Tritium Washout by Precipitation : A Numerical Eulerian Stationary Model. Dimiter Atanassov National Institute of Meteorology and Hydrology 1784 Sofia,

Tritium Washout by Precipitation : Tritium Washout by Precipitation :

A Numerical Eulerian A Numerical Eulerian Stationary Model.Stationary Model.

Dimiter Atanassov Dimiter Atanassov

National Institute of Meteorology and HydrologyNational Institute of Meteorology and Hydrology1784 Sofia, 66 Tzarigradsko Chaussee1784 Sofia, 66 Tzarigradsko Chaussee;;

4000 Plovdiv, 139 Rouski Str.;4000 Plovdiv, 139 Rouski Str.;BulgariaBulgaria

[email protected]

Presented at: IAEA EMRAS Tritium/C-14 Working GroupPresented at: IAEA EMRAS Tritium/C-14 Working Group

2007 Spring Meeting, Bucharest2007 Spring Meeting, Bucharest

mailto:[email protected]

Rain Scavenging of Tritiated Water Vapour: A Numerical Eulerian Rain Scavenging of Tritiated Water Vapour: A Numerical Eulerian Stationary ModelStationary Model

The study was conducted at the Center of Excellence ofproject

IDRANAP(Inter-Disciplinary Research and Applications

based on Nuclear and Atomic Physics)

at National Institute of Physics and Nuclear Engineering "Horia Hulubei“, Bucharest-Magurele, Romania

Work package WP3 : The impact of tritium releases on environment and population,

Coordinator: Dan Galeriu

supporting by the FP 5 of the European Commission


ContentContent

• Introduction - the present state of the HTO washout problem, intention of the present study

• Individual raindrop problem

• Modelling of HTO washout from the atmosphere

• Sensitivity analysis of the model

• Some results and analysis

• Conclusion - future development of the model

Rain Scavenging of HTO Vapour…Rain Scavenging of HTO Vapour…Introduction - Introduction - the present state…intention of the study

The concept established by Jeremy M. Hales in 1972 is the basis of almost all

studies on washout of gaseous tritium (HTO) from the atmosphere. The concept is

not substantially modified till today.

At microphysical level, concerning the scavenging by individual raindrop, the

present study will closely follow the Hales’ approach. The knowledge on this topic will

be reviewed below, specifying exactly which assumptions will be accepted here.

Concerning the HTO washout in a rain event, there are two peculiarities of the

traditional approach, that will be reversed here.

1) Following Hales, all authors combine the washout model with a model for

dispersion of the gaseous HTO in the atmosphere. Gaussian plume dispersion

models are usually used, which allows analytic expressions for the washout output

characteristics to be obtained. The present washout model will be formulated

separately, without inclusion of a dispersion model. In this way, any kind of

dispersion model, including the most advance ones could be used.

4

Rain Scavenging of HTO Vapour…Rain Scavenging of HTO Vapour…Introduction - Introduction - the present state…intention of the study

2) The traditional final goal of the HTO washout studies is to determine some

"universal" coefficients (washout ratio and washout coefficient). After have ones

been established, they are applied for description of the washout process in more

general models, as coefficients in simple expressions, in order to determine the

washout output characteristics like downward flux and concentration in the rain

water. The problem in this approach is that the mentioned “universal” coefficients

accumulate the shortcomings of the Gaussian models. Second, any attempt to

determine the coefficients more precisely, lead the necessity to considered them as

functions of height of the HTO source, distance from it, atmospheric stability, air

temperature, etc…, whish is usually insolvable task in real situations.

A new, numerical Eulerian approach, free from the mentioned above shortcoming

will be proposed in the present study. The model will directly calculate the

demanded output characteristics, making needless the mentioned above “universal”

coefficients. The washout coefficient is considered here in order to express the

present results in a conventional way and for comparison with other studies.

5


ContentContent







Rain Scavenging of HTO… individual raindrop problem

bx xxkN 0

• liquid-phase step, and

It is convenient to consider the scavenging of gaseous HTO by an individual

0yykN by

• gas-phase step

One of the most significant peculiarities of the Hales’ works is the combination

eb yyKN

ey

mole-fraction - bx

HTO scavenging by individual raindrop

where: is gas-phase mole-fraction that would exist in equilibrium with the liquid

of the two steps, introducing an “overall” mass-transfer coefficient

K

0x

by

bx

0y

trough the drop’s surface :

raindrop as a consequence of two steps, represented by the following equations

NFor the HTO flux



be xfy

be xHy

The general relationship

in the special case where the system obeys Henry’s law, as the system water-HTO, reduces to the following linear relationship:

where is the Henry’s law constant.

The advantages are that the problem becomes one step problem and that the difficulty in determining the interfacial pollutant concentrations at the drop's surface is overcame. The disadvantage is in definition of the overall mass-transfer coefficient

H

yx kk

H

K

11

The equilibrium relationships can be employed to express the overall mass transfer

coefficient in terms of the andxk yk



Here, following Ogram(1985), the mass of gaseous and liquid HTO is expressed in

The previously discussed equations for the flux of HTO across the drop’s surface can be employed to express the condensation/evaporation of gaseous HTO to/from a raindrop by the following equation:

CCK

dt

dCg

d

6

MR

H

T

].[ 3cmgCconcentration of the liquid phase HTO in the drop, ].[ 3cmgCg

][d cmof the gas phase HTO in the drop’s environment, is the drop’s diameter,

is a dimensionless coefficient, constant with respect to

constant and on density and molecular weight of water. depending on temperature, Henry’s low constant, universal gas

C

][st is the time,term of concentration instead of mole-fraction; is

is concentration


• gas-phase limiting situationThe transport of liquid HTO within the drop is due by circulation of the liquid water

inside (convection) and by molecular diffusion. For a drop falling in the real

atmosphere, the convection motion inside are stimulated by the friction with the air,

and the HTO is mixed up so rapidly that the liquid-phase mass-transfer coefficient

becomes large enough (kx=) and the overall mass-transfer coefficient K= ky. In

this case the condensation of the HTO will be governed by the availability of the

gas-phase HTO in the air.

