Technical Not*
                                    ORP/lV-77-4
          GENERALIZED MODEL OF
THE TIME-DEPENDENT WEATHERING HALF-LIFE
       OF THE RESUSPENSION FACTOR
              FEBRUARY 1977

      U.S. ENVIRONMENTAL PROTECTION AGENCY
         OFFICE OF RADIATION PROGRAMS
              LAS VEGAS FACILITY
           LAS VEGAS, NEVADA  89114

-------
                                   Technical  Note

                                    ORP/LV-77-4
             GENERALIZED MODEL OF


    THE TIME-DEPENDENT WEATHERING HALF-LIFE


          OF THE RESUSPENSION FACTOR
          GEORGE V. OKSZA-CHOCIMOWSKI
                 FEBRUARY 1977
Office of Radiation Programs-Las Vegas Facility
     U.S. Environmental Protection Agency
           Las Vegas, Nevada  89114

-------
                                  PREFACE


     The Office of Radiation Programs of the U.S.  Environmental  Protection
Agency carries out a national program designed to  evaluate population exposure
to ionizing and non-ionizing radiation, and to promote development of controls
necessary to protect the public health and safety.

     Radioactive contaminants discharged from facilities in the nuclear fuel
cycle may deposit on the ground surface, to be subsequently resuspended and,
possibly, inhaled by members of the general public.  This report introduces
time-dependent models of empirical parameters commonly used in characterizing
the extent and duration of the significant hazards posed by resuspension.
Readers of this report are encouraged to inform the Office of Radiation Programs
of any errors or omissions.  Comments or requests  for further information are
invited.
                                   Donald VI. Hendricks
                                   Director, Office of
                                 Radiation Programs, LVF
                                        ii

-------
                             TABLE OF CONTENTS

                                                                Page
ABSTRACT                                                          iV
LIST OF FIGURES                                                    V
LIST OF TABLES                                                  vlii
LIST OF SYMBOLS                                                   ix
ACKNOWLEDGMENT                                                    xi
INTRODUCTION                                                       1
RESUSPENSION FACTOR MODELS                                         5
TIME-DEPENDENT HALF-TIME MODEL BASED ON ANSPAUGH'S MODEL
     OF THE RESUSPENSION FACTOR                                   12
BASES OF PROPOSED INTERIM MODEL OF THE TIME-DEPENDENT
     HALF-TIME                                                    17
     Initial Values of T,  (t, Rf(o)/Rf(»))                        17
     Values of T^(t, Rf(o)/Rf(~)) Within One Year After
       Deposition                                                 18
     Values of T^ (t, Rf(o)/Rf(»)) After One Year From
       Deposition                                                 22
     Final Values of 1% (t, Rf(o)/Rf(oo))                          25
PROPOSED INTERIM MODEL OF TIME-DEPENDENT HALF-TIME                27
APPLICATION                                                       33
SUMMARY AND COMMENTS                                              40
REFERENCES                                                        41
                                    ill

-------
                                  ABSTRACT
     A generalized model has been developed to predict the changes with time
in the weathering half-life of the resuspension factor for plutonium 239 and
other long-lived radioactive contaminants.   The model  is largely based on
assumptions and empirical data presented by other authors, and is applicable
to a wide range of average conditions akin  to those for which data is available.
These conditions are parametrically described as ratios of initial and final
resuspension factors, valid for a given locality.

     As a direct application of the above model of time-dependent half-life,
the corresponding general model of time-dependent resuspension factor, also by
the present author, is introduced in the report.  Included in the latter are
graphs of both models for a range of conditions, as well as graphic comparisons
of specific cases of these general models with models  proposed by others.
                                       iv

-------
                               LIST OF FIGURES

Number                                                                Page
  1       Half-Time T.  as function of time and local  conditions
          (expressed as ratios of initial-to-final  resuspension
          factors, R^(o)/Rf(«)), as derived (Equation (14))  from
          Anspaugh's model of Rf(t).                                   14

  2       Tungsten-181  air activity as a function of time at
          five downwind stations within six miles that were
          closest to the line of maximum deposition from Pro-
          ject Schooner [From Anspaugh et al.  (1973)].                 18

  3       Median weekly air concentrations, corresponding to
          three isopleths, from Operation Plumbbob [From Wilson
          et al. (I960)].                                              20

  4       Least-square fit to gross-gamma air activity levels
          three to eleven months following Baneberry venting
          [From Anspaugh et al. (1973)].                               21

  5       Contour representation of plutonium-239 distribution
          at Rocky Flats [From Volchok (1971)].                         23

  6       Airborne plutonium at Sampling Station S-8.  Adapted
          from Sehmel and Orgill (1974).                               24

  7       Half-time T,  as function of time and local  conditions
                     %
          (expressed as ratios of initial-to-final  resuspension
          factors Rf(o)/Rf(»)) according to proposed model and
          "Case 2" constants (Table 2).  Curves  numbered accord-
          ing to ratio Rf(o)/Rf(~), in orders of magnitude,  i.e.,
          "2" corresponds to Rf(o)/Rf(~) = 102,  etc.                    29

-------
Number                                                                Page
  8       Half-time T,  as function of time and local conditions
                     *5
          (expressed as ratios of initial-to-final resuspension
          factors Rf(o)/Rf(»)) according to proposed model
          (Equation (33)) and "Case 1" constants (Table 2).
          Curves shown correspond to upper and lower limits  of
          range of values presumed possible for Rf(o)/Rf(»).           31

  9       Resuspension factors as functions of time based on
          the proposed model of time-dependent half-time T, ,
                                                          %
          assuming various initial values but the same final
          resuspension factors.  The first captioned ordinate
          represents Rf(t) values presumed possible based on
          Table 1.  Some of the higher values do not apply to
          outdoors pollutant resuspension by wind, under nor-
          mal conditions, but have been included for the pur-
          pose of demonstrating applications of the general
          model.  The second captioned ordinate pertains to
          the generalized case.                                        3*

 10       Resuspension factors as functions of time based on
          the proposed model of time-dependent half-time T,,
          assuming the same initial value but various final
          resuspension factors.  The first captioned ordinate
          shows values presumed possible based on Table 1.
                                 -2  -1
          The initial value of 10   m   is highly improbable
          for outdoor pollutant resuspension by wind, under
          normal conditions, but serves to illustrate appli-
          cations of the model for the full range of values
          postulated in the model development.  The second
          captioned ordinate pertains to the generalized
                                                                       Ot)
          case.
                                        vi

-------
Number                                                           Page
 11       Three expressions of the time-dependent resuspension
          factor based on the proposed model  of the time-depen-
          dent half-time T^, with initial  and final values as
          assumed in previous models with  which these expres-
          sions are compared.  See also Table 3.                  37
 12       Two forms of the proposed model of weathering half-
          time T^ as function of time and local  conditions
          (expressed) as ratios of initial-to-final  resus-
          pension factors Rf(o)/R-(»), compared(T)with
          Anspaugh's model and(J)with Langham's  and Kathren's
          models.  Values of RAo) and Rf(») employed in the
          proposed model correspond to those used in the com-
          parison models.                                         38
                                      vii

-------
                               LIST OF TABLES
Number                                                           Page
  1       Resuspension factors for plutonium and other
          radiolsotopes [From Mishima (1964)]                      3
          Examples of constants used in general  model             28

          Comparison of proposed model (Case 1)  of the
          time-dependent weathering half-time T,  with
          previous models, implicit or derived,  and
          corresponding models of the time-dependent
          resuspension factor, including numerical
          values                                                  36
                                    viii

-------
                               LIST OF SYMBOLS

     The numbers in parentheses following the description  of  each  symbol
refer to the equations in which the symbols are first used or defined.   The
symbols have been separated into these groups, according  to their  primary
function in the context of this report, as follows:   constant parameters,
time-dependent parameters or variables, and special  values of time.   Some
explanations are included.

     A  =  constant coefficient, days, in (31)
     B  =  constant, dimensionless, in (31)
     C  =  constant coefficient, day  , in (31)
     D  =  constant exponent, dimensionless, in (31)
     K  =  constant coefficient, (days)   , in (11)
     Rf =  constant resuspension factor, m  , in (1), (4)
     Rf(o)/Rf(«)  =  ratio of initial-to-final resuspension factors,
                     determined or predicted for a given  site, in
                     (14), (24)
     S. =  total contamination per unit area, regardless  of depth
                                                         2
           at which contaminant is found, constant,  uCi/m , in (8)
     T^ =  weathering half-life, also "half-time", days,  in (2)
     A^ =  constant attenuation factor, days  , in (2)

     C(t)  =  time-dependent airborne concentration, yCi/m ,  in  (3)
     R*(t) =  time-dependent resuspension factor, m~ , in (8)
                                                         2
     S,(t) =  time-dependent surface contamination,  yCi/m , in (4)
      a
     Ti (t) =  time-dependent weathering half-life, days,  in (12)
     TjJt-)=  empirically obtained weathering half-times,  days,
              corresponding to indexed values of time t-,  where
              i = 1, 2, 3, in (25), (26), 27)
     T, (t-.-,)   = minimum value of weathering half-life,  for values
      *2  mi n
                  of t > t,, (see infra), days, in (18).   Literally,
                          uS
                  the expression signifies "the value of  T^ at a
                  time tmin (see infra) corresponding to  a minimum
                  value of T,".  Such awkward notation was adapted
                            %
                  to maintain continuity of exposition in the text
     A(t)  =  time-dependent attenuation factor, days" ,  in (12)
                                     ix

-------
t   =  time, days
t   =  time at which values of T,  become infinitely large,  days,
       in (16), (17)
ti  =  indexed values of time corresponding to empirically
       observed weathering half-times, days, in (25),  (26),
       (27)
 min=  time, days for t > tfls, at which the weathering half-time
       Tj, reaches a minimum, in (18)

-------
                              ACKNOWLEDGMENT
     The author gratefully acknowledges the assistance and advice of numerous
individuals in the preparation of this report.   Special  recognition is  extended
to Dr.  W. Wood, of the Office of Radiation Programs,  Criteria and Standards
Division, of the Environmental Protection Agency,  and to Mr.  V.  E.  Andrews,
Dr. P.  N. Lem and Mr.  H. K. Maunu of the Office of Research and  Development,
Environmental Monitoring and Support Laboratory (EMSL) - Las Vegas, of  the
same agency.

