X/Q for Releases From Area Sources

2009 RETS-REMP Workshop

Jim Key

Key Solutions, Inc.

www.keysolutionsinc.com

Concerns

• Industry Tritium Issues Have Revealed Many Unanalyzed Dose Pathways– Storm Drains– Ground Water– Service Water– Discharge Basins or Lakes With Little Water

Turnover

Evaporation From Area Sources

• Has Been Mostly Ignored

• Tritium Concentrations in Bodies of Water Can Continue to Build Up

• Release from Such Sources are Estimated to be 10 Ci/yr and Higher

Application of Gaussian Model to Release from Area Sources

• Simplify Gaussian Model As Follows– Ground Level Release– Ground Level Receptor

• Modify From Point Source Geometry to Square Area Geometry

Examine

• Point Source Plume Centerline

• Point Source Sector Average

• Area Source Plume Centerline

• Area Source Sector Average

Standard Gaussian Model

General Gaussian X/Q

2y

2

y 2

yExp

2

1

u

1Downwind Factor

Crosswind Factor

Vertical Factor

2z

2

2z

2

z 2

zHExp

2

zHExp

2

1

General Gaussian X/Q

2y

2

2z

2

2z

2

zy

2

yExp

2

zHExp

2

zHExp

u2

1Q

Horizontal and VerticalParameters

y(x) and z(x) are functions of

– Downwind Distance – x– Atmospheric Stability – Pasquill Category

y Lateral

Diffusion Coefficients

z

Vertical Diffusion

Coefficients

Atmospheric Stability CategoriesStability

Category Condition Description

Lapse

RateA Extremely Unstable Sunny Summer Weather -1.9

B Moderately Stable Sunny and Warm -1.9 to –1.7

C Slightly Unstable Average Day -1.7 to –1.5

D Neutral Overcast Day or Night -1.5 to –0.5

E Slightly Stable Average Night -0.5 to 1.5

F Moderately Stable Clear Night 1.5 to 4.0

G Highly Stable Added by NRC > 4.0

Simplifications

• Ground Level Release– Set H = 0

• Ground Level Receptor– Set z = 0

• Plume Centerline– Set y = 0

Ground Level ConcentrationGround Level Receptor

Plume CenterlinePoint Source

xxu

1Q

zy

Point Source Geometry

Receptor

Wind

Point Source

x

Sector Averaged Concentration

• Wind Directions in Each Sector are Distributed Randomly Over Period of Interest

• Calculate Average Value of /Q for Sector Length

Calculate Average Value of Function Over Sector Length

ab

xF

)x(F

b

ab,aAVE

Find Average Value of /Qover Sector Arc Length

LengthArc

dyQLengthArc

AveQ

16

x2Length Arc

Crosswind Integrated Concentration

dy2

yExp

16x2

1

u

1Q

2a

2a2y

2

zyAve

This term is cannot be integrated analytically

Easier to Use…

2y2

y

2

2dy2

yExp

From Standard Math Tables

Crosswind Integrated Concentration

2y

zy

2

16x2

1

u

1Q

• Function Of Only– Downwind Distance – x– Wind Speed - u

xu

03.2Q

z

Ground Level ConcentrationGround Level Receptor

Sector AveragePoint Source

Time-Averaged Concentration

• Wind Directions in Each Sector are Distributed Randomly Over Period of Interest

• Calculate X/Q Using Joint Frequency Distribution: f(,S,N) Direction– S Stability Class– N Wind Speed Class

Time-Averaged Concentration

• Allowed By NRC Guidance– Reg Guides 1.109– NUREGs 0133, 0472, 0473, 1301, 1302

• Less Scatter and Variability Than Real Data

• Dose Models Are Based On 1 Year Annual Exposure

/Q Variability

• Real Time/Short Term /Q– Factors of 3 to 10

• Long Term /Q– Factors of 2 to 4

From NCRP Report No. 76

Applying JFD Data to X/Q

• Use Average Wind Speed (Not Max Wind Speed)

• Determine yo for Each Stability Class

• Determine Virtual Distance (Xv) for Each Stability Class

Calculate X/Q Using:

Frequency theis N,S,f

Index Class Speed Wind theis N

Index ClassStability theis S

IndexSector theis

Where

N,S,fxu

03.2,xQ

N,S vNzS

Now Consider Area Source• Simplifications

– Ground Level Release– Ground Level Receptor

• Assume Point Source at Center of Release– Very Conservative– Does not consider that source is initially

distributed over large surface area.