• stagnant drop case If there is no convection and the diffusion is the sole mechanism for transport, the

mixing inside the drop is weak. Then, there exists a thin surface layer whit high

liquid HTO pollutant concentrations, which are in equilibrium with the outside gas-

phase HTO. In this case the stagnation in the inside drop mixing will govern the

condensation.

Limiting situations - determination of the Overall Mass-Transfer Coefficient

10


The case of stagnant drop, represents the slowest rate of mass-transfer possible.

The gas-phase limiting case, represents the most intensive rate of mass-transfer

possible. These, therefore, provide lower and upper limits for washout behavior.

All atmospheric washout behavior should fall somewhere between these two limits.

Hales (1972a,b) and his coworkers Dana et al. (1978) always considered the two

limiting cases, gas-phase limiting and stagnant drop case.

The next authors (Ogram (1985), Abrol (1990), Belot (1998)) considered only the

more likely in the atmosphere gas-phase limiting case.

The gas-phase limiting case will be the only case considered in the present study.

HTO scavenging by individual raindropLimiting situations - determination of the Overall Mass-Transfer Coefficient

11


Determination of the Overall Mass-Transfer Coefficient

Following Bird et al.(1960), Hales and all next authors determine the gas-phase mass-transfer coefficients by the following semi-empirical expression, referred to as the Froessling equation:

3/12/1Re6.02d

ScD

kK gy

3/12/1vd

6.02d g

g

D

D

Finally, the HTO concentration in the drop turns to be a function of the following parameters:

][ 12 scmDgdiffusion coefficient of HTO, ][ 12 scm

is the drop’s velocity.

Re andwhere: is gas phaseis kinematic viscosity of the air,

Sc are Reynolds and Schmidt numbers,

].[ 1scmv

TpTKMRHTCtfC g ,,D,,d,,,,,,,d, gv

RpTMCHtfC g ,,,,,,,,d,~ v


Empirical formulas for raindrop’s downfall velocity

n

bz ea

d

1v1) Best (1950)

a=191, b=0.0158, n=1.754 for d < 0.015cm.

after Belot(1998)a=932, b=0.0885, n=1.174 for d > 0.015cm.

2) Best (1980) a=958, b=0.177, n=1.174 for all d.after Ogram(1985)

3) Kesler (1969)

1/2d1300v z after Ogram(1985)

4) Hales (1973)32 4023d-4507.4d-d4355v z

after Ogram(1985)

Common assumptions in the literature dedicated to HTO washout :

1) Changes of the drop’s size during the downfall is not taken into account 2) The drop’s velocity does not change during its downfall 3) drops are falling down strictly in vertical direction


Fig.2.1 Raindrop’s downfall velocity as a function of drop’s diameter, according to different authors

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1drop's diameter [cm]

velo

city

[cm

/s]

Best ( 1)

Best ( 2)

Kessler ( 3)

Hales ( 4)


Fig. 2.2. Time for passing through a 10m layer, according to the different drop’s downfall velocity formulas

0.1

1

10

100

1000

10000

100000

1000000

10000000

100000000

1E-06 1E-05 0.0001 0.001 0.01 0.1 1drop's diameter [cm]

time

[s] Best ( 1)

Best ( 2)

Kessler ( 3)

Hales ( 4)


Fig. 2.2. Time for passing through a 10m layer, according to the different drop’s downfall velocity formulas

0

2

4

6

8

10

12

14

0.01 0.1 1drop's diameter [cm]

time

[s]

Best ( 1)

Best ( 2)

Kessler ( 3)

Hales ( 4)


Solution of the individual raindrop problem

The basic equation for condensation / evaporation of HTO to / from the raindrop is an ordinary linear differential equation of the following type and it solution is:

If pTCg ,, and diameter d are constant to time , the solution reduces to :

where )(0 tCC is the concentration at the initial moment 0t

tends to the equilibrium concentration gCThe drop’s concentration )(tC

gCC 0gCC 0 condensation is going on; If evaporation is going on.if

t

a

t

xdxxA

t

adxxA

dxee xBaCtC '''')''(')'(

)'()()(BACdt

dC

)(d

6

d

6

0

01)(

ttK

g

K

eCeCtC


Fig.2.6 Non-dimensional liquid HTO concentration in raindrops of different diameters as a function of time; T = 150C, p=850 hPa.

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500 600 700 800

time [s]

conc

entr

atio

n d=0.02cm

d=0.4cm

d=0.9cm


Fig.2.8. Sensitivity to temperature: non-dimensional HTO concentration in raindrops of diameter 0.02 and 0.9cm, as a function of time,

for temperatures T= 0, 15, and 300C ; p=850hPa. The concentration curve for d=0.4cm and T= 150C is also presented.

