Topic 11- Radiation Dosimetry

Gaussian Plume Model

Studies at the Surface Friction Layer

• C = concentration
• Q = source strength
• s_z s_y = crosswind & vertical plume standard deviations
• = mean wind speedF
• h = effective stack height
• x, y = downwind and crosswind distances
• z = height above ground
• This calculates the concentration downwind of gases & particles with negligible settling velocity (< 20µm)

Gaussian Plume Model

•    Used to describe ground-level concentration downwind from a continuously emitting point source:
• C = concentration
• Q = source strength
• s_z s_y = crosswind & vertical plume standard deviations
• = mean wind speed
• h = effective stack height
• x, y = downwind and crosswind distances

Numerical Values for Lateral Diffusion (s_y)

Numerical Values for Vertical Diffusion (s_z)

Normalized Equations Ground Level

Normalized Equations 30 m Stack

Normalized Equations, 100 m Stack

Effects of Buildings on Plume Dispersion

Effects of Terrain on Plume Dispersion

Presenting Wind Data

Recording Wind Data

Computer Models for Atmospheric Calculations

• Computer codes are preferred method
  • AIRDOS
  • CAP88
  • COMPLY
  • ISCLT
  • TRAC
  • MESODIF
• Gaussian Plume Model
• Puff Trajectory Model
• Other

Troposhperic and Stratospheric Behavior

• Previously discussed near surface emissions
  • Diffusion over tens to hundreds of kilometers
  • Beyond that no interest for industrial or research
  • Nuclear explosions or major violent accidents are different
    • Chernobyl spread globally

Meridional transport

Tropospheric and Stratospheric Behavior

• Residence of stratospheric aerosols depends on
  • Altitude
  • Time of year
  • Latitude dependent
• Russian explosions into stratosphere had mean residence time of < 6 mo
• Explosions into mid stratosphere in tropics had 2-3 year residence
• But, 5-10 y at 100 km elevation

Transport in Troposphere and Stratosphere

• Aerosols introduced into troposphere are distributed by planetary winds
  • Deposited by rain scavenging
  • Example, ⁹⁰Sr and rainfall
  • Mean residence time of dust is ~ 30 days
  • Rainfall removes particles primarily by rainout (droplet formation around the particle)
  • Ocean spray scavenging may be a factor

Elevated Releases - Qualitative

Characterizing Turbulent Diffusion

• Gradient Transport Theory
  •  transport at a fixed point
  •  similar to molecular diffusion
• Statistical Theory
  • study history of motion
  • determine statistical properties needed to represent diffusion
• For large diffusion times, both generate the Gaussian distribution

Instantaneous Releases

• Explosion or puff
• Puff diffuses in 3 dimensions
• Formula estimates ground-level air concentrations downwind
• Where
  • Q_T = total quantity of material released
  • s_x,y,z standard deviation of puffs
  • u, mean wind speed
  • t = time in sec after release

Gaussian Diffusion of Single Puffs

Predictive equations for Diffusion Values Between 100 and 4000 m

Continuous Releases, Infinite Number of Clouds

Gaussian Plume Diffusion

Gaussian Plume, Reflection at Surface

Long-term average air concentrations

• At a given point
• Must account for changes in
  • wind speed
  • win direction
  • atmospheric stability

Seasonal concentration Calculations

• Avoids use of s_z
• Requires frequency of each stability class
  
• Where
  • f(S)  = frequency of stability class S
  • f(N) = frequency wind speed class N
  • f  = frequency of wind in sector
  • s_z(S) = vertical dispersion class S
  • u(N) = mean wind speed for class N

Prerequisites and Assumptions

• Model is valid assuming:
  • Homogeneity of turbulence
    • Requires uniformity of topography
  • Stationary turbulence
    • Requires transport in= transport out
    • (I.E. Source & turbulence constant)
    • Valid for hours at a time
  • Long diffusion times
  • Spatially constant basic flow
    • Wind velocity unchanged with height
    • OK up to ~ 150 m

Stationary Diffusion

Effect of Diffusion Times

Assumptions, Continued

• Nonzero wind speed
  • Need to neglect diffusion in x direction
  • Or use Gaussian puff model
• Total reflection of the plume
  • Ignore ground deposition (depletion)
  • Conservative approach

Practical Consequences of Using Gaussian Plume

• Theoretical assumptions, etc rarely met
• Empirical studies of diffusion parameters make model predictions ~ reasonable
• Allows model to be used for estimating long periods, varying turbulent states

Diffusion Parameters

• Model expressed in terms of s_y s_z
• Selection of appropriate values subjective & controversial
• Major tests done to determine parameters, variety of tracers (radioactive and stable)
• Several parameter systems defined for short term, long term, ground level, etc
• All have some relevance
•  Be careful

Practical Applications of Gaussian Plume

• C (Ci/m³) - limited use
• Need normalized, time-integrated air concentration
• E(Ci-s/m³): estimates dose in ith sector
  • Inhalation
  • Submersion
  • Total deposition

Sector Averaged Calculations

• N = total probability of wind in all frequencies, stabilities, & class (i.e., 100%)
• k = wind speed in class k
• j = stability in class j
• n_jk = percentage of time of occurrence of wind direction(i), speed(k), and stability class (j)
•  _k = representative windspeed in class k
•  s_zj = diffusion parameter for stability class j
• x = downwind distance
• For n=16 sectors

Atmospheric Transport, Continued

• Dispersion of aerosols vs gases
• Plume depletion and enhancement mechanisms
  • Deposition
  • Resuspension
• Tropospheric and stratospheric behavior