     The author, although recognizing the assistance  of others,  accepts full
responsibility for the contents of this report.
                                      XI

-------
                                INTRODUCTION

     Particulate matter deposited on a surface can be resuspended by mechani-
cal action or wind forces, phenomena familiar to anyone who has erased a
blackboard, or complained about dust on a windy day.  When material thus
prone to resuspension includes radioactive particles, as in localities
contaminated, in some cases, by nearby nuclear facilities, continuous occu-
pancy of such areas may present a potential health hazard, ascribed to the
possible inhalation of radioactive particulates of respirable size.

     The extent to which radioactive contaminants can be resuspended from
the ground or other environmental surfaces is often characterized by an
empirically determined quantity, the "resuspension factor".  This is defined
as the ratio of volumetric airborne pollutant concentration at some given
height to the areal concentration of the pollutant on the ground.  The height
of interest may be specified to be within the "breathing zone" of the average
individual, from which respirable aerosols may penetrate into the indivi-
dual's lungs.  Coupling thusly obtained resuspension factors with some real-
istic assumptions regarding the fraction of respirable-size particles pre-
sent in the total concentration of breathing height, these resuspension fac-
tors may provide a rough index of potential inhalation concentration due to
a given ground concentration of contaminant.

     Some criticism can be leveled at the concept of resuspension factor in
general.  In the first place, it assumes that the air concentration above a
contaminated surface is directly proportional to this surface contamination
level, rather than on the extent of ground contamination upwind of the sam-
pling site, as would be more logical and, in fact, was found to be true
(Stewart, 1967; Mishima, 1964).   In the second place, the resuspension fac-
tor, as a constant of proportionality relating air and ground concentrations,
is meant to include the effects of a myriad parameters, such as wind velo-
city, surface roughness, physical and chemical characteristics of both pollu-
tant and soil surface, vegetation cover, etc.  Whereas some of these factors
may change little at a particular site, some others, such as wind velocity,
tend to be less constant.  Consequently, the resuspension factor must be

                                      1

-------
considered to be, in a strict sense, an empirically determined value  applying
only to specific or prevailing conditions at a given site,  at a given time or
for a specified time period, respectively.  Thus,  it is not surprising that
the measured val
(Mishima, 1964).
                                      -21        13  1
the measured values of R  vary from 10  m   to 10   m  ,  as  shown in  Table 1
     In view of the described inaccuracies, continued use of resuspension
factors may be justifiable only under such conditions as would tend to mini-
mize the liabilities of the concept, or allow to compensate for these adverse
characteristics.  Ideally, these conditions would include uniform distribu-
tion, unchanging with time, of the contaminant over a large ("infinite")
area, in the absence of additionally contributing "sources" of significance
(inadequately operating nuclear facilities) or "sinks" (rivers, lakes, other
large bodies of water) and of unpredictable or extreme climatic conditions.
To simplify discussion, such optional conditions are assumed to exist for
all cases considered in this report, and to have existed whenever actual
numerical values used in the report were originally obtained.

     Evaluation of potential health effects from continuous occupancy of  a
contaminated region involves both estimations of the amount of material apt
to be resuspended and the duration of the hazard.  The latter necessitates
establishing some reliable or credible models of the decrease of airborne
concentration with time, which may be effected by describing correspondingly
varying time-dependent resuspension factors.  The present report cites a  number
of observations indicating that concentrations of resuspended materials do,
indeed, decrease with time, and several models based on such observations.

     The models mentioned above had to be, to considerable degree, site-
specific (corresponding to conditions an exact replica of which would not be
found at any other site) and also "date-specific" (produced by circumstances
that would never be exactly repeated, at a given site, at any later time).
The relative abundance of independent observations, when contrasted with  the
paucity of reliable data on which accurate models could be based, suggested
the advantage of developing a general model that would partake of the fea-
tures and assumptions of previous models, while retaining enough flexibility
to allow incorporation of new data.

-------
TABLE  1     RESUSPENSION  FACTORS  FOR  PLUTONIUM AND OTHER  RADIOISOTOPES
                [FROM  MISHIMA  (1964)]
                                   Conditions of Rcnujptnglon
            Average Hc«\ifpension F»dni  in ArcidcnU Involving Plutonium
            Vnhlrnlnr Truffle (N>-vi»ln)
            People \lorklnc or  Acltvr in n Closed Arra
            Dirly nin.il. Sulmrlon. or Mi tinpnlilan Areas
            People XVnrkini; or  /irlivc in .in Oprn Arr-ji
            Isolated  Aie*

            nefiuspcnalr-n of Ar.ed  I'luloniiim Deposit (0.74 to 152 gCI/m2) from  "Plumbob"


            Plutonium Oxide, Nn Movement
            Plutonium Oxldr, H slvps/miii
            Pluloniinii Oxide. 3C ilip^/mm
            Plutonium Nitrate  No Movemrm
            Plutonium N'nrale.  M  slepi/min
            Plutonium Nitrate.  .Ifi  slept/mm

            Plulonnim Oxide. Change Rnnm (>3000 ft2). 9 air chanfcs/hr. 0.01  uCI/ni*.
            4 to C |iTisoni, active, in JTCJ
                "Loose" Cnnlannn.inon (pfitunjtetl liy smear a)
                "Loose" Cnnlan.inalion (cMlmMi'H b> unlrr-rall«. Personal ! ampler.  No Ventilation
            Personnel Traflir In a Small Unvrntilalcd Roam
            Proposed rtesuspenqion Factors foi Plutonium Oxide:
                Ould.xir-: (iiinesn  nt cnnrliiion.i)
                Outdoors (mnricr.ilc ailmlv)
                Indoors (quiescent co.vlitmns)
                Inrlor.rs (mndrrair acllvilv)

            Krom Crater  of lowir Shot. No Artldclal Disturbance
            Survey of Hnail. No >\HiMri.il Diihirbnncp
            Sur\cy ol lload. L.mdrovci . ;)-l).i> ^ 4
            Survo\ of Ho.id. I anclroicr. D-Dav > 1
            Surxcy of fload. T.iillxiard of I.androvcr, U-Dny « 7
            Survey of rto.io. D-l)^> •  1 and 2
            Sample Collrr'.inn in Ca!i of I aninovi-r,  ll-lloiir + 5
            Sample Colirrlioii in Calj of I nn irovur,  ll-lloin « 8
            Ur:inii.m S«in,ilr Downulnd 01 Ciulcr, Sample Height. I ft Above Ground
            Uiarmnn Sain;.">- Do^nuind of Crater, Dust Stirred Up. Sample Height: 1 (I
            Uranium Sample Dnunwind of Crater, Sampln Height. 2 II
            Plutonium Sani|ilcil I ft Ahnxc  Ground.
                Vclnculai Dust
                T'eilcslri.Tn Dust
            Iodine-13!,  DnLlnird (Chamberlain t Stanbury)

            Iodine-131,  Oprn (Chamberlain 6 Slanbury)
            Yllrlum-ni. 0-R a P.irticlos. N.iiural lurhiilcnce,  Sampled 1 ft Above Ground;
                Giounri Com animal ion I evil  1.0
                Ground Cnniamination 1 eve I  S 8
                Grnunil Cont.iiiiinatinn I i-vel  74 6
            Polonium-210, 0-lt u r.irlirlc'-.  Natural 1 uHjulrncr, Sampled I ft Above Ground:
                Ground Coiilnmm.ilinn I <-\el  0. C uCi/m"
                Grounil Conl animal ion I ctcl  j uCi/m~
                (1. 0-4 u I'ariicles, Natur.il Turliuh-nce.  Sampled I ft Above Ground:
                Gruuncl Cnnl.iiniiutinn I e\il  11^ i;/Jii^
                Grass Conl.iinin.ilinn I rvcl 70 u/m^
                Concrrtc  Contamination Level IRO c/m^