0

0.2

0.4

0.6

0.8

1

1.2

1 10 100 1000 10000time [s]

conc

entr

atio

n 0.9cm

300C

0.9cm

00C

0.9cm

150C

0.02cm

300C

0.02cm

150C

0.02cm

00C

0.4cm

150C


Fig.2.7. Sensitivity to atmospheric pressure: non-dimensional HTO concentration in raindrops of different diameter,

as a function of time; T= 150C, p=850 and 100hPa. The short vertical lines mark the time, which is necessary a drop with

the corresponding diameter to pass through a 10m vertical layer. according to the downfall velocity formula of Best1

0

0.2

0.4

0.6

0.8

1

1.2

1 10 100 1000 10000time [s]

concentr

ation

0.02cm850hPa

0.02cm1000hPa

0.4cm850hPa

0.9cm850hPa

passing time d=0.4cm

0.9cm1000hPa

Rain Scavenging of HTO… HTO condensation at microphysical level

Drop size distribution (DSD)

1) Marshall-Palmer (MP):

after Belot(1998)

2) Sekhorn-Srivastava (SS): after Ogram(1985)

3) Best (B):

= 0.08 21.0d41exp R after Ogram(1985)

14.0d38exp R37.0R= 0.07

3d6

)d(w= )(dw

d

0.5W

25.2

2

d

c=

25.2

2

dexp

c

Drop size distribution : number of drops

)d(n

is within interval of unit length around

:

where:

In the all formulas the rainfall rate

is the total (non "spectral")W846.00236.0 R=

232.0000216.0 Rc cm,

[mm.h-1 =36000 ml.cm-2.s-1} is the sole inputR

)d(n

)d(n

volumetric liquid water contain of air.

parameter determining the DSD.

dn per unit space volume which diameter

d .

Rain Scavenging of HTO… Rain Scavenging of HTO… HTO condensation at microphysical level

Fig.3.1. Drop size distribution (DSD) for the rainfall rate of 0.5 and 30 mm/h, according different authors

1E-171E-151E-131E-111E-091E-071E-050.001

0.110

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9drop's diameter [cm]

num

bers

.cm

-3.c

m1

Marshall-Palmer Sekhorn-Srivastava Best


Rain Characteristics

dw =

The spectral volumetric liquid water flux [cm3.cm-2.s-1.cm-1]=[ml.cm-2.s-1.cm-1] and the total over the spectrum one [cm3.cm-2.s-1]=[ml.cm-2.s-1] could be calculated in the following way:

=

)d(n

Assuming the drops are spherical, the spectral volumetric liquid water content in the air [cm3.cm-3.cm-1]=[ml.cm-3.cm-1] and the total over the spectrum one [cm3.cm-3]=[ml.cm-3], by definition are:

3d6

d)d(

0

dw

0

3 d)d(6

dnd

=W =

)d(r )d(vz dw d)d(0

dr

0

d)d()d(v dwz=calcR =


Rain Characteristics - “normalization” procedure

=

The total rainfall rate appears in the calculations as an argument through the empirical formulas for DSD. In the model's applications, the rainfall rate will be taken from the observations. When is necessary to underline this, the rainfall rate will be denoted as - "measured" total rainfall rate. Because the formulas for drop’s downfall velocity and DSD are, in principle, non-coincidence empirical formulas, the calculated rainfall rate could not equal the "measured" rainfall rate, appearing as an argument in the mentioned formulas. In this way, the calculated spectral liquid water content of the air and the spectral rainfall rate do not correspond to the measurements, in sense that integral of the spectral rainfall rate over the spectrum, i.e. does not equal the measured value

)d(normr

=

measR

calcR measR

normalized to the measurements by the following way: The values of , calculated from the previous formulas, should be

corrected, and calcrcalcw

)d(calcr )d(calcr+calc

calcmeas

R

RR

)d(normw dv

d

z

normr

If are used, the calculated valueandnormw normr

This procedure represents a normalization to the input data. Its effect will be discussed below.

will equal the measured value

measRcalcR

.

.

24


Fig.3.2. Spectral volumetric liquid water content of the air, according to different DSD formulas,

in case of rainfall rate of 0.5 and 30 mm/h.

1E-17

1E-15

1E-13

1E-11

1E-09

1E-07

0.00001


[ml.c

m-3

]



Fig.3.3. Spectral rainfall rate, according to different DSD formulas

and drop’s downfall velocity formula of Best1, in case of total rainfall rate of 0.5 and 30 mm/h.

1E-171E-151E-131E-111E-091E-071E-050.001

0.110


[mm

.h-1

]



ContentContent







Rain Scavenging of HTO… Modelling of HTO washout from the atmosphere

Common assumptions, concerning rain phenomenon :

• the raindrops are spherical and their size remains constant during the downfall , • the drops are falling down in vertical direction with a constant to time velocity ,• there exists a level , that is suggested to be the level from which all the raindrops start their downfall, usually, it is assumed to be the cloud's bottom , • the raindrop spectrum is known, and is not changing with height , • the processes are stationary : the rain spectral characteristics, the geometry of the rain cloud, the geometry of the gaseous HTO cloud are not changing, during the typical time for which the smallest raindrops are falling down from the rainfall top level to the ground surface.

The listed assumptions make possible:

- the surface rainfall rate and the rainfall top level to be the only needed input data,

- the HTO washout models to be developed as 1-dimensional vertical models.

In the reality, satisfaction of most of these assumptions is not likely. Nevertheless, they are used in HTO washout studies and will be used here, because any refusal of some of them would lead necessity of :

- inclusion of a cloud model (water condensation, raindrop growth, in-cloud streams…)

- additional input information

Assumptions and input data, concerning the rain phenomenon

28


Meteorological data input :

• vertical profiles of the atmospheric pressure and atmospheric temperature.

Gas HTO data input :• the present model needs the profile of the gaseous HTO over the considered site.

