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 groundlevel 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

Livermore site 
Direction 
Calm
0.00.9 
1.02.9 
3.04.9 
5.06.9 
>7.0 
Total 
N 
1.7 
0.4 
0.3 
0.3 
0.1 
2.8 
NNE 
1.7 
1.8 
2.0 
0.5 
0.1 
6.1 
NE 
1.7 
2.5 
1.7 
0.1 
0.0 
6.0 
ENE 
1.7 
1.3 
0.2 
0.0 
0.0 
3.2 
E 
1.7 
0.9 
0.1 
0.0 
0.0 
2.7 
ESE 
1.7 
1.1 
0.0 
0.0 
0.0 
2.8 
SE 
1.7 
0.7 
0.1 
0.0 
0.0 
2.5 
SSE 
1.7 
0.8 
0.3 
0.0 
0.0 
2.8 
S 
1.7 
4.1 
0.7 
0.3 
0.2 
7.0 
SSW 
1.7 
7.0 
2.0 
1.2 
0.3 
12.2 
SW 
1.7 
6.8 
5.6 
1.8 
0.2 
16.1 
WSW 
1.7 
7.4 
5.3 
0.9 
0.0 
15.3 
W 
1.7 
3.6 
5.1 
2.2 
0.1 
12.7 
WNW 
1.7 
0.9 
0.3 
0.0 
0.0 
2.9 
NW 
1.7 
0.6 
0.1 
0.1 
0.0 
2.4 
NNW 
1.7 
0.6 
0.1 
0.1 
0.0 
2.5 
Total 
27.2 
40.4 
23.9 
7.4 
1.0 
100.0 
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 23 year residence
 But, 510 y at 100 km elevation
Transport in Troposphere and Stratosphere
 Aerosols introduced into troposphere are distributed by planetary
winds
 Deposited by rain scavenging
 Example, ^{90}Sr 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 groundlevel 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
Parameter (m) 
Stability Class 
Power Function 
s_{y} 
Unstable 
0.14X^{0.92} 
Neutral 
0.06X^{0.92} 
Very Stable 
0.02X^{0.92} 
s_{z} 
Unstable 
0.53X^{0.92} 
Neutral 
0.15X^{0.92} 
Very Stable 
0.04X^{0.92} 
s'_{x}
is presumed to behave as s'_{y} 
Continuous Releases, Infinite Number of Clouds
Gaussian Plume Diffusion
Gaussian Plume, Reflection at Surface
Longterm 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/m^{3})  limited use
 Need normalized, timeintegrated air concentration
 E(Cis/m^{3}): 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
 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/m^{3}
 Mass particulate/m^{2}
 Activity/m^{3}
 Activity/m^{2}
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/m^{2})
 Dt = time increment (s)
 C = mean ground level air concentration (g/m^{3})
 w = deposition rate (g/m^{2}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 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 (incloud scavenging)
 Can involve submicron particles
 Washout
 Below cloud scavenging
 Effective for particles >1µm
 Lamda = washout coefficient
 Lamda = 1.6 x 10^{4}r_{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
Principle Transport Mechanism 
Particle Diameter (µm) 
Relative Erodibility 
Airborne Transport 
< 20 
Nonerodible, except at very high wind speeds 
20  50 
Difficultly erodible 
Saltation 
50  500 
Highly erodible 
500  1000 
Difficultly erodible 
Surface Creep 
> 1000 
Nonerodible except at high wind speeds 
Resuspension Mechanisms
Approaches to Estimating Resuspension
 Mass loading
 ^{ }X = air concentration in µCi/m^{3}
 M = air dust load in g/m^{3}
 C_{r} = concentration in the resuspendable fraction
of the soil in µCi/g
Massloading
Resuspension Factor
 Alternative to Mass Loading
 R = X/S
 R = resuspension factor, m^{1}
 X = air concentration, µCi/m^{3}
 ^{ }S = surface deposition, µCi/m^{2}
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