            Plutonium Omilc, Samplmi; jlclRhl  *> fl
                I Inor Lc\i'l 0. I uCi/m".  Nn Circulation
                Floor l.cvrl 24 C uCi/nj'.  No Clrculfltion
                t lixir Level 0. I uCi/m2   I .in
                Floor Level 0.91 uCi/m^.  Ian
                I*loor Li'vrl 0  DUG i,Ci/m*.  Tan and Dolly
                Kloor Level l.^uCum2.  Can and Dolly
                Floor l.rvel I. 3 bCilm*.  After TrM
                Door Level I. I i.Ci/m'.  After lest

            Uranium, Samnlinq llcirhl-  '•  (1
                Kluor I uvfl 0  00£uCi/i«',  No Circulation
                Floor l.cvrl n.Ti uCi/m-. Nn (,'irctilnlmn
                I- Ivor Lcvrl 0.0 IS ».Ci/in',  I .m
                Floor Lcvrl I. I uCi/ni-.  I in
                Floor Lexcl 0. II uCi/iu'. l>«llv
                Floor l.c\cl I  3 uCi/m',  Dolly
                Floor I eiel 0.11 uCI/ni*. Tan and  Roily
                Floor Level 1. 0 uCI/in^.  Fan and Utjlly
                     Reeunpenelon Fictor
                           * in-;
                           x 10'*
                           x l
-------
     The present report describes what may well be called a "back-door"  ap-
proach in developing a general model of the time-dependent resuspension  fac-
tor.  It is rooted initially in a collation of known or accepted facts about
rates of decrease of airborne contamination, as characterized by the "decay
half-life" or "half-time" of such decrease, originally assumed to be exponen-
tial.  Based on these facts, a general model of the "time-dependent weathering
half-life of the resuspension factor" is developed, followed by the corres-
ponding general model of the resuspension factor itself.  Conditions repre-
sentative of the various sites from which data is available are operationally
described by "ratios of initial-to-final resuspension factors", as observed
or expected at each of the given sites.

-------
                         RESUSPENSION FACTOR MODELS

     One of the earliest attempts to devise a simple means  of  predicting  the
extent of resuspension of pollutant from a previously contaminated  surface
dates back to 1956.  P.  S.  Harris and W.  H. Langham  correlated surface  de-
position and air concentration measurements of piutoniurn at the Nevada  Test
Site (NTS) by defining a quantity known as the "resuspension factor", Rp
(Langham, 1971)

     R   _     Air Concentration (pCi plutom'um/m )               /^
               Surface Deposition (uCi plutom'um/m )

From measurements made at two different times following a contaminating event,
under circumstances involving "extensive vehicular traffic", it was concluded
that Rp = 7 x 10  m   applied to "disturbed Nevada desert conditions".

     In addition, an "attenuation factor", XA, was calculated  to describe the
exponential decline in air concentration with time,  due to progressive  reduc-
tion of the amount of contaminant available for resuspension.   For  the  condi-
tions previously described, AA corresponded to a "half-time",  T, , of 35 days
(Langham, 1969, 1971)


     XA   =    -r^  = srlsfs  =   °-0198 da*s -1               (2)

Other values of T,  have been estimated or proposed,  such as T,  = 40 days, for
prevailing conditions at NTS (Langham, 1971), and T,  = 45 days, by  Kathren
(1968), in his lung dose model.

     Applying the "attenuation factor" concept, the  time-dependent  concentra-
tion of resuspended contaminant may be represented by the following expression:

          C(t)  =  C(o)  e'V

-------
                                                                     3
where     C(t)  =  airborne contaminant concentration  at  time  t,  yCi/m
          C(o)  =  initial airborne contaminant concentration,  at some
                                                         3
                   arbitrarily assigned time t = o,  yCi/m
            A.  =  attenuation factor, days
                                1
                =  0.0198 days   , disturbed NTS conditions  (Langham, 1971)
                =  0.0173 days   , prevailing NTS conditions (Langham,  1971)
                =  0.0154 days   , proposed by Kathren (1968)
             t  =  time, days

     From the definition of resuspension factor, the air  concentration  C(t)
at a given time, t may be related  to the existant surface concentration of
contaminant, S_(t), by the following version of Equation  (1).
              a

          C(t)  =  RF Sa(t)                                 (4)

where       Rf  =  resuspension factor, constant at a given  location, for
             T                             _i
                   prevailing conditions, m
                                                                        2
         Sa(t)  =  surface concentration of contaminant at  time t, pCi/m
          a

For some initial time t = 0, Equation (4) becomes

          C(o)  =  RF Sa(o)                                 (5)

where    S(o)  =  initial surface concentration of contaminant,  at time
                               2
                   t = o, yCi/m

     Combining Equations (3) and (5), the change with time  in the airborne
pollutant concentration can be seen to be proportional to an exponentially
decreasing surface concentration.

          C(t)  =  C(o) e^A* = RP Sa(o) e'W
                                 I  O
This expression serves to emphasize an important assumption, implicit  in the
concept of a "constant resuspension factor", as expressed by Equations  (1)
and (4).  That is, the amount of contaminant found in resuspension above some

-------
given surface will represent a constant fraction of the amount of contaminant
available on that surface.  Consequently, all  other factors  remaining  equal,
any reduction in the concentration of airborne pollutant must be  merely  due
to a corresponding reduction in the concentration of contaminant  present on
the surface.  From Equations (4) and (6), this progressive reduction in  the
surface concentration of pollutant must clearly be

         Sa(t)  =  Sfl(o) e'V                              (7)

     In reality, such behavior of a surface contaminant would seldom,  if ever,
result in maintaining the relationship described by Equation (4), namely that
a time-invariant fraction of pollutant on a surface is resuspended.   Such
assumed relationship takes into account only one of the processes whereby
a pollutant becomes unavailable for resuspension, which are commonly grouped
under the general term "weathering" (Anspaugh, 1975).  These include not only
the transport of small contaminant-bearing particles downward into the soil
by "percolation", but also the "cementing" of such particles into or onto
larger ones, on the surface, by the forces of adhesion or cohesion.   Whereas
the first of the processes mentioned would undoubtedly result in  a decrease
in surface contamination, it is equally clear that the second would not.
Consequently, the surface concentration of contaminant Sa(t) must consist
                                                        a
both of "uncemented" and "cemented" particles, the latter being unavailable
for resuspension.

     To justify the continued use of Equation (6), it would be necessary to
postulate that the "uncemented" particles would constitute a constant fraction
"k" (less than 1.0) of the "unpercolated" particles remaining on  the surface.
This constant fraction could then be incorporated, conceptually,  into the
empirically determined constant resuspension factor, and the relationship
indicated by Equation (6) maintained.

     Such a postulate, unfortunately,  would imply that a constant fraction
"1 - k" of the "unpercolated" particles on the surface must consist of
"cemented" particles.  For this fraction to remain, indeed,  constant,  the
surface concentration of "cemented" particles would have to decrease in

-------
proportion to the reduction in "uncemented" particles surface concentration.
However, this entails a contradiction since, by definition of the processes
involved, a "cemented" particle cannot "percolate" into the ground.

     The inconsistency described in the preceding discussion can be  avoided
by defining a "time-dependent fraction 'k(t)1, available for resuspension,
of the surface concentration of pollutant S (t)".  As direct consequence of
such definition, the functions C(t) and S.(t) would no longer parallel  each
                                         a
other, as expressed by Equations (3) and (7); the behavior of C(t) would
depend on the product of two time-dependent functions, k(t) and S3(t).
                         ~ """" ~                                     a
Furthermore, resuspension factors determined empirically as ratios of air-
borne to surface concentrations would include, implicitly, the time-dependent
fraction k(t).  For most practical purposes, this would mean that the general
relationship expressed by Equation (4) is inaccurate, and should be  replaced
by a formulation describing the decaying airborne concentration C(t) as the
product of a time-dependent resuspension factor and a time-dependent surface
concentration.

     The added complexity of such formulation justifies seeking simpler ex-
pressions, one of which may be introduced by considering an alternative to
the use of "surface concentration" as the contaminant source-term.  Note
that the extent to which a given "surface" has been contaminated is  deter-
mined by means of samples taken to some depth, which varies according to
technique (Bernhardt, 1976).  However, regardless of technique, such sampling
is limited to the top layer of soil, containing that portion of the  total
contamination which is presumed to be, at least in part, available for resus-
pension.  Thus, "surface concentration" of contaminant should be differen-
tiated from TOTAL activity or contamination present in the soil - regardless
of depth - per unit surface area, since a fraction of the latter may be pre-
sent at depths greater than that of the easily erodible surface layer.

     The results of such differentiation should be examined in the framework
of conditions best suited to the application of the resuspension factor con-
cept, i.e. uniform deposition of pollutant over a large - ideally "infinite" -
area.  Under such conditions, the net effects of redistribution and  losses of

                                    8

-------
contaminant material by saltation, creep and resuspension would  be  negligible.
Ignoring, for purposes of this discussion, the effects  of radioactive decay,
the only remaining factor of importance is that of "percolation" of contamin-
ant into the soil.  As already discussed, this would have significant effect
on the "surface concentration" S (t).   However, by definition,  penetration of
pollutant to greater depths into the soil would not affect TOTAL CONTAMINATION
PER UNIT AREA S..   Thus, once deposition of contaminant has concluded, S. may
be assumed to remain constant.