The traditional HTO washout studies incorporate Gausian dispersion models. The HTO profile is calculated by the Gausian model, the input data for which are the HTO emission rate and the parameters for the HTO source: its height, distance to it, …

The model’s construction :

A modern HTO atmospheric transport modelling system should consist at least of the

following models: a model describing the dispersion of gaseous HTO in the space, dry

deposition and wet deposition (washout) models, and a meteorological preprocessor,

ensuring the necessary meteorological input for the mentioned models. The present

washout model is constructed as a subroutine of a such transport modelling system.

The washout model takes the gaseous HTO profile from the dispersion model,

calculates the washout, and returns the gaseous HTO profile, corrected by the

scavenged HTO by the rainfall. This exchange between the dispersion and the

washout models takes place at each time step, over each horizontal grid cell (point).

The model’s construction; meteorological and HTO input data

29


The domain of the model is between

the soil surface and the level Hrain

from which the drops start their

downfall. The uniform vertical grid is

defined. At the top level Hrain=z(N) we

assume the liquid HTO into the

raindrops is in equilibrium with the

surrounding gaseous HTO CN=Cequil.

The basic equation (1) is applied

layer by layer downward the level

Hrain, separately for all drop size

intervals. All parameters in equation

(1) are assumed constant within a

grid layer. This makes possible the

analytic solution (2) to be applied to

a drop for the time it is passing

through the grid layer.

HTO washout calculations - numerical scheme

)(d

6

d

6

0

01)(

ttK

g

K

eCeCtC CCK

dt

dCg

d

6

(1) (2)

h(i) grid z(i) grid

equilN CC

1Np , 1NT , 1NCg , h (N-1)

z (N-1), 1NC , 1Nf

2Np , 2NT , 2NCg , h (N-2)

z (N-2), 2NC , 2Nf

z (3), 3C , 3f

2p , 2T , 2Cg, h (2)

z (2), 2C , 2f

1p, 1T, 1Cg, h (1)

z (1)=0, 1C , 1f

Hrain = z (N)

30

Rain Scavenging of HTO… Rain Scavenging of HTO… Modelling of HTO washout from the atmosphere

HTO washout calculations - numerical scheme

Before to apply (2) to a drop with diameter d in layer i, its passing time t is

determined, according to the accepted formula for drops' downfall velocity.

The concentration C(d,i), calculated for this drop, after it has spent time t in the i-th

layer, is used as initial condition for the next i-1-st layer. The calculations for the last

1-st layer, the layer above the ground, give the spectral mass concentration of liquid

HTO in raindrops at the surface C(d,1).

31


HTO washout output characteristics

dC=

=

)d(n

Spectral and total mass concentration of liquid HTO in the air, i.e. mass of liquid HTO per unit space volume are:

3

6d

d)d()d(

0

dCwV =

)d(r )d(vz dw d)d(0

dfJ =

wC

Total mass concentration of liquid HTO in liquid water in the air,i.e. the total mass of liquid HTO per unit volume of water in the air is:

[g.cm-3]

=W

VcC = d)d()d(

1

0

dCwW

[g.cm-3] = [g.ml-1]

Spectral and total mass flux of liquid HTO, i.e. spectral and total mass of liquid HTO

passing through unit surface area perpendicular to

per unit time are :

dC)d(f = [g.cm-2.s –1 .cm –1] dC [g.cm-2.s –1]

zv

[g.cm-3 .cm-1]

[g.cm-3]= [g.ml-1]

If the flux , the result RJ of liquid HTO at the ground surface is divided to the rainfall rate is the mass concentration of liquid HTO in the falling rainwater:

=R

JfC



Mass concentration of liquid HTO in liquid water in the air:

=W

VcC = d)d()d()d(v

d)d()d(v

1

0

0

dCw

dw

z

z

Mass concentration of liquid HTO in the falling rainwater:

=R

JfCd)d()d(

d)d(

1

0

0

dCw

dw

=

Pumping Passive sampling

The differences obtained by model calculation will be shown later. The difference is expected to be detected in measurements by the following different technics:



=

Washout ratio:

1g

f

C

C

rW

Washout coefficient:

=HTO

JcW

0

)( dzzCgHTO =, [s-1] ,

The washout ratio and washout coefficient are invented as universal characteristics of the washout process. After they have ones been determined in theoretical studies, they are used to determine , and . Usually, they are coefficients in simple formulas like that:

1grf CWC ;

1gr RCWJ ,

HTOWJ c

cCfC J

WRCC WRCeC ,

Some comments:• The shortcomings of the washout ratio is obvious - only the surface is taken into account.• “The washout coefficient approach is most useful when the scavenging process can be considered irreversible. In case of HTO, irreversible conditions would only be expected to hold at very short travel distance. At longer travel distances a washout coefficient, based on irreversible scavenging, may significantly overestimate plum depletion. The “true” washout coefficient will therefore be a parameter that is a function of downwind distance”- Ogram(1985). • Hales(1972) suggested to overcome the problem by redefining the washout coefficient in term of a reversible process.• Belot (1998) introduced a reversible washout coefficient and considered it as a function of downwind distance and source’s height.

gC

34


HTO washout output characteristics - summary

Two types of outputs can be distinguished:

• Absolute characteristics:

liquid HTO concentration in the water in the air [g.ml-1] , liquid HTO concentration in the rainfall [g.ml-1] , and the downward flux of liquid HTO [g.cm-2.s –1].

•Relative characteristics:

like washout ratio [ ] washout coefficient [s –1].

J

cW

cC

fC

rW

The present model is designed to be used for direct calculation of the absolute characteristics; the washout coefficient will be considered below only in order to discuss the washout coefficient concept.


