Effluent Diffusion

Meteorology and Atmospheric DispersionQaisar Nadeem

Department of Nuclear Engineering, PIEAS Pakistan1

Diffusion of Effluents

Gaussian Plume Model



Effluent Diffusion

• An effluent released at some point into theatmosphere not only moves in a gross way, dueto the various temperature conditions.

• Individual particles in the effluent becomeincreasingly separated from one another as theresult of local atmospheric turbulence in aprocess called “turbulent diffusion”.

• Turbulent diffusion results from the successivecollisions of individual particles, while thediffusion of pollutants is due to the cumulativeeffects of turbulent eddies in the atmosphere



Diffusion Equation

Where

–χ is the concentration of some effluent as a function

of space and time. and

–K is diffusion coefficient.

Only valid if atmosphere is isotropic and at rest.

2 dK

dt



Diffusion Equation for Non Isotropic

Atmosphere

2 2 2

2 2 2x y z

d d d dK K K

dx dy dz dt

With a wind blowing at an average speed v in the x-direction.

2 2 2

2 2 2x y z

d d d d dK K K v

dx dy dz dt dx



Effluent Diffusion

• Assumption

– Point source

– Located at origin

– Constant release rate (Q́ )

χ is not a function of time.

Movement of effluent in the direction of wind is due to wind

itself. (Kx=0)2 2

2 2y z

d d dK K v

dy dz dx



Solution of Diffusion Equation

The effluent moving in x-direction spreads out in

Gaussian Distribution in the y and z-directions.

The standard deviations of these distributions are

given by:

2 2'exp

44 y zy z

Q v y z

x K KK K

2 y

y

xK

v

2 zz

xK

v




Writing in terms of σy and σz (which are

functions of x)

σy and σz are called, respectively, the horizontal

and vertical dispersion coefficients.

2 2

2 2

'exp

2 2 2y z y z

Q y z

v




(Elevated Releases)

If the effluents are released at some altitude h

into the atmosphere above the ground.

Ground Level Concentration (at z=0):

2 22 2

2 2 2 2

'exp exp

2 2 2 2 2y z y z y z

z h z hQ y y

v

2 2

2 2

'exp

2 2y z y z

Q y h

v



Solution of Diffusion Equation(Largest ground level concentration)

Along the centerline of the plume (y=0):

Exponential factor is never greater than unity.

Effluent concentration at all points is always greater

along the plume with a ground level release (h=0) then

when the effluents are released at some altitude.

2

2

'exp

2y z z

Q h

v

'

y z

Q

v

Conservative estimate when release

height is not known.



Diffusion of Radioactive Effluents

• Radioactivity is emitted from nuclear powerplants in puffs, rather than at constant rate.

• The effluent concentration at any point atground level then rises in time to somemaximum value and subsequently falls tozero as the puff passes.

• The total radiation dose from such a puff isproportional to the time integral of χ overthe passage of the puff.

T t dt




Here, Q is the total amount of effluent releases in

the puff.–σy and σz , should increase as root of x from the point of

emission.

–However experimental data show that σy and σz increase

much more rapidly

2

2exp

2y z z

Q h

v

The diffusion model for atmospheric dispersion is not

an exact description of the phenomenon.



Pasquill Conditions

• σy and σz for six different atmospheric

conditions (A-F)

• Derived from experimental data.

• Less stable conditions have higher values of

both σy and σz than stable conditions.

Seventh Category

– Type: G

– Extremely stable

3

5z zG F

2

3y yG F



Pasquill Conditions

Horizontal dispersion coefficient σy as a function of distance from

source for the various Pasquill conditions . (From D. H. Slade,

Editor, Meteorology and Atomic Energy-1968. Washington, D.C.: US

Atomic Energy Commission, 1968.)

Vertical dispersion coefficient az as a function of distance from source for the various Pasquill conditions. (From D. H. Slade, Editor, Meteorology and Atomic Ener gy-1 968. Washington, D.C. : US Atomic Energy Commission, 1968.)



Easy way to determine σy and σz

for computer coding

• Standard Terrain



Easy way to determine σy and σz

for computer coding

• City Terrain

Home Work:Plot σy and σz using MATLAB and compare with figures given on slide 13.



How to find Pasquill Categories???

Surface Wind

Speeda

m/s

Day

Incoming Solar Radiation

Night

Cloudinesse

Strongb Moderatec SlightdCloudy

(≥4/8)

Clear

(≤3/8)

< 2 A A-Bf B E F

2-3 A-B B C E F

3-5 B B-C C D E

5-6 C C-D D D D

> 6 C D D D Da Surface wind speed is measured at 10 m above the groundb Corresponds to clear summer day with sun higher than 60⁰ above the horizon.c Corresponds to a summer day with a few broken clouds, or a clear day with sun 35-60⁰ above horizon.d Corresponds to a fall afternoon, or a cloudy summer day, or clear summer day with the sun 15-35⁰.e Cloudinees is defined as the fraction of sky covered by clouds.f For A-B, B-C, or C-D conditions, average values should be obtained for each.

