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)

    F/AD? CD2V/2?

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

    6. steady state 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


    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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