ContentContent







Rain Scavenging of HTO… Rain Scavenging of HTO… Sensitivity analysisSensitivity analysis

Local sensitivity criterion

Typical variability of the factors governing the washout process

symbol definition units range typical Numerical parameters

dRmin minimum raindrop size cm 0.000005 - 0.1 0.02 dz vertical grid step in the rain layer m 2 - 50 10

Cloud and Rain parameters R rain rate mm.h-1 0.5 - 30 3

zv downfall drop velocity formulas B, K B

DSD drop size distribution formulas M P, SS, B M P Atmospheric parameters

T temperature 0C 0 - 30 15 p atmospheric pressure hPa 1000 - 500 950

HTO gaseous cloud parameters )(zCg vertical profile of the HTO gas

(distance from the souse )

m

(100 - 33000)

2/,...,,;,...,,;

,...,,;,...,,;

12

12

tttttt

tttttt

dcbaAdcbaA

dcbaAdcbaA

,...,,, dcba

S =

If

will be used below as an assessment criterion for the local sensitivity of to the parameter .

A

is an output characteristic of the washout model, depending on the parameters

the quantity:

A

a

100%


• Calculation of will be performed many times for the same values , , for different values of where the lasts will take the values, formulas and profiles from the previous table.

• It is easy to prove, that the criterion will have the same value for , for andfor ; they are considered at ones in the following sensitivity analysis. The sensitivity of is considered separately.

• A simple Gaussian plum dispersion model is used only in order to generate profiles at 10 distances from the source, from 100m to 33000m. The source is at 60m height, the emission rate is 1000g.s-1, the wind speed is 10m.s-1 and the atmospheric stability is neutral. The sensitivity to the distance from the source could be considered as a sensitivity to the profile and magnitude of the gaseous HTO; close to the source, the vertical gradient of is significant, while far away from the source, tends to a constant of small magnitude.

S 1a 2a,...,, ttt dcb

S fCA JAcWA

cC

zCg

zCg

zCg

38


Sensitivity to DSD. The result of criterion S, maximum by absolute value with respect to the

distance to the source is presented. The temperature is constant to z , equal to 150C.

Sensitivity analysis

cC fC,J,Wc

R[mm/h] 3.0 0.5 30.0 3.0 0.5 30.0

zv of : Best Kessler Best Kessler Best Kessler Best Kessler Best Kessler Best Kessler

SS - MP 0.18 0.11 -12. -12. 22. 21. 0.13 0.13 -15. -13. 22. 22. B - MP -52. -50. -48. -47. -52. -50. -44. -45. -46. -46. -46. -48. B - SS -52. -50. -37. -36. -72. -69. -44. -46. -32. -34. -66. -69.

Sensitivity to the temperature - The value of criterion S is calculated for temperatures of 00C

and 150C; for the different DSDs, rain rates and downfall velocity formulas.

cC fC,J,Wc

R[mm/h] 3.0 0.5 30.0 3.0 0.5 30.0

zv of : Best Kessler Best Kessler Best Kessler Best Kessler Best Kessler Best Kessler

MP 49. 46. 62. 59. 32. 30. 38. 40. 53. 54. 23. 24. SS 49. 46. 58. 55. 37. 35. 38. 40. 48. 49. 27. 29. B 29. 27. 41. 39. 18. 17. 22. 23. 34. 35. 13. 14.


To the numerical parameters

• Consideration of raindrops which diameter is smaller then 0.02cm changes the washout outputs les then 1%. dRmin =0.02cm is used as a lower limit of integration.

• An increase of the vertical grid step dz from 2m to 10m changes the values of Cc, Cf

and J by -18% from its value in case of dz=2m; the value of Wc is changed by about

3%. An increase of dz from 2m to 20m changes the values of Cc , Cf and J by -86%

from its value in case of dz=2m; the value of Wc is changed by about 9%. These

assessments are the maximum ones for variation of temperature, pressure, DSD, drops downfall velocity in their ranges set in the table. They are obtained at a distance of 100m downwind the source. At a distance of 400m and more, the all assessments are less then 1%. A vertical grid step dz=10m could be recommended as appropriate one.

• Omission of the normalization procedure lead an error in J and Wc determination

which magnitude is up to 62% (in case of R=30.0mm.h-1, vz according to Best and

DSD of SS). The procedure does not affect Cf, as the effect on J and R is

compensated. The conclusion is that in any case the normalization procedure must be applied.

Summary of the sensitivity analysis

40


To the atmospheric and rain parameters

• Sensitivity to drop size distribution. The value of the criterion S varies from less then

1% (between DSD of SS and MP, in case of R=3mm.h-1, vz according to Kesler) up to

72% for A=Cc (between DSD of B and MP, in case of R=30mm.h-1, vz according to

Best) and 69% for A= Cf , J and Wc (between DSD of B and MP, in case of

R=30mm.h-1, vz according to Best).

• Sensitivity to drop’s downfall velocity. The maximum value of the criterion S is 8% (in case of R=3mm.h-1, DSD of MP and SS).

• Sensitivity to the temperature. The value of the criterion S varies from 13% (in case

of R=30mm.h-1, vz according to Best) up to 62% for A=Cc (in case of DSD of MP,

R=0.5mm.h-1, vz according to Best) and 54% for A= Cf , J and Wc (in case of DSD of

MP, R=0.5mm.h-1, vz according to Kesler).

• The effect caused by typical variations in atmospheric pressure is less the 2%.

CommentThe influence of the rain parameters and the temperature on the washout process is significant. Unlike the temperature which is usually well known, the rain parameters are difficult to be established precisely.

Summary of the sensitivity analysis

41


ContentContent







Rain Scavenging of HTO… Rain Scavenging of HTO… Some results and analysisSome results and analysis

A part of the parameters considered in the sensitivity analysis, after being once fixed, will be not varied in the model’s application. What will vary is: the rainfall rate, the profile of the gaseous HTO and the temperature profile. The Marshall-Palmer DSD and the Best’s formula for drops' downfall velocity are used in the following considerations.