• Plume Centerline• Sector Average

Area Source For Plume Centerline Assumes

• Ground Level Release

• Ground Level Receptor

• Simple Geometry

Simple Geometry for NearField Area Source

WindArea

Source

Receptor

2b

2a

Calculate Average Value of Function Over An Area

1212

x

x

y

yy,yx,xAREA yyxx

y,xQ

)y,x(Q

2

1

2

1

2121

• Integration Over Area of Source

• Calculates Plume Centerline Concentration

Ground Level Concentration

x2

yexp

xxu

1Q

2y

2

zy

Near field conditions or large area sources require that we consider y(x) and z(x) as

functions of x

Problem to Solve

1212

x

x

y

y2y

2

zy

AREA

yyxx

dydxx2

yexp

xxu1

Q

2

1

2

1

Problem to Solve - 2

a

a

b

b2y

2

zy

AREA

dydxx2

yexp

xx

1

ub2a2

1

Q

• Cannot Be Solved Analytically• Use Error Function for Integral Over dy

Error FunctionErf

x

0

t dte2

)x(Erf2

Error FunctionIdentities

)x(Erf)x(Erf

)x(Erf2

dtex

0

t2

a

a zy

b

b

2

y

dxxx

1dy

x2

yexp

uba4

1

Problem to Solve - 3

x2

bErf

y

Replace With

a

a zy

y

a

a zyy

dxxx

x2bErf

uba4

1

or

dxxx

1

x2

bErf

uba4

1

Problem to Solve - 4

a

a zy

y

Areadx

xx

x2bErf

uba4

1Q

Problem to Solve - 5

• Reduced to Integral of dx• Integrate Using Simpson’s Rule

Area Source For Sector Average

• Similar Development for Point Source Results In -

a

a zArea

dxux2ba

1Q

• Cannot Be Integrated Analytically• Integrate Using Simpson’s Rule

– Simpler Function to Integrate Numerically

Simple Case

• Calculate X/Q Assuming– Ground Level Release– Emission Source is One Mile Square– Receptor is Due West ½ Mile from Center of

Source (i.e. at Boundary)– Assume Worst Case Met Conditions

• Extremely Stabile (Class G)• Calm Conditions (0.04 m/s)• Least Dispersion

Example 1

• Ground Level Release• Emission Source is One Mile Square• Receptor is Due West ½ Mile from Center

of Source (i.e. at Area Boundary)• Assume Worst Case Met Conditions

– Extremely Stabile (Class G)– Calm Conditions (0.04 m/s)– Least Dispersion

Point Source vsArea Source

Receptor

Wind

Point Source

1600 meters

Area Source

Example 1 Calculations

Geometry /Q (m/sec2)

Point Source Plume Centerline 9.2E-02

Point Source Sector Average 1.5E-02

Area Source Plume Centerline 6.6E-03

Area Source Sector Average 3.2E-03

Source = 1 Square Mile

Receptor at Source Boundary

Simple X/Q for Area Source

• u = 0.022 m/s

• x = 20,800 m zG = 7.5 m

04E76.5800,20022.07.7

03.2QX

Geometry for Example 2

1600 meters

ReceptorPoint Source

Wind

3200 meters

Example 2 Calculations

Geometry /Q (m/sec2)

Point Source Plume Centerline 8.1E-03

Point Source Sector Average 1.2E-03

Area Source Plume Centerline 9.2E-06

Area Source Sector Average 5.7E-04

Source = 1 Square Mile

Receptor 2 Miles From Boundary

Point Source vs Area SourceX/Q

• Larger Sources – Expect Greater Difference

• As Distance to Receptor Increases Difference Slowly Decreases

Plume Centerline Comparison

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

800 1800 2800 3800 4800 5800 6800 7800 8800 9800

Distance (meters)

X/Q

(m

/sec

3 )

Area Source Plume Centerline

Point Source Plume Centerline

Sector Average Comparison

1.E-07

1.E-06

1.E-05

1.E-04

800 1800 2800 3800 4800 5800 6800 7800 8800 9800

x (meters)

X/Q

(m

/se

c3)

Area Source Sector Average

Point Source Sector Average

Sector Average Comparison

1.E-07

1.E-06

1.E-05

1.E-04

800 1800 2800 3800 4800 5800 6800 7800 8800 9800

x (meters)

X/Q

(m

/se

c3)

Area Source Sector Average

Point Source Sector Average

Point Source Sector Average

ALOHA