     This suggests an alternative definition for "k(t)M, as that "time-depen-
dent" fraction, available for resuspension, of the total contamination per
unit area S.".  The obvious advantage of determining such a new "resuspen-
dible fraction" is that it would permit expressing the decay with time of
the airborne pollutant concentration as the product of a constant resuspen-
sion factor and a constant "total contamination per unit area"  times a time-
dependent resuspendible fraction of the latter.

     Anspaugh et al. (1974, 1975) were of the opinion that such time-depen-
dent fraction could not be realistically determined, and that it would be
more advantageous to define a time-dependent resuspension factor R^(t)
assuming a constant value for the soil concentration, equal to the total
deposition per unit area SA> regardless of distribution with depth.  Ex-
pressing (4) in this format, it becomes

          C(t)  =  Rf(t) SA                                 (8)

where    Rf(t)  =  Rf(0) e'V
                                                     •j
  with    C(t)  =  air concentration at time t, yCi/m
         Rf(t)  =  time-dependent resuspension factor, m
         MO)  =  initial resuspension factor, at time t = 0
          T                            _i
                   (deposition time), m
            A.  =  attenuation constant, days
                                                                     2
            S^  =  total soil activity per unit area, constant, yCi/m
             t  =  time, days

-------
     Using (8) and (9) with values of AA corresponding to half-times  of  35
days (Langham, 1971) or 45 days (Kathren, 1968),  relationships  observed  for
up to several weeks after a contaminating event may be approximated reason-
ably well.  However, Anspaugh et al.  (1975)  quoted a number of  observations
indicating that such models are less  accurate for longer periods  of time.
These include a half-time of 10 weeks determined  by Anspaugh et al. (1973)
from observations made 12-40 weeks after accidental venting of  an underground
explosion, a half-time of about 9 months obtained by Sehmel  and Orgill,  (1974)
                                               -Q   1
at Rocky Flats, and a resuspension factor of 10   m   at a location contamina-
ted 17 years previously.  The significance of this last observation can  be
best demonstrated by rewriting (9) as follows:

          Rf(0)  =  Rf(t) eV                              (10)

where     Rf(t)  =  10"9 m"1

  with        t  =  17 years x 3s  =  6205  days

                 .  0.693
                 =  — —
      .       .
   and       x.
Solving (10) for R^(o) with 35 days < T,  < 45 days results in values of
         32  -l     .                Id XT
3.16 x 10   m   < R-(o) < 2.275 X 10   m  , which are clearly impossible,
                                                          — l          -?  — l
since it is highly unlikely that R^(o) should exceed 1.0 m  , with 10"  m"
being the largest value reported to date (Mishima, 1964).

     The obvious implication of the preceding discussion is that half-time
T,  must increase with time, and be considerably greater than  35 or 45 days
at 17 years post deposition.  The direct observations of Anspaugh and Sehmel
and Orgill mentioned above, also support this conclusion.  Based on these
observations, and on the assumption that T,  - 35 days is valid during the
                                          %
first 10 weeks after deposition, Anspaugh et al . (1974) developed a "time-
dependent" model of the resuspension factor conforming to the following con-
straints:  "1)  The apparent half-time of decrease during the first 10 weeks
should approximate a value of 5 weeks and should approximately double over
                                      10

-------
                                                                   -4  -1
the next 30 weeks; 2)  The initial resuspension factor should be 10   m  ;
and 3)  The resuspension factor 17 years after the contaminating event should
approximate 10"9 m  ."  This model can be represented by (11).
          Rf(t)  =  Rf(0) e ' * v «•   +  Rf(«)              (11)

where     R*(t)  =  time-dependent resuspension factor, at time t, m"
           T                                                     _•!
          R,(o)  =  initial resuspension factor, at time t = 0, m
           T           yi   i
                 =  10"  m" , as specified by Anspaugh et al. (1974)
          Rr(~)  =  final resuspension factor, at time "t = »", m"
                       o   1
                 =  10   m  , as specified by Anspaugh et al. (1974)
              t  =  time from deposition, days
              K  =  constant coefficient = 0.15 (days)"*3

                                          -9  -1
The "final resuspension factor" R*(») = 10   m  , in the above expression,
reflects the expectation that there would be no further measurable decrease
in the resuspension process after 17 years, which, in 1974, was "the longest
period post deposition for which measurements (had) been reported".  At such
time, the mechanical behavior of the aged pollutant deposit would not differ,
presumably, from that of the native soil itself, by virtue of the two having
become intimately associated.

     As Anspaugh pointed out, this model "was derived from a composite of
numerous experiments," and "contains no fundamental understanding of the
resuspension process," but intends merely to describe it.  Nevertheless, two
basic assumptions are implicit in the model formulation, as follows:

     1)  The half time T,  is time-dependent,

     2)  The resuspension factor Rf(t) reaches a limiting value, Rf(»), at a
long ("infinite") time after deposition (t = "»").

Should these two assumptions be accepted as valid, some generality may be
attached to Anspaugh1s model as expressed by (11), derived as it was from
"numerous experiments", and thus, at many locations; "there have not been
measurements at any individual source over such long time-periods" (Anspaugh
et al. 1974, 1975).

                                        11

-------
             TIME-DEPENDENT HALF-TIME MODEL BASED ON ANSPAUGH'S
                             RESUSPENSION FACTOR

     There is evident need to model "weathering" processes on a fundamental
basis, particularly, as regards restispension, In view of the conceptual  con-
nection between "weathering" and the time-dependent behavior of the half-time,
T, (t).  The present author makes no pretense of providing such a solid founda-
 .
tion, but merely attempts to produce a general empirical model of half-time
as function of time and local conditions, represented by "initial" and "final"
resuspension factors, Rf(o) and Rf(»), respectively.  This model is based on
observations and models reported by previous authors, with liberal use of the
assumptions inherent in their development.

     An obvious starting point for a tentative model is provided by Equations
(9) and (11).  Incorporating into the former the postulated time-dependence
of the half-time and hence that of the attenuation factor x(t), previously
denoted as A., it may be formally rewritten as

                                           _ 0.693 t
          Rf(t) = Rf(0) e ' X(t) * = Rf(0)e  VtJ

Equation (12) may be interpreted as being mainly a definition of T,(t).   With
some rearrangement, it becomes

         T m  _    0.693 t
         Vt}  "	RTt)                               (13)
Replacing Rf(t) in the above equation with Anspaugh's model as expressed by
(11) results in equation (14)

                 Rf(0)\_       - 0.693 t	.
                                         "E51           04>
                            in   e      -   R
                                      12

-------
Note that, in a strict sence, T,  is no longer a function solely of time but
                               *£
also depends on local conditions, parameterized by initial and final resuspen-
sion factors as obtained or extrapolated from actual measurements in a given
                                                                          -4
area.  While Anspaugh and his coworkers assigned specifically values of 10
 -1       -9  -1
m   and 10   m  , respectively, to these factors, both higher and lower values
have been observed.  Table 1 (Mishima, 1964) presents resuspension factors
                                             21                    31
ranging over 11 orders of magnitude, from 10   m   (interiors) and 10   m
(disturbed exterior conditions) to 10    m   (aged deposit).  Although some of
these values would be normally quite inappropriate in a study concerning speci-
fically wind resuspension, their use may be allowed for the purposes of devel-
oping a general model.  In particular, the ratio of the lowest to the highest
values in Table 1 can be assumed to represent a credible lower limit of the
ratio Rf(°°)/Rf(o) in Equation (14).  A tentative upper limit for this ratio
may then be provided via the assumption that Rf(o) and Rf(») must differ by
at least one order of magnitude.   Consequently, the value of Rf(<»)/Rf(o) in
Equation (14), applied generally, should vary over 10 orders of magnitude,
provided constraints are met.

                                      10'1,                 (15)
          where     Rf(o) >. 10 Rf(») (assumption)

                                          -k
Using Anspaugh 's value for "K" of 0.15 day  , the constraints imposed on T,
                                                                          *5
(a value of 35 days during the first 10 weeks, double the value over the next
30 weeks) are clearly met, regardless of the value of Rf(»)/Rf(o), as seen in
Figure 1.  In fact, any ratio Rf(«)/Rf(o) <_ 10   should satisfy this require-
ment.

     Nevertheless, Equation (14) has one serious drawback, which becomes ap-
parent upon examination of the denominator in the right-hand side of this
equation.  For certain specific values of time t, this denominator will ap-
proach zero and consequently the half-time will tend to + or - infinity (»)
along a vertical asymptote (Figure 1).
                                     13

-------
LSf.
                                                   1 1
                                                  4-H-
                                                Rf<0)/llfH. MMIo o« taMM to Final
                                                   Rf(e)/MfH = 101 oorrMondi to In*
r-    Rf      i •    i
             FIGURE 1.  Half-time Ty? as function of time and local conditions (expressed as ratios of Inltial-to-flnal resuspenslon factors. Hk(0)/RfH). as
             derived (Equation (14)) from  Anspaugh's model of Rf(t).