Six Cg(z) distributions have been manually determined and the model has been run with

them instead to use the Gaussian dispersion model. The value of Cg is constant with z

in the first three “profiles”, but the magnitudes of Cg are too different: 1.0e-6, 1.0e-9 and

1.0e-21 g.cm-3, correspondingly for “profiles” №1,2,3. In the case of the next 3 profiles,

profiles №4,5,6, the amount of gaseous HTO in a vertical column is one and the same

= 0.6997e-5 g.cm-2, but their vertical distribution is quite different.

Fig.5.1

Model application

profile № 4 profile № 5 profile № 6

0E+0 5E-10 1E-9

Cg, g/ cm3

100

300

0

200

400

z , m

0E+0 5E-10 1E-9

Cg, g/ cm 3

100

300

0

200

400

0E+0 5E-10 1E-9

Cg, g/ cm 3

100

300

0

200

400

43


Spectral view of the washout process

0 .0 E + 0 5 .0 E -5 1 .0 E -4 1 .5 E -4 2 .0 E -4 2 .5 E -4

H T O co n cen tra tio n in to th e d ro p s , g /m l

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

z , m

T em p era tu re 1 5 d eg ree

d ro p s d iam e te r

E q u ilib riu m

0 .0 2 cm

0 .0 7 5 cm

0 .2 5 cm

0 .0 E + 0 5 .0 E -5 1 .0 E -4 1 .5 E -4 2 .0 E -4 2 .5 E -4

H T O co n cen tra tio n in to th e d ro p s , g /m l

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0T em p era tu re 0 d eg ree

Fig.5.2 Changes of HTO concentration into the raindrops of different diameter (0.02, 0.075, 0.25cm) during their downfall, for case of the gaseous HTO profile №5 and for two constant to z temperatures of 150C and 00C. The rainfall rate is 3mm.h-1, the DSD is according to Marshall-Palmer, the downfall velocity is according to Best.


• The equilibrium concentration is bigger and the drops can accumulate more HTO in the case of lower temperatures (compare the equilibrium curves - bolt). However, the processes of condensation and evaporation of HTO to and from the drops are more intensive in higher temperatures.

- During the downfall of the smallest drops (0.02cm), the HTO concentration in them follows closely the equilibrium curve in the case of 150C and lags behind it in the case of 00C. - The HTO concentration into the drops of diameter 0.075cm reaches close values (3.59e-05 g.ml-1 and 4.06e-05 g.ml-1, for the cases 150C and 00C, correspondingly) around the bottom of the HTO gaseous layer, but below that layer, the lose of HTO by evaporation is more intensive in the case of 150C. - For the big raindrops, the considered layer of gaseous HTO is obviously too thin and these drops accumulate a small part of the tritium they can accumulate.

•A change of the thickness or the position of the gaseous HTO layer can change the results substantially.

Spectral view of the washout process

45


Typical behaviour of the washout outputs - constant gaseous HTO profile

Fig.5.3 Washout characteristics for Cg= 1.0e-6 g.cm-3=const (“profile” №1), as a function of rainfall rate ( 0.5, 3.0, 30 mm.h-1), for temperatures 0 0C, 15 0C and 30 0C .

0.e+00

1.e-01

2.e-01

0.1 1 10 100

rain rate mm/h

a) HTO concentration in the water in the air cC

0.e+00

1.e-01

2.e-01

0.1 1 10 100

rain rate mm/h

b) HTO concentration in the rain water fC

1.e-07

1.e-06

1.e-05

1.e-04

1.e-03

0.1 1 10 100

rain rate mm/h

t=30Ct=15Ct=0C

c) HTO flux on the surface J

1.e-07

1.e-06

1.e-05

1.e-04

1.e-03

1.e-02

0.1 1 10 100rain rate mm/h

d)HTO washout coefficient Wc

The results for the two other cases of Cg(z)=const just scale with Cg : the bigger amount

of the gaseous tritium in the atmosphere leads higher concentrations Cc, Cf and higher

flux J; the washout coefficient remains the same.

Comments on the results

The interesting conclusion from these examples is that the both concentrations Cc and

Cf are almost independent on the rainfall rate R (Fig.5.3 a,b). The flux and the washout

coefficient are increasing linearly with the increasing of R in logarithmic axes (c,d).

46


a) HTO concentration in the water in the air

Cc [g/ml]

b) HTO concentration in rainfall

Cf [g/ml]

Typical behaviour of the washout outputs

Fig.5.4. Washout characteristics as a function of rainfall rate (0.5, 3. , 30. mm.h-1),

for Cg(z) profiles №4,5,6 with equable column HTO amount =0.7g/cm2; for 0,15,300C.


