0 0.66 0.035 1.1?10

b-4b10.0 6.6 2.3 8.5?10

b100 66 127 0.065b

D. High Reynolds Number

The relationship between drag force and velocity is no longer linear at high

Reynolds number. Therefore, the determination of the terminal settling velocity is

more complicated than those in Stokes flow. There are two different methods to

determine the terminal settling velocity: iteration and graphical method.

1. Iteration method

C?f(Re) D

1/24?dg??pp??V? and C is function of Reynolds number Ds??3C?D??

Therefore, the iteration method is to start with a guess for V, and then compute s

the C as given at the beginning in this chapter. The computed C is then used DD

to calculate the V as the above equation. The iteration is repeated until the s

difference between V and V is within acceptable error. ss

2. Graphical method

C is function of Reynolds number as shown at the beginning in this chapter D

3dg4??pp2 C??K?constReD23?

Therefore, the interception of the two lines as shown below is the answer.

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