-------
                    T^-n- -   as t  -v  tas+                (16)


                    T,  -»• - »   as t  -»•  t  -                (17)
                     *5                   35
          where     t^ =  TT  In  1 -
                     as    n
For t < t  . T,  will be negative.  For t > t  . T,  will have a minimum at some
         as   *s                             as   *2

specific time t_. , which implies that T,  will decrease with time for values
               mm                      *%

exceeding ta, but less than t ._, and increase thereafter.  Thus, every posi-
           aa                INI n

tive value of t,  will occur twice, with exception of T, (t_. ).  These obser-
               *5                                      ^   nil n

vations are summarized below.
           < o                for t < tm.n                  (18)
     d \  > 0                for t > t  .                   (19)

     dt
     o > T,  > - »             for 0 > t > t,e               (20)
          *5                                as




                                                            (21)
     Equations (16) through (22) illustrate the deficiencies of Equation (14),


formally described as "the time-dependent half-time model derived from Ans-


paugh's model of the resuspension factor".  These may be itemized as follows:





1)   The discontinuity at t = t,,, described by Equations (16) and  (17), has
                               aa

     no physical parallel.





2)   The decrease with time of T, indicated by Equation (18) is contrary to
                                T

     empirical evidence.





3)   The minimum value of T^ at tm1n, implied by Equations (18), (19), (21),


     and (22), contradicts the expectation of there being only one  (absolute)


     minimum T,  at t = o.






                                        15

-------
4)   The negative values of T,  given by Equation (20) are clearly impossible.

5)   The behavior of T,  expressed by Equation (21), beginning with a value
     "close to infinity", at some short time after deposition, and decreasing
     thereafter, is highly unlikely.

In sununa, only Equations (19) and (21) conform to physical reality, thus
limiting the applicability of the model to times t greater than t^ .  Note
that the value of tmin depends both on the ratio Rf(~)/Rf(o) and the constant
coefficient "K".  For a value K = 0.15 /  vtlays, the model would be generally
satisfactory with t.  < 2 days, for all ratios Rf(»)/R^(o) <_ 10" .  However,
smaller values of "K" would result in higher values of t. , and a further
decrease in the domain of applicability of the model.  Such limitations sug-
gest the need for alternative formulations of a general model.  One such ex-
pression is developed in the next section.
                                       16

-------
ERRATUM: The expression describing t   ,  following Equation (17)  on page 15, has
                                    dS

         been printed Incompletely. The correct expression is as  follows

-------
         BASES OF PROPOSED INTERIM MODEL OF TIME-DEPENDENT HALF-TIME

     Two basic assumptions are implicit in Anspaugh's formulation of the time-
dependent resuspension factor (11).  They are:  1)  that the half-time T,  is
time-dependent, T,(t); and 2)  that the resuspension factor R*(t) approaches
a limiting value Rf(«), applicable to "aged" deposits such that "t ->• »".  The
present author proposes to formulate an interim generalized model of T,  as
a function of time and local conditions (as reflected by Rf(0)/Rf(«)) by first
reviewing the facts at his disposal in the light of these two assumptions.

Initial Values of Ttj (t, RJ(0)/RJ(n°)).  Measurements of half-time T, at several
	% ->	T-»—"—T1 "                              *5
locations and at various times after a contaminating event suggest that the
half-time increases with time post deposition (Anspaugh et al. 1973, 1974,
1975).  Since conceptually "weathering" is the only phenomenon (or group of
phenomena) affecting the resuspension factor, this time-dependent behavior
of the half-time must reflect a dependence on weathering as well.  Further-
more, since both weathering and half-time increase with time after deposition,
an initial time following a contaminating event, when the extent of weathering
is small, the half-time is correspondingly short.  For the case of "just de-
posited pollutant", at time t = 0, when the process of weathering begins, the
half-time could be extremely small; it would be finite, however, since forces
of adhesion between the contaminant and the native soil become effective upon
contact.  Thus,

                              \ (0) > 0                    (23)

     Values of Tj(t) measured within days to several months after deposition
appear to be roughly similar, in spite of the different local conditions under
which these measurements were made (Anspaugh et al., 1973, 1974; Wilson et al.
1960).  However, as the related resuspension factors approach final values re-
flecting these and other conditions, the half-times at the corresponding lo-
cales may differ from each other by one order of magnitude, as will be seen
in the discussion of "Final Values of T,".  This would indicate a convergence
of half-time values as time approaches t = 0.  Lacking other data, one single
initial half-time at t = 0, valid at all locations under all average condi-
tions (excluding severe disturbances) may be postulated.

                                     17

-------
0,  [Rf(0)/Rf(»)]
                                           for all  i  =  1,2,3...
(24)
       where  [Rf(0)/Rf(»)J     •   conditions  at  location i
                           1
                                    generalized  location index
                                    initial  half-time,  independent of location
Values of TV  (t,  Rf(0)/Rf(")) Within One Year  After Deposition.  Several  values
of half-time  have been reported.  Anspaugh et  al.  (1973) determined a half-time
T,  = 38 days  from a least squares fit to averaged  measurements made from  3  days
to roughly 8  weeks after a nuclear cratering event, Project Schooner  (Figure 2).
              10"
  o
  a.
  >
  t
  2

  q:

  $
                                     • AVCNAGE OF STATIONS 5 AND II
                                     • AVERAGE OF STATIONS 23,27 AND
                                  LEAST SQUARES FIT T „ • 38 DAYS
                    KK>  /»U  JOO 4UO  5OO  6OO  TOO  SOU  9OO  1000  MOO  1200  1300
                                     HOURS POST SHOT
FIGURE 2.  Tungsten-181 air activity as  a  function of time at five  downwind
stations within six miles that were closest to the line of maximum  deposition
from Project  Schooner.  Data were normalized to the first sample taken  after
70 hours had  elapsed following the detonation [From Anspaugh et al.  (1973)].
                                      18

-------
Wilson et al. (1960) obtained a half-time value of 35 days as a "best apparent
fit" to median data obtained between approximately 3 and 23 weeks  after a non-
critical high explosive detonation involving plutonium,  Operation  Plumbbob
(Figure 3).  Olafson and Larson (1961) reported the measurement period as
beginning 18 days after the event and continuing to 160  days.  Stewart (1967)
described the decay in the average airborne concentration as having a half-life
of about 37 days.  Following other experiments, Langham  (1971) reported
"attenuation factors" of 35 and 40 days estimated at the Nevada Test Site,
although the time periods to which these are strictly applicable are uncertain.

     Interpreting this data somewhat loosely, it would appear that a half-time
of 35 to 38 days, applicable to various conditions, should be expected between
3 and 160 days after deposition.

                    35 days <_ T^) <_ 38 days                   (25)

          where 3 days <_ t-j <_ 160 days, time after deposition

          and T^tj)  =  empirically determined half-time, days, presumed
                         to apply to a contaminant deposit present in the
                         soil for a time tj

                 t,   =  time post deposition, days

     In addition, Anspaugh et al. (1973) calculated a half-time of 76 days over
a period extending from roughly 11 to 45 weeks after the accidental venting of
an underground nuclear explosion (Baneberry venting, Figure 4).  When corrected
for background variation, the half-time was found to be  66 days.
                                        19

-------
ovsuu
4000
2000
^ 1000
£ 800
ki
5!
% 400
* 200
-j
2
e
i 100
J; so
I 40
£
20



•
iv


>

\
\ " .
^X, 1000 AG/M8
L . ^s^T'/e • 5 WEEKS
x. >A AA
"^ ^ "X.
i x_ X_ ^
1 >U ^ 4.
^ ffl mix mi miX. A



_
i

>v >^ 100 AG/M'-i X. A
X. X--— ; 	 X
V ^ ~ \J^a5 WEEKS \
x». © © x. x.
XT' m X_ "^ T.
"• ' m Au/Mi -X. ^v ID A ><

T'/2o5WEEKS \ ® N^n,, A A"
i m ibv \ •»
1 _ x ra x A
\ mi,?. -x. v nn
j ^ /aX X. m
-p

LACEMENT DAY • tt!K»~ ^^X

© ^SJ^ ffl^*
) <^X_
i-

A 1000 AG/M* © on ^v
© 10 AG/M* ..
an ©
1 1 1 1 1 1 1 1 1 1 1 1
              -2  0  2  4   6   8   10  12  14  16  18  20

                         (START UP OF AIR SAMPLES • TO )
FIGURE 3.  Median weekly air concentrations,  corresponding to  three
isopleths, from Project Plumbbob  [From Wilson et al. (I960)].
                                 20

-------
                                  =   66 days
                                                    (26)
          where 77 days <_ t2 £  315  days, time after deposition

          and  T, (t2)  =  empirically determined half-time,  days, presumed
                          to apply  to  a contaminant deposit present in the
                          soil  for  a  time t~
                       =  time post deposition, days
              10"
              10"
              10
                                       STATION 4
                          A - vol.! B.IO- USX Cwil.dcnct L<«f I

                          • 1.0 iio.n.0 I-ill
6O   80
                    80 I  li»   IS   i«u  I 160   ItiO   MO  I 2X>   Sf   J60  [MO  JOO
                  March I  ftpul I   Mgy  I  June I   July  I  Augutl I Stpftmter I October I
FIGURE 4.   Least-squares fit  to  gross-gamma air activity levels 3 to  11 months
following  Baneberry venting  [From Anspaugh et al.  (1973)].
     For  the purposes of the  interim model, the values expressed in  (25)  and
(26) will  be considered to apply generally.  Additional data, when available,
will serve to corroborate, extend or even reduce  the  time periods over which
these half-times are expected to apply.
                                       21

-------
Values of T.  (t, Rf(0)/RfH) After One Year From Deposition.   Sehmel  and
Orgill (1974) reported an "average weathering half-life...from April  1971
to October 1972... of about 9 months" from measurements made at sampling
station S-8, Rocky Flats, near an area contaminated with plutonium from
leaking drums in previous years.  The age of the deposit, however, 1s un-
certain.  It can only be determined to be within a very broad range of time
values, as the following review of the pertinent facts should indicate:

     1)  Drums containing cutting oil contaminated with plutonium "were
placed in outside storage from 1958 until 1968.   Initial leakage was de-
tected in 1964" (Krey and Hardy, 1970).  Apparently the drums contaminated
the adjacent soil for a period of four years, from 1964 to 1968.  However,
the possibility that leaks had developed earlier and gone undetected should
not be excluded.