A = Very stable D = Neutral

B = Moderately unstable E = Slightly stable

C = Slightly unstable F = Stable

Regardless of wind speed, class D should by assumed for overcast conditions, day or night.



How to find Pasquill Categories???

Relationship between Pasquill category and temperature gradient and standard

deviation of the angle of the vane

From Regulatory Guide 1.23, USNRC (1980)



Effluent Dispersion

• χv / Q` rises to a maximum value.

• Decreases more or less exponentially.

• For more unstable conditions (A,B)

maxima occurs near the source point

and then decreases rapidly

• For stable conditions (E,F), peak is

located much further away from the

source.

• In the dispersion of effluents from

nuclear power plants, the concentration

of the effluent is usually higher in the

more important, populated off-site

regions under stable than under

unstable conditions.

• Stable conditions are often assumed

calculations of such effluent dispersion.The quantity χv / Q' at ground level, for effluents emitted at a height of 30 m, as a functionof distance from the source. (From D. H. Slade, Editor, Meteorology and Atomic Energy-1968. Washington, D.C.: US Atomic Energy Commission, 1968.)



Location of Maximum Concentration

Lets find location ofmaximum for variousatmospheric conditions

Considering the equation

Derivative is put equal tozero

Before differentiating Consider that both y and z

are function of x, thereforerelated as y = az

Taking Log. of both side of theequation is taken

Where C is composite ofconstants

Now differentiating w.r.t. x

and so

/ 2

2exp

2y z z

Q h

v

Ch

z

z ln2

ln2ln2

2

021

3

2

dx

dh

dx

d z

zz

2 22 or 2z zh h



Maximum Concentration

The maximum value of concentration is

where (yz)max means that y and z are to be

evaluated at the value of x determined from

/

max

max e y z

Q

v

2 2 2

or 2

z

z

h

h



Problem

Estimate the location of the maximum concentration

of a non-radioactive effluent released at a height of

30m under type F conditions

Solution:

– From figure/equation z = 21.2 m, that corresponds to

x = 1900 m

3021.2

2 2z

hm




• As the plume disperses in the atmosphere materialshall decay w.r.t its decay constant half life.

• Q/ is replaced with Qo/ exp(-t) where,

– Qo/ = rate of emission of radioactivity from source

– = decay constant– t = time required to reach the point of observation

• Assuming that• The effluent moves only with wind in x- direction and does

not meander, then

• The effluent conc. is given by:

The equation overestimates the value of at x (due to plume meander)

2

2exp

2

o

y z z

Q x h

v v



Deposition and Fallout

• The amount of radioactivity in an effluent plume may also

decrease with distance from the source as some of the

radioactivity falls out or diffuses out of the plume and deposits

on the ground.

• This effect is most pronounced for some types of radionuclides,

in particular, the isotopes of iodine, during periods of rain.

• Water droplets tend to pick up the radioactivity and carry it

directly to the ground, a process known as washout, or wet

deposition.

• Once radioactivity is deposited on the ground, it may provide a

significant source of radiation exposure.

• Radioactivity falling on food stuffs, pasture land, or bodies of

water may enter into the human food chain.




• Deposition rate per unit area of the ground istaken to be proportional to the effluentconcentration and is written as:

• where

– vd is a proportionality constant.

– If Rd has the units of Ci/m2-sec and χ is in Ci/m3,then vd has the units of m/sec and is called thedeposition velocity.

• χ is evaluated at or near ground level, and thevalue of vd is obtained from experiment.

d dR v




• The dry-deposition velocity for iodine rangesfrom about 0.002 to 0.01 m/sec;

• It is somewhat smaller for particulate fallout.

• The value of vd for wet deposition depends onthe height of the plume.

• For a nominal height of 103 m, vd isapproximately 0.2 m/sec for iodine and 0.1m/sec for particulates.

• The noble gases are not subject to either dry orwet deposition.




• Consider now the concentration of an effluentsubject to deposition as a function ofdistance from the source.

• Along the centerline of the plume, χ will bea function of only x and z .

• The total amount of effluent (radioactivity)in a volume element of the plume of unitwidth in the y-direction and thickness dx isgiven by the integral

z 0

dx x,z dz




• The rate at which this effluent decreases due

to deposition is then

• Noting that dx / dt = v, we find that the

average wind speed, above equation can be

written as

d

z 0

ddx x,z dz x,0 v dx

dt

d

z 0

x,0 v dxd x,z dz

v



Effective Plume Height

• Effective height of the plume is defined by

the relation

0

1z x,z dz

x,0

2 22 2

2 2 2 2

'exp exp

2 2 2 2 2y z y z y z

z h z hQ y y

v

2

2zz

hz exp22




• Introducing effective plume height into the equation below

• We get

• Where

• χ decreases exponentially with distance, due to deposition.

d

z 0

x,0 v dxd x,z dz

v

d x,0 x,0 dx

dv

vz




• Substituting this result into equation below

• We get

2

2exp

2

o

y z z

Q x h

v v

2

2exp

2

o

y z z

Q x h

v v

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Effluent Diffusion