0.e+0

1.e-5

2.e-5

3.e-5

4.e-5

5.e-5

0.1 1 10 100

rain rate mm/h

0.e+0

1.e-5

2.e-5

3.e-5

4.e-5

5.e-5


0.e+0

1.e-5

2.e-5

3.e-5

4.e-5

5.e-5

0.1 1 10 100

rain rate mm/h

0.e+0

1.e-5

2.e-5

3.e-5

4.e-5

5.e-5


0.e+0

1.e-5

2.e-5

3.e-5

4.e-5

5.e-5

0.1 1 10 100

rain rate mm/h

0.e+0

1.e-5

2.e-5

3.e-5

4.e-5

5.e-5


0E+0 5E-10 1E-9

Cg, g/ cm 3

100

300

0

200

400

z , m

0E+0 5E-10 1E-9

Cg, g/ cm3

100

300

0

200

400

0E+0 5E-10 1E-9

Cg, g/ cm3

100

300

0

200

400

4

5

6


Now, when Cg(z) is not a constant, the concentrations CC and Cf depend in a complex

way on the rainfall rate R, on the Cg(z) profile and on the temperature. The both

concentrations CC and Cf are decreasing with increasing of R for temperature of 00C,

in different degree for the different Cg(z) profiles. For temperature of 300C, CC and Cf

are also decreasing with the increasing of R for profile №4, but for elevated gaseous

tritium clouds (profile №5), they are weakly increasing with the increasing of R. For

the profile №6 and temperature of 300C, CC and Cf depend weekly and

indeterminately on R, similarly to the case of Cg(z) = const, but for temperature of 00C

and 150C they are decreasing with increasing of R.

For all cases, the lower temperatures lead higher concentrations CC and Cf, higher

flux J and washout coefficient WC. For the concentrations this effect is more

significant for smaller rainfall rate R than for a bigger one. For the flux J and WC, the

effect is more significant for bigger rainfall than for a smaller rain rate.

Comments on the results

48


c) HTO flux on the ground

J [g/cm2.s]

linear y axis

c) HTO flux on the ground

J [g/cm2.s]

log y axis



0.e+00

2.e-09

4.e-09

6.e-09

8.e-09


0.e+00

2.e-09

4.e-09

6.e-09

8.e-09


1.e-11

1.e-10

1.e-09

1.e-08


1.e-11

1.e-10

1.e-09

1.e-08


1.e-11

1.e-10

1.e-09

1.e-08


0.e+00

2.e-09

4.e-09

6.e-09

8.e-09


0E+0 5E-10 1E-9

Cg, g/ cm 3

100

300

0

200

400

z , m

0E+0 5E-10 1E-9

Cg, g/ cm3

100

300

0

200

400

0E+0 5E-10 1E-9

Cg, g/ cm3

100

300

0

200

400

4

5

6




d) Washout coefficient

WC [1/s]

linear y axis

d) Washout coefficient

WC [1/s]

log y axis



0E+0 5E-10 1E-9

Cg, g/ cm 3

100

300

0

200

400

z , m

0E+0 5E-10 1E-9

Cg, g/ cm3

100

300

0

200

400

0E+0 5E-10 1E-9

Cg, g/ cm3

100

300

0

200

400

4

5

6



0.e+0

3.e-4

6.e-4

9.e-4

1.e-3


0.e+0

3.e-4

6.e-4

9.e-4

1.e-3


0.e+0

3.e-4

6.e-4

9.e-4

1.e-3


1.e-6

1.e-5

1.e-4

1.e-3

1.e-2


1.e-6

1.e-5

1.e-4

1.e-3

1.e-2


1.e-6

1.e-5

1.e-4

1.e-3

1.e-2



The dependence of J and WC on R is approximately linear in logarithmic axes.

However, if Cg is function of z, unlike the cases of Cg=const, the curve J(R) and

WC(R) has different slope for different temperatures and for different Cg(z) profiles.

Any attempt to define J or WC as a simple function of R will lead up to a necessity to

define different functions for the different temperatures and different Cg(z) profiles. In

case of incorporation of a Gaussian dispersion model, the last means to J and WC as

a functions of height of the source, distance to it, as well as a functions of atmospheric conditions.

It should be taken into account, that in case of logarithmic y-axis, a comparison of the results for different Cg

profiles is risky. For example, for R =0.5mm.h-1, the difference between the HTO flux for temperature 00C and 300C is bigger for profile №4 (=2.71e-10 g.sm-2.s-1), than for profile №5 (=1.83e-10 g.sm-2.s-1), despite it seems just opposite on the figure (Fig.5.4c - logarithmic y-axis). By this reason, the HTO flux and the washout coefficient is presented twice on Fig.5.4c,d : once in a linear y-axis and second in a logarithmic y-axis.

Obviously, the processes are so complex, that any change of the gaseous HTO layer thickness, a change of its height, a change of temperatures can change the tendencies and the results. In reality, the gaseous tritium and the temperature could be complex functions of z. A change of the rainfall rate will, on the other hand, change the rain spectral characteristics. The final effect from such changes on the washout outputs is difficult to be predicted without the help of a model.

Comments of the results

51



Fig.5.5. Washout characteristics for different Cg(z) distributions:

№1- 1.e-6, №2 - 1.e-9, №3 -1.e-21 g.cm-3 = const ; profiles №4,5,6 with equable column HTO amount = 0.7g/cm2, for rainfall rate ( 0.5, 3.0, 30 mm.h-1), temperature 15 0C.

0.0e+0

1.0e-5

2.0e-5

3.0e-5

3 4 5 6 7

gas HTO profiles

0.0e+0

1.0e-5

2.0e-5

3.0e-5

3 4 5 6 7

gas HTO profiles

3030.5

0.0e+0

2.5e-9

5.0e-9

7.5e-9

3 4 5 6 7gas HTO profiles

0.0e+0

2.5e-4

5.0e-4

7.5e-4

1.0e-3

0 1 2 3 4 5 6 7gas HTO profiles

a) HTO concentration in the water in the air

b) HTO concentration in the rain water

c) HTO flux on the surface

d) HTO washout coefficient

The significance of the elevation and the shape of the gaseous tritium cloud is obvious also from Fig.5.5. The concentrations CC and Cf the flux J are bigger when the cloud

of gaseous HTO is close to the ground (profile №4), because when it is high, the under layer evaporation causes a decrease of their values. The both concentrations CC and

Cf are more sensitive to the rainfall rate in case of surface gaseous HTO cloud.