     2)  A contour representation of plutonium-239 distribution at Rocky
Flats shows a "hot spot as defined by the contours, just adjacent to the
area where the leaking drums had been stored" (Volchok, 1971).  Thus, there
is little doubt that the plutonium concentrations measured by Sehmel and
Orgill "near the original oil storage area" have the leaking drums as the
original, though not necessarily immediate, source (Figure 5).

     3)  As implied by 2) and Figure 5, the leaking plutonium was subse-
quently dispersed.  According to Martell (1970), it was "redistributed by
winds, mainly in the period between spring 1967, when Dow started to move
the drums for reprocessing, and September 1969,  when a four-inch thick as-
phalt slab was placed over the contaminated area."  Therefore, the 1971-1972
observations of Sehmel and Orgill cannot be related directly to the original
contaminated area in the immediate vicinity of the drums, since this source
was no longer in existence as of 1969.  By elimination, the immediate source
of the plutonium concentrations measured by these researchers was that con-
taminant redistributed by wind in the preceding  years.  Martell states that
most of this redistribution took place in 1967-1969, 19 months to 4 years
prior to the first measurement of Sehmel and Orgill.  However, as mentioned
in 1), leakage and possibly redistribution started much earlier, at least
as early as 1964.
                                      22

-------
                        PLUTONIUM-239 CONTO
FIGURE 5.  Contour  representation of plutonium - 239  distribution at
Rocky Flats.   [From Volchok (1971)]
                                   23

-------
The difficulty of assigning  a  proper "weathering age" to material redistribu-
ted continuously from a  source,  no longer in existence, which had been subject
to continuous deposition of  pollutant over an uncertain period of years should
be obvious.  The task increases  in complexity considering that, in all proba-
bility, deposition did not proceed uniformly with time, and that redistribu-
tion most certainly did  not.   However, one additional fact further complicates
the issue, as described  below.

     4)  Sehmel and Orgill began their data collection in April 1971.  How-
ever, "In mid-March 1971, a  ditch was dug east of the original oil storage
area and west of sampling station S-8" (in the area of heaviest contamination)
following which they observed  "increased airborne activity" (Figure 6), due to
the fact that "ditch digging significantly increased the availability of plu-
tonium for resuspension" (Sehmel and Orgill, 1974).  The resulting "resuspen-
sion source change" may  be equated to a "source freshening" (present author's
expression), suggesting  that the measured half-life of nine months could, in
fact, correspond to a much fresher source than what the as yet undetermined
chronological age of the deposit would indicate.
                                   REPORTED AIRBORNE
                                 PLUTONIUM CONCENTRATIONS
                 1 -
                  J  J A S 0 N 0  J F M
                      WO
AMJJASONOJFMAMJJASOND
    1971                  W77
                   VCCCIATION      DITCH/   V
                  PROGRAM (COM     Mill
                                      W MR AW MASS SODIM
FIGURE 6.  Airborne  Plutonium at Sampling Station S-8.  Adapted from Sehmel
and Orgill (1974).
                                      24

-------
     In conclusion, very limited use may be made of the 9-months half-time
subject of this discussion.   To include all possibilities (even the most pes-
simistic ones), the age of the deposit corresponding to this half-life must
be tentatively assumed to be between 19 months or less and an admittedly ex-
treme 14 years or more from deposition.

                    T%(t3) =  270 days                      (27)

          where  19 months (or less) <_ t., <_ 14 years (or more)

          and  T^(t~)  =  empirically determined half-time, days,
                          presumed to apply to a contaminant de-
                          posit present in the soil for a time t-,

                  t3   =  time post deposition

Such a wide range of weathering ages does not permit using (27) as a constraint
for the interim model.  For modeling purposes, the observations of Sehmel and
Orgill may be construed as furnishing proof that half-lives of 9-months do, in-
deed, exist, and are assumed to apply generally (given sufficient time) under
all conditions Rf(0)/Rf(«).
"Final" Values of T^ (t. Rf(0)/Rf(°°).  Postulating that the function Rf(t) =
Rf(Q) exp {-[0.693/T, (t)]  t} approaches a limiting value R^(») at "sufficient-
 I                   *5                                     T
ly large" values of time t is equivalent to assuming that, as t "approaches
infinity", the time-dependent weathering half-life Tj, also approaches infinity,
but along an oblique asymptote defined by Equation (28).

               \ [t, Rf(0)/Rf(-)]  =  4™-.
                                      inf  f!   1           {28)
                                      I ll I  n / A  I
In a strict sense, this implies that T.  is not solely a function of time, but
depends also on average local conditions, parameterized by the ratio of ini-
tial and final resuspension factors, Rf(0)/Rf(»).  Referring to Table 1, it

                                      25

-------
                                              -2  -1*
may be assumed that Rf(0) can be as high as 10   m   , and that values as low
      131
as 10    m"  are possible for R^(»).   Furthermore, regardless of the actual
values of Rf(0) and Rf-(
-------
             PROPOSED INTERIM MODEL OF TIME-DEPENDENT HALF-TIME

     A simple generalized model  of the weathering half-life of the resuspen-
sion factor, as a function of time (t) and of average local conditions
(Rf(0)/Rf(«), is presented.  The model is based on the facts and assumptions
discussed in the previous section, using them as constraints, when applicable.
It is of the general form
           Rf(0)/Rf(»))  =  A In (1  + B + CtD) +  ^
            f     f                               ^KfV^       (3D
          Where  A  =  constant coefficient, days
                 B  =  constant, dimensionless
                 C  =  constant coefficient, day"
                 D  =  constant exponent, dimensionless
              MO) =  initial resuspension factor, m  , at a given location
               '                                     -i
              Rf(») =  final resuspension factor, m  , at the same location
                 t  =  time, days

The second term of the expression serves to meet the requirement that Rf(t)
approach the expected Rr(») at long times after initial contamination.   The
first term of the equation forces T. (t) to the observed or expected values
at deposition and shortly thereafter.   The specific choice of constants A,
C, B, and D, will obviously determine what these values are and/or when they
are attained, according to the model.   Referring to (23), (25), (26) for ex-
planation of the symbols used, a group of relationships may be roughly sketched.

                              V0)  * fo (A'B)

                              W = fl 
-------
Note that the constant B has no role other than determining T,(0) since, as
T, (0) Is assumed to be small, 1t Is expected that B « 1.

     By making judicious choices of constants, a range of Initial values
T, (0) may be postulated while satisfying the model constraints.  Two sets of
such constants are presented 1n Table 2 as examples of applications of the
general model (31).

            TABLE 2.  EXAMPLES OF CONSTANTS USED IN GENERAL MODEL
                                         Case 1
                                                     Case 2
               A (days)
                                    28
36
               B (dimensionless)
                                    4 x 10
                                               -2
4 x 10
                                                           -3
               C (days'0)
               D (dimensionless)
                                    1/3
1/4
Using "Case 2" constants in (31) with a range of ratios R4:(0)/R-(») from 101
     11
to 10   results in the family of curves shown in Figure 7.  The relevant fea-
tures of the model in the present application are summarized below.
               T. (0) = 3.5 hours  at t = 0
                     « 25 days    at t = 1 day
               35 days <_ Tj(t|) <_ 38 days
     Where      5 days <_ t, <_ 12 days
Where
               ytg)  =  66 days
               60 days <. t2 <. 276 days
                                                       (32)

                                                       (33)

                                                       (34)


                                                       (35)
                                    28

-------
p-
                                                <-*1  Rf(o)/R{(»), Ratio of Initial to Final Re*uspen«lon Factor
                                             i T :       R«(o)/RfH = 10' correspond* to HIM marked 1
                                                       Rf(o)/Rf(») = 10* correspond* to HIM marked 2, etc.
                                                          TIIIM From Deposition (days)
        0.
              FI6URE 7. Hilf-tlmt Tyz u function of tlmi and locil ctndltlww Hxpreised it ratiw of InJtlil-to-flnaJ rnuptailM fadwi Bf(0|/llfHJ
              iccorrHng to propestd modtl and "Cwi r cwwtanti (TiblB 2). CUI^M numb*rwl seeding to ridoR(|0|/RfK In ort^t*f
                            to Bf|0}/RfH.lO' »tc-

-------
Comparing (32), (34), (35) with Equations (23), (25), and (26), it is seen
that the constraints imposed by the latter three expressions are essentially
met.  The same applies for the conditions of "Case 1", for which the model
produces the following results:

                     = 1 day   at t = 0                     (36)

                     * 20 days at t = 1 day                 (37)

               35 days <. T^^J <. 21 days                   (38)
     Where     10 days <_ t-j <_ 21 days

               T^(t2)  =  66 days                           (39)
     Where     67 days i tg £ 309 days

A family of curves corresponding to various ratios Rr(0)/R*(<») can be obtained,
analogous to those of Figure 7.  Figure 8 shows two of these curves, for the
upper and lower limits of the range of values presumed possible for
Rf(0)/Rf(«), 1011 and 101, respectively.