Gaussian Plume Model

• Advantages
  • Simple
  • Can be hand calculated
• Limitations
  • Flat terrain
  • Predictions valid within factor of 2 -3
  • Valid only to 10 km downwind
  • Cannot account for curvature in wind direction
  • Much less valid for complex terrain

Plume Enhancement & Depletion Mechanisms

• Depletion mechanisms
  • Dry deposition
    • Gravitational settling
    • Impaction
  • Washout
    • Precipitation washes dust from air
  • Rainout
    • Dust serves as condensation nuclei
    • Process removes most submicron particles from atmosphere

Particles

• What are they?
  • Distinct portion of solid , liquid, or gas larger than single molecule
• Size classification based on how particles are measured (e.g., Sieve sizes, settling velocities ..)
• Classifications
  • Based on media
  • Colloids - 1 to 400 nm
  • Solids dispersed in gas - smoke/aerosol
  • Liquids in gas - fog, aerosol

Properties of Colloids

• Enhanced adsorption
  • Ability to concentrate substances on their surfaces)
  • Large surface area to volume ratio
• Electric charge (positive or negative)
• Described by concentration
  • Mass particulate/m³
  • Mass particulate/m²
  • Activity/m³
  • Activity/m²

Particle Sizes & Rates of Fall

Gravitational Settling

• Falling particle accelerates with gravity until constant velocity reached
• Terminal velocity balance of
  • Resistance offered by fluid medium (acts opposite to weight of particle)
  • Radius/density determine weight & downward force
  • Opposing (aerodynamic drag) f(size, velocity, density, viscosity, resistance to sheer stress)

Gravitational Settling of Particles

Gravitational Settling

• Applicable to particles in fluid or air
• F_R = resultant force on particle
• F_E = external force on particle (e.G., Gravity or centrifugal force)
• F_B = buoyancy force
• F_D= friction or drag force, opposing settling of particle
• u_t = terminal settling velocity

Particle Gravitational Settling

• Assumptions
  • spherical particle
  • terminal velocity (u_t)
  • laminar flow
  • r_p = density of particle
  • r = density of air
  • g = gravitational settling
  • d = particle diameter
  • µm = absolute viscosity of air

Terminal Fall Velocity for Smooth Spheres

Particle Settling - Dry Deposition

• <20 µm particle size or gases
• Deposit at rates > gravitational settling
• Other mechanisms responsible:
  • Surface impaction
  • Electrostatic attraction
  • Adsorption
  • Chemical interaction
• Determined experimentally as ratio between deposition and air concentration

Dry Deposition

• Deposition velocity:
  
• V_d , deposition velocity, m/s
• DS = mass of particles per unit of surface area (g/m²)
• Dt =  time increment (s)
• C =   mean ground level air concentration  (g/m³)
• w =  deposition rate (g/m²-s)

Deposition Velocities

• Empirical approach acceptable
  • Loss mechanisms ignored or accounted for
  • Experiment period is short
• V_d is dependent upon
  • Particle size
  • Particle density
  • Shape
  • Electrostatic charge
  • Surface chemistry

Deposition Velocities, continued

• Surface parameters of importance include:
  • Texture
  • Roughness
  • Presence of hairs or other projections
  • Electrostatic charge
  • Surface chemistry
  • Effective surface area
  • Surface orientation

Deposition Velocity Values

• Extremely variable
• Five orders of magnitude range for dusts (10^-3 - 10^-8 cm/s)
• Less variability for gases          
  • (2 x 10^-3 - 26 cm/s)
• 2 cm/s for reactive iodine on grasses
• 0.1 cm/s for particles less than  4 µm
• 0.018 cm/s for unreactive gases

Deposition Velocity Caveats

• This is not a true velocity;
• A measure of dry removal processes

Wet Deposition

• Rain and snowfall - precipitation scavenging
  • Rainout (in-cloud scavenging)
    • Can involve submicron particles
  • Washout
    • Below cloud scavenging
    • Effective for particles >1µm
    
    • Lamda = washout coefficient
    • Lamda = 1.6 x 10^-4r_p^0.8
    • R_p= rainfall rate, mm/h

Washout Coefficients

Cloud Depletion From Washout

• x = downwind distance, m
• u = mean wind speed acting on  the plume
  • (From whicker & Schultz, radioecology, 1982)

Plume Enhancement Mechanisms

• Resuspension
• Small particles (< 50µm) elevated from ground surface
• Subset of wind erosion
• Atmospheric variables include
  • Velocity
  • Turbulence
  • Density = f(temp, press., Humid.)
  • Viscosity

Plume Enhancement Mechanisms

• Soil variables include:
  • Texture (particle size distribution)
  • Cohesiveness
  • Moisture content
  • Density
  • Plant cover
  • Ground surface roughness
  • Topography

Soil Sizes and Erodibility

Resuspension Mechanisms

Approaches to Estimating Resuspension

• Mass loading
  
•   X = air concentration in µCi/m³
• M = air dust load in g/m³
• C_r = concentration in the resuspendable fraction of the soil in µCi/g

Mass-loading

Resuspension Factor

• Alternative to Mass Loading
• R = X/S
• R = resuspension factor, m^-1
• X = air concentration, µCi/m³
• S = surface deposition, µCi/m²

Caveats on Plume Depletion/Enhancement

• Extreme variability in ranges
• Multiple approaches available
• Variety of physical processes influence
• Site specific parameters may be developed based on experiments

Atmospheric Model Recap

• Generally conservative
  • Tend to over predict concentration
  • Under predict plume spread
  • Model accuracy limited to factor of 2 at best
• Complex dispersion models
  • More physically realistic treatment
  • Limited predictive improvement
  • But they cost more and are more time consuming