52


Fig.6 Washout coefficient for different Cg(z): №1- 1.e-6, №2 - 1.e-9, №3 -1.e-21 g.cm-3 = const; № 4,5,6 with equable HTO amount = 0.7g/cm2, for rainfall rate 3.0mm.h-1, for different temperatures 0 0C, 15 0C, 30 0C.


Fig.5.5d is a good demonstration of advantages and disadvantages of the washout coefficient concept. If the gaseous HTO is distributed homogeneously over z, the value of the washout coefficient is one and the same, despite the big difference in the amount of gaseous tritium in the air. However, if the gaseous HTO is distributed in different manner over z, even the total amount of it is one and the same (profiles №4,5,6), the

values of WC are too different.

The washout coefficient depends significantly on the temperature - Fig.6.

That is way, if one wishes to use the washout coefficient concept, the

coefficient WC should be a function

of temperature and the Cg(z) profile,

except of the rainfall rate.

0.0e+0

5.0e-5

1.0e-4

1.5e-4

2.0e-4

2.5e-4

0 1 2 3 4 5 6 7gas HTO profiles

t=30C

t=15C

t=0C

53


Source height : H=60m H=30m H=10m

Comparison of the present’s model and analytic solution [Belot(1998)]

Washout coefficient as a function of downwind distance, according to the present numerical model and according to the analytic solution of Belot(1998) for different height of the gaseous HTO source.

Reasons for the differences:- numerical approximation - dependence of Henry’s low coefficient, gas phase diffusion coefficient of HTO and kinematic viscosity of air on temperature and atmospheric pressure. - normalization procedure

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

distance (m)

analytic

irrev

norm

calc

irrev.

zo = 1 mt = 18°C

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

distance (m)

analytic

irrev

norm

calc

irrev.

zo = 1 mt = 18°C

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

distance (m)

analytic

irrev

norm

calc

irrev.

zo = 1 mt = 18°C


ContentContent







Rain Scavenging of HTO… Rain Scavenging of HTO… Conclusion - future development of the modelConclusion - future development of the model

Conclusion

The study follows the approach established by Hales (1972a,b) concerning the processes in an individual raindrop, but concerning the washout process, some innovations are made: 1) A new, numerical Eulerian approach to the washout process is proposed. The corresponding model describes only the washout process, without to consider the dispersion. The model is designed to be used as a numerical subroutine in general models dealing with HTO problems. The present model can be combined with any advance dispersion model. As a result, the washout model becomes free from the non-inherent duty to take into account the height of the source, distance from it, etc., and these tasks are forwarded to the dispersion model, which is by idea constructed to deal with them. 2) In the traditional studies, the goal is to determine the relative characteristics like washout coefficient and ratio. However, later in applications, these characteristics are used as coefficients in simple formulas in order to determine the absolute washout characteristics like HTO downward flux and concentration in the rain water. The present model is constructed to determine directly the absolute washout characteristics.

56


Conclusion

Different rain drop’s size distributions and downfall velocities formulas have been considered. A sensitivity analysis to all of them and to other model’s parameters has been performed. The analysis established recommendable values for some numerical parameters and procedures. A vertical grid step of 10m is an appropriate choice, especially close to a HTO source. The calculated spectral liquid water content in the air and the liquid water flux have to be normalized to the measured rainfall rate. The sensitivity analysis has also shown that both the rain parameters and the temperature are influencing significantly the washout process. Unlike the temperature which is usually well known, the rain parameters are difficult to be established precisely, that could make the model's outputs substantially wrong.


Conclusion The washout coefficient WC can be determined as a relatively simple function of the

rainfall rate R in case of constant vertical distribution of the gaseous HTO Cg(z).

However, if Cg(z) is not constant, the relationship between WC and R becomes

dependent on Cg(z) , i.e. on the characteristics of the HTO source and atmospheric

conditions and the dependence on the temperature becomes more complex. It is not easy to define such a function even theoretically; what is more, in practice there could be more then one HTO source and the temperature is changing with height. Obviously, the washout processes are too complex to be described comprehensively by simple parametresations like the washout coefficient concept. A proposed here alternative is to use the present model, determining directly HTO concentration in the rainwater CC and Cf and HTO downward flux J. Anyway, the

washout coefficient is needed mainly to determine exactly these characteristics. On the other hand, the present model gives the washout coefficient also, if anybody needs it, but it will take realistic different values at different sites at different moments.


Conclusion The present study makes some steps toward modernization of the HTO washout simulations, however significant shortcomings still remain. More of them are related to the description of the cloud-rain processes: the assumption for stationarity of the processes, the assumption that the raindrops are falling down without changing their size and velocity are unrealistic. The solution of these problems is related with provision of a better input information for the cloud-rain processes. The all washout models for now, including the present one, use only the rainfall rate as the sole input data for the rain phenomena. Today is easy to get much more information; the radars seem to be the most promising source of data. They are able to ensure information for the cloud and rain drops distribution in space and time, for their spectrum and movements (sometimes the drops are moving even upward). A substantial reconstruction of the modeling approach will be necessary in order the models to be able to incorporate this information.


Future development of the model

1) Modification of the program code as a subroutine in order to be used in general HTO models.

2) Validation of the model’s results on field measurements.

3) Determination of site , climatic specific cloud-rain characteristics.

4) Extension of the model for cases of fog and snowfall.

5) Inclusion of the radar measurements as input data

ENDEND

thank you for attentionthank you for attention

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Tritium Washout by Precipitation : A Numerical Eulerian Stationary Model. Dimiter Atanassov National Institute of Meteorology and Hydrology 1784 Sofia,