     The extent to which the model conforms to the requirements set by (28)
can be gaged by first examining the latter.  Clearly, this equation is equiva-
lent to

               lim     ST,(t, Rf(0)/R-H)  =    0.693       (40)
                        *5     T     T         	n ln\
Taking the partial derivative of the general model (31) with respect to time t,
results in

                3T (t, Rf(0)/Rf(»))  =  ACDt0"1 +   0.693   (41)
                                      30

-------
CO
                               CtlSTIUlTS:

                                           , wiis«t tt it. < tm >, trmmt rim *t»

                                   V»*4ui.AiuMi» itii.ui;$). CWIBT

                               (J) V-ttria;i. kis0ii|A *t H.I1I73I, uittiiii lEiiiit
                                                                                                                                                       100000
                                                                          Time From Deposition (day*)

                          FIGURE 8. Half-time T vz as function of time and local conditions (expressed as ratios of initial-to-final resuspenslon factors  R (0)/RfH)
                          according to proposed model (Equation (33)) and "Case 1" constants (Table 2). Curves shown correspond to upper and lower limits of range
                          of values presumed possible for Rj(0)/RfM.

-------
As t Increases, the first term In (41) grows progressively smaller, vanishing

entirely for values of t "approaching Infinity".




                         11m   %,   (41)    =    0.693      (42)
Clearly, the constraint expressed by (40) is met.  Therefore, as time in-


creases, the half-life T,  approaches one of the oblique asymptotes indicated
                        *5

by (28).  Two such asymptotes are shown in Figure 8.
                                       32

-------
                                 APPLICATION

     The primary application of the model is in describing time-effected
changes in the magnitude of airborne contamination through the dependence of
the rate of change of the resuspension factor on the modelled variable, the
time-dependent weathering half-life.  Figures 9 and 10 depict the behavior
of the resuspension factor as a function of time for different ratios
Rf(0)/Rf(«), as predicted by Equation (12) and the present model  (31) of
yt, Rf(0)/Rf(«) (Case 2).

     The intent in modelling the time-dependent behavior of the weathering
half-life was to provide a simple predictive tool of reasonable accuracy and
flexibility enabling it to meet a wide range of average local conditions.
However, the present author lacks the necessary data to determine the degree
to which the predictive ability and thus usefulness of the tool may have been
marred by the inevitable "trade-offs" between flexibility and accuracy.  For
this reason, examples of the model applicability are limited to a comparison
of the behavior of the half-time and resuspension factor as functions of time
as predicted by the model with those predicted by or derived from other models
(Figures 11 and 12).  Table 3 describes the salient features of these models,
specifically Langham's (1956-1971), Kathren's (1968) and Anspaugh's (1974), as
well as those of the present model (Case 1), including constants used in the
latter for comparison purposes.
                                      33

-------
CO
          10*
          W* • •;• 10*
                                                     Rf(o}/RfH. Ratio of Initial to Final Resuapenaton Factor
                                                        Rf
-------
GO
171
           10"
           10-'
           10-'
           Iff


           10 •


           Ifr


           10'
           io*   2
           10-"
           10"
                                   Rf(o)/Rf(»), Ratio of Initial to Final Resuspenslon Factor
                                       R|(o)/Rf(x) = 10' corresponds to line marked 1
                                       Rf(o)/Rf(x) = 10> corresponds to line marked 2, etc.
           10"    10"
                            1   2    34    5   6    7   8   0   10  11   12   13  14  18   16   17  18   10   20  21
                                                                            HIM From D«po*ttlon
                                                                                                                              24  25   28  27   28   20   20
       FI8URE10. Resuspenslon factors as functions of time based on proposed model of time-dependent half-time Ty, assuming the same Initial value but various final
                  resuspsnslon factors.  First captloned ordlnate shows values presumed possible based on Table 1. J  The Initial value! of 10^nr1 Is highly improbable for
                  outdoors pollutant resuspenslon by wind under normal conditions, but'serves to illustrate applications of the model for the full range of values
                  postulated In the model development  Second captloned ordlnate pertains to generalized case.)

-------
     TABLE 3.  COMPARISON OF PROPOSED MODEL (CASE 1) OF THE TIME - DEPENDENT WEATHERING HALF - TIME T^ WITH
     PREVIOUS MODELS, IMPLICIT OR DERIVED,  AND CORRESPONDING MODELS OF THE TIME - DEPENDENT RESUPENSION FACTOR,
     INCLUDING NUMERICAL VALUES

RESUSPENSION
FACTOR
Rf(t)
HALF-TIME
Ty.(t)
INITIAL
VALUE
Rf(o)
FINAL
VALUE
Rf(«)
RESUSPENSION
FACTOR MODELS, V
SPECIFIC VALUES E
THEIR AUTHORS. £
LANGHAM'S
MODEL
-0.693 t
Rf(o)e T'/'
35 days
(constant)
10 -sm-1
0
lO^nV'e'00"""'"1'
VITH
MPLOYED BY
,EE FIGURE 11.
KATHREN'S
MODEL
-0.693 I
R,(o)e T%
45 days
(constant)
10-«m-'
0

©
irr4irrle'001S4d"~1t

ANSPAUGH'S
MODEL
-0.15 ,-
days 'A Vl
R|(o)e y ^,(30)
-0.693 1
.J^vr.Msn
L Rf(o)J
(derived)
10-*m"
10-»m-'


10'4m"1e'oli"»V'tl''l+10''m-1
PRESENT PROPOSED MODEL
(CASE 1)
[ 0.693 1
J / | Vj \ 0 693 t
-^28ln(1.04- J — v)+ r
D /«>« I daVs / m ((0
Rf(o)e \ / inj^
f1/3 0.693
28 days In (1.04+ ,,-/,)+ -R|((
'"b.c
1
%_
)
U '
t
3)
e)J
Values employed in present proposed model (or
purposes of comparison with each of the preceding
models. With these values, the model acquires the
following forms:
^ I" 0.6931
lO-'m-'e L28daysm(1.04+day~1V/J)
j" 0.693t
10 -m -e L28days ln (l-OA-day'/'t'7'
CD j" 0.693t
j
.1
)J
1
10-4m"e l28daysln(1.04+dBy'V») +0.0602tJ
a\

-------
                                             Ll(LM + Mf^t*1) + l.llltt

                                            r*»v'
FIGURE 11.  Three  expressions of the time-dependent resuspension
factor based on  the  proposed model of the time-dependent  half-time
Ty?/ with initial and final values as assumed in previous  models
with which these expressions are compared.  See also Table  3.
                                37

-------
10>i
 10"-
 w-

10"
-
IIP-
                   HALF-TIME DERIVED FROM ANSPAUGH'S MODEL:  T% •
                   PROPOSED MODEL :  T% = 28 day* In (1.04 + d«r v> t^ ) +
                   FROM KATHREN'S MODEL:   T% = 45 days
                   FROM LANQHAM'S MODEL:
3S days
                FROM KATMREirS VODEL
                FROM LANQHAV8 MODEL
                                 nr*
                                                          1.0
                                                                                  10*
                                                                                         1     1
                                                                                        10*    10*
                                                    1
                                                    10»
   R6URE 12. Two forma of the proposed modal of weathering half-time Tyt as function of time and local conditions (expressed as ratios of
   Infflal-bMlnsI rasuspanslon factsrs Rf(Ol/RfH I. compared © with Anaoaitgli'a medal and ©with Langham'a and Kadtrejfs mntete,
   Values  of R|(0)  and  Rff»)  employed  In  ttw  proposed  model correspead to then used  In  the cenBsrlsen medelt.

-------
     As clearly shown In Figure 11, the proposed model (Case 1) produces a
time-dependent resuspension factor that behaves similarly to Anspaugh's model,
for the same conditions R<(0) and R,(»).  One obvious difference Is In the
rates at which both models approach the limiting value Rf(«).  Anspaugh's
model Includes the "assumption that there may be no further measurable de-
crease In the resuspension process after 17 years which Is the longest period
past deposition for which measurements have been reported".  No such assump-
tion is made in the present proposed general model, according to which the
limit value R^(~) may be approximated before or after 17 years, depending
functionally on the ratio Rf(0)/Rf(») (Figure 9).

     The versatility of the proposed model is further illustrated in Figure
11 by comparing it with Kathren's and Langham's model.  With Rf(«) = 0, as
implied in the latter two models, the proposed model matches their behavior
fairly closely.

