Modeling Road Dust Spread: Particle Motion and Turbulent Diffusion, Slides of Computational Techniques

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Environmental Modeling of the

Spread of Road Dust

2

Problem Definition

In places where roads are usually covered with ice and snowin winter, studded tires became common in the 1960’s. Thestuds clear the roads, leaving bare pavement for the studs toeat into. A plume of dust forms over the roadway andspreads out along the side of the road. For example, inNorway (with only 4,000,000 people), over 300,000 tons ofasphalt and rock dust used to be generated every winter, thatcaused a serious health hazard (lung cancer).We want to determine how the dust cloud spreads.

4

Individual Dust Particle Motion

Assume that we have still air and only one dust particle inmotion. Letr= density of the airair^ v^ = velocity of the particleA = apparent cross section of the particlef^ = friction factor of the airk^ = kinematic viscosity of the airThe magnitude of the force corresponding to the interaction ofthe air is given byF = .5 r

(^2) vA fair Example: spherical particle of diameter D, A = (

2 /4) D.

5

Let the Reynolds number be given by Re = Dv/k.The friction factor f = f(Re). A Stokes formula can be usedwhen Re<.5. Theoretically,(1)^

f(Re) = 24/Re. For .5 < Re < 2x

5 , an empirical formula for f(Re) is given by f(Re) = .4 + 6/( 1 + Re

1/2^ ) + 24/Re.

Consider a spherical particle when (1) holds. The interaction ofthe air reduces toF = 3

rDvkair^ Note that the magnitude of F is proportional to v.

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When the particle is a homogeneous sphere of density r

,p^

C = 18 ( kD

-2^ ) ( r/rair

),^ a constant.p

When the vertical wind velocity component is zero, then theparticle vertical velocity ( vz

= z^ ) is given by.^

vz(t) = g/C + ( z^

• g/C ) e 0

-ct ,^ g = gravity.

When t goes to infinity,| v^ (t) | is bounded by g/C.t The limit velocity is the particle velocity when there is nomomentum along the z axis. This is the free fall velocity vf

In our case,(2)^

v= ( gDf^

2 r)/( 18krp^

).air^

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Trajectory of One Particle

We use the following constants:k = 1.5x

-5^2 m/s r= 1 kg/mair^

3 r= 4x10p^

3 3 kg/m^ g = 9.81 m/s

2 See Fig. 1 (meters).In the case of of free fall velocity, the upper bound on theparticle diameter is 37 microns to ensure that Re < 0.5. This issignificant since particles with diameters less than 10 micronspose a serious health hazard.

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Other boundary conditions could have beenReflecting:

^ (x,0,t) = 0z^ Mixture:

^ (x,0,t) +z^

(x,0,t) = 0 The initial conditions give us a source of particles at heighth:^ (x,z,0) =

(x)^ (z-h) U, v, D^ f^ x

, and D^ z^

are normally functions of x, z, and t. To get an analytic solution, we assume that they are constants.

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Now consider a dimensionless version of (3). Apply thetransformationt = Tt

• *x = Xx^ z = Zz^ to get^ ^ (UT/X )t^

^ - ( vx*^

T/Z )^ f^

=z* ( D^ T/Xx^

2 )^ xx^

• ( D^ T/Zz^

2 )^ zz

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Let h* = hU ( D

-1/2D )x z

. The equations can be rewritten as

^ +^ ^ t^ x

^ - v ^ fz =^ ^ xx^ +^ ^ zz^

(PDE)

(x,0,t) = 0

(BC)

(x,z,0) =

(x)^ (z-h*)

(IC)

A classical mirror image method yields the solution^ =^ 

(x,z-h,t) - exp( v

^ h )^ (x,z+h,t),f

Where^ (x,z,t*) =

-1( 4t* ) exp( -( (x-t)

2 + ( z*+v

^2 t )) / ( 4t* ) )f

14

Numerical Experiments

Concentration of varying diameter particles 1m aboveground. Wind constant at 2 m/sec and continuous source unitstrength at 0.5m.

16

Data from Figure 3 after 5 and 10 Seconds Particles travel at a constant speed, but the concentration drops.

17

Different Wind Speeds

Concentration of 37 micron diameter particles 1m aboveground after 5 seconds. Wind constant at 2, 5, and 10 m/secand continuous source unit strength at 0.5m.

19

Particle Concentration Contours

Concentration of 37 micron diameter particles. Continuoussource unit strength at 0.5m.

20

Possible Improvements

Nonconstants for things like D

, D^ , …x^ z^

A Lagrangian approach instead of an Eulerian oneIndividual particle motions are simulated by a MonteCarlo methodn bodiesCollisions of particles can be modeledWe can have more complicated initial distributions ofparticles (e.g., log, actual measurements, random, etc.)Drawback: No analytic solution possible in general.