     Figure 12 shows the time-dependent behavior of the model (Case 1), com-
paring it with that of the half-time derived from Anspaugh's model and the
values used by Langham and Kathren.  Again, the values Rf(0) and Rf(°°) were
chosen to be those of the models with which the comparisons are made, respec-
tively.  Besides having certain advantages, evident upon examining the graph,
the model has one obvious drawback, that of having inflection points at t < 1.0
day and at t > 100 days.  These were not Intended to represent any fundamental
notions of the time-dependent behavior of the weathering half-life, but reflect
merely the limitations of the model, when shown logarithmically.  The first
inflection point is due to the requirement that T,  be other than 0 at time
t = 0.  The second inflection point occurs when both terms of (31) approach
comparable values and the second term begins to exert dominance.
                                     39

-------
                            SUMMARY AND COMMENTS

     The proposed generalized model is based on several  simple assumptions and
limited empirical data, obtained at various locations and under different con-
ditions, as reported by a number of researchers, some of whom Interpreted the
data differently.  The model is intended to be general,  that is, able to con-
form to different average conditions, when these conditions are expressed as
initial and final resuspension factors.  This generality is achieved primarily
through the second term of the Equation (31), which Includes the ratio
Rf(0)/Rf(»).  Additional flexibility is provided by the  first term of the ex-
pression, which can be adapted to fit a number of empirical observations with-
out materially altering the general model.  At present,  Equation (31) used in
conjunction with the constants of "Case 1" appears to best accommodate the
existing data.  Additional data, when available, may serve to further refine
the model.

     The model has limitations and deficiencies, but the greatest drawbacks to
be encountered in using it are those of the resuspension factor concept per se,
it being best suited for large (ideally "infinite") areas, uniformly contamina-
ted, where average local conditions are maintained or vary uniformly with time,
unaltered by unscheduled severe disturbances.  The degree to which any model
of the resuspension factor (or of the weathering half-life) is successful is
clearly linked to the extent to which the above requirements are met.
                                       40

-------
                                 REFERENCES
Anspaugh et al., 1973                   L. R. Anspaugh, P.  L.  Phelps,
                                        N. C. Kennedy, and  H.  G.  Booth,
"Wind Driven Redistribution of Surface Deposited Radioactivity"  Environ-
mental Behaviour of Radipnuclides Released in the Nuclear Industry  Pro-
ceedings of a Symposium Organized by the International Atomic Energy
Agency, the OECD Nuclear Energy Agency and the World Health Organization
and held in Aix-en-Provence, 14-18 May 1973, IAEA, Vienna,  1973


Anspaugh et al., 1974                   L. R. Anspaugh, J.  H.  Shinn, and
                                        D. W. Wilson "Evaluation of the
Resuspension Pathway Toward Protective Guidelines for Soil  Contamination
with Radioactivity"  Lawrence Livermore Laboratory, Biomedical Division,
IAEA-SM-184/13. UCRL-75250  Preprint/Proceedings of the IAEA/WHO Symposium
on Radiological Safety Evaluation of Population Doses and Application of
Radiological Safety Standards to Man and the Environment, Portoroz, Yugo-
slavia, 1974
Anspaugh et al., 1975                   L. R. Anspaugh, J. H. Shinn, and
                                        P. L. Phelps  "Resuspension and
Redistribution of Plutonium in Soils"  Lawrence Livermore Laboratory,
UCRL-76419  Preprint/Proceedings, Second Annual Life Sciences Symposium,
Plutonium-Health Implications for Man, Los Alamos, New Mexico, 1975
Bernhardt, 1976                         David E. Bernhardt  "Evaluation
                                        of Sample Collection and Analysis
Techniques for Environmental Plutonium"  Technical Note ORP/LV-76-5.  U.S.
Environmental Protection Agency, Office of Radiation Programs-Las Vegas
Facility, Las Vegas, Nevada, April 1976


Kathren, 1968                           R. L. Kathren  "Towards Interim
                                        Acceptable Surface Contamination
Levels for Environmental Pu02"  Battelle Pacific Northwest Laboratory,
BNWL-SA-1510, in Proceeding of Symposium on Radiological Protection of
the Public in a Nuclear Mass Disaster  (Strahlenschutz der Berolkerung
bei Einer Nuklear katosteophenlpp. 460-470, Interlaken, Switzerland,
T955
Krey and Hardy, 1970                    P. W. Krey and E. P. Hardy
                                        "Plutonium in Soil Around the
Rocky Flats Plant"  U.S. Atomic Energy Commission, Health and Safety
Laboratory, HASL-235  Health and Safety (TID-4500). New York, New York,
WO
                                       41

-------
Langham, 1971                           W.  H.  Langham  "Plutonium Distri-
                                        bution as a Problem in Environmen-
tal Science"  Los Alamos Scientific Laboratory. LA-4756. UC-41. Los Alamos,
New Mexico, December 1971
Martell et al., 1970                    E.  A.  Martell, P.  A.  Goldan. J. J.
                                        Kranshaar, D.  W.  Shea and R. H.
Williams  "Fire Damage"  Environment. Vol.  12, No. 4,  May 1970


Mishima, 1964                           J.  Mishima  "A Review of Research
                                        on Plutonium Releases During Over-
heating and Fires"  U.S. Atomic Energy Commission. HW-83668  UC-41. Health
and Safety  (TID-4500, 37th Ed.), Ricnland, Washington, 1964


Olafson and Larson, 1961                J.  H.  Olafson  and K.  H.  Larson
                                        "Plutonium, Its Biology and Environ-
mental Persistence"  University of California, Los Angeles, School of Medi-
cine. Dept. of Biophysics and Nuclear Medicine, (Tl0-4300, 16th Ed.)Los
Angeles, California, 1961


Sehmel and Orgill, 1974                 G.  A.  Sehmel and M. M. Orgill  "Re-
                                        suspension Source Change at RocKy
Flats"  BatteHe Pacific Northwest Laboratories Annual Report for 1973 to
the U.S.A.E.G. Division of Biomedical and Environmental Research, Part 3.
Atmospheric Sciences -BNHL-1850- Pt. 3. UC-11. pp. 212-214, Richland.
Washington, 1974


Stewart, 1967                           K.  Stewart  "The Resuspension of
                                        Particulate Material  from Surfaces"
Surface Contamination, pp. 63-74, B. R. Fish,  Editor,  Pergamon Press, 1967


Volchok, 1971                           H.  L.  Volchok   "Resuspension of
                                        Plutonium-239  in the Vicinity of
Rocky Flats"  Proceedings of Environmental  Plutonium Symposium.   Eric B.
Fowler, Richard W. Henderson, Morris F. MilUgan, Editors.  Los Alamos
Scientific Laboratory, Los Alamos, New Mexico, 1971

                                 \
Wilson et al., 1961                     R.  H.  Wilson,  R.  G. Thomas and
                                        J.  N.  Stannard  "Biomedical and
Aerosol Studies Associated with a Field Release of Plutonium"  University
of Rochester Atomic Energy Project, WT-1511.  Operation Plumbbob - Test
Group 57, Program 72, Rochester, New York,  November 1960

-------
                                   TECHNICAL REPORT DATA
                            (Please read Inunctions on the reverse before completing)
 1. REPORT NO
  ORP/LV-77-4
                                                           3. RECIPIENT'S ACCESSION NO.
 4. TITLE AND SUBTITLE
  Generalized Model of  the  Time-Dependent Weathering
  Half-Life of the Resuspension Factor
                                                            5. REPORT DATE
             6. PERFORMING ORGANIZATION CODE
 7 AUTHOR

   George V. Oksza-Chocitnowski
             8. PERFORMING ORGANIZATION REPORT NO.
 9 PERFORMING ORGANIZATION NAME AND ADDRESS
   Office of Radiation Programs-Las Vegas Facility
   U.S.  Environmental Protection Agency
   P.O.  Box 15027
   Las Vegas, Nevada 89114
                                                            10. PROGRAM ELEMENT NO.
             11. CONTRACT/GRANT NO.
 12. SPONSORING AGENCY NAME AND ADDRESS
   Same as above
                                                            13. TYPE OF REPORT AND PERIOD COVERED
             14. SPONSORING AGENCY CODE
 IS. SUPPLEMENTARY NOTES
 16. ABSTRACT
   A generalized model  has been developed to  predict the changes with  time in the
   weathering half-life of the resuspension factor for plutonium 239 and other long-
   lived radioactive  contaminants.  The model is largely based on  assumptions and
   empirical data presented by other authors, and is applicable to a wide range of
   average conditions.   These conditions are  parametrically described  as ratios of
   initial and final  resuspension factors, valid for a given locality.   Based on the
   above model of time-dependent half-life, the corresponding general  model of time-
   dependent resuspension factor is developed and presented in the report.  Graphs
   of both models for a range of conditions and graphic comparisons of specific cases
   of these models with existant models are included in the report.
 7.
                                KEY WORDS AND DOCUMENT ANALYSIS
                  DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS  C.  COSATI Field/Group
   Plutonium Isotopes
   Radioactivity
   Airborne Contaminants
   Resuspension Models
Plutonium 239
Alpha particles
Resuspension
Resuspension factor
Weathering half-life
1802
1808
1302
1201
 8. DISTRIBUTION STATEMENT
      Release  to  public
                                              19. SECURITY CLASS (This Report)
                                                Unclassified
                           21. NO. OF PAGES

                               53
20. SECURITY CLASS (Thispage)
  Unclassified
                                                                         22. PRICE
EPA Form 2220-1 (9-73)

-------