United States
Environmental Protection
Agency
Air and Energy Engineering
Research Laboratory
Research Triangle Park. NC 27711
Research and Development
EPA/600/SR-94/201
& EPA Project Summary
January 1995
Development of a
Lumped-Parameter Model of
Indoor Radon Concentrations
Kirk K. Nielson, Vern C. Rogers, and Rodger B. Holt
The report describes a simplified,
lumped-parameter model to character-
ize indoor radon concentrations from
data that are more readily available than
those required for existing mathemati-
cal models. The lumped-parameter
model was developed from numerous
sensitivity analyses with the more de-
tailed RAETRAD model and from analy-
ses of trends from empirical data sets.
The model analyses established radon
dependence on soil parameters, house
size, floor cracks and openings, and
indoor air pressures. The empirical
analyses estimated house air infiltra-
tion properties, concrete slab diffusion
properties, sub-slab ventilation effec-
tiveness, and floor crack areas.
The lumped-parameter model was
defined by simplifying these theoreti-
cal and empirical trends into a single
equation. The equation expresses net
soil-related indoor radon concentra-
tions as a function of the sub-slab ra-
don concentration, which defines the
radon source strength, and a number
of house parameters that characterize
the radon entry and accumulation char-
acteristics.
The model was validated by com-
parison to radon measurements at the
Florida Radon Research Program ra-
don test cells, by comparisons with
soil radon potential mapping calcula-
tions, and by comparisons with indoor
radon data at more than 60 houses.
The test-cell comparisons exhibited an
average agreement within 3% for the
floating-slab cell and within 17% for
the slab-in-stem-wall cell. The compari-
sons with soli radon potential mapping
calculations showed a relative standard
deviation of about 30%. The compari-
sons with house radon data depended
on sub-slab ventilation system but av-
eraged within a factor of 2 for both
slab-in-stem-wall houses and mono-
lithic slab houses.
This Project Summary was developed
by EPA's Air and Energy Engineering
Research Laboratory, Research Tri-
angle Park, NC, to announce key find-
Ings of the research project that Is fully
documented In a separate report of the
same title (see Project Report ordering
information at back).
Introduction
A lumped-parameter model has been
developed to provide a simple means of
estimating indoor radon concentrations from
data that are more commonly available than
those required for existing, detailed math-
ematical models. It was developed under
the Florida Radon Research Program
(FRRP) to simplify evaluations of different
construction options for attenuating indoor
radon entry and accumulation. The FRRP
technically supports the Florida Department
of Community Affairs development of
radon-resistant building construction stan-
dards. The lumped-parameter model con-
sists of an empirical correlation of long-term
average indoor radon concentrations with
site parameters that include sub-slab radon
levels and soil water contents, and house
parameters that include house width and
height, age, slab crack location, slab water/
cement ratio, stem-wall type, indoor pres-
sures driving air infiltration through the su-
perstructure and the slab, and sub-slab
ventilation effectiveness. The use of default
Printed on Recycled Paper
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values for some of these parameters fur-
ther simplifies the model with little loss of
accuracy.
Theoretical Basis and
Development
The lumped-parameter model was de-
veloped from numerous sensitivity analy-
ses with the more detailed RAETRAD
model and from analyses of trends from
empirical data sets that characterized build-
Ing construction and performance. The
theoretical RAETRAD analyses character-
ized radon entry by diffusion through the
intact floor slab and by both diffusive and
advective entry through cracks and open-
Ings in the building floor and foundation
walls. The RAETRAD analyses suggested
simplified approximations to express in-
door radon as a function of radon source
strength, house radon entry rates, and
house ventilation parameters. Radon
source strength was defined in terms of
sub-stab radon concentration, and house
radon entry rates were defined from floor
openings, pressure driving forces, and slab
diffush/ity.
From RAETRAD sensitivity analyses,
the interactive effects of soil type, house
size, ftoor crack size and location, and
indoor pressures were determined. House
size had a relatively small, inverse effect
when the slab radon entry rate was nor-
malized by the house area and sub-slab
radon concentration. Sub-slab radon deple-
tion from leakage through cracks also
caused a small, linear dependence of slab
radon entry on crack location. Radon en-
try through cracks also was relatively uni-
form for a given soil when normalized by
the crack area and sub-slab radon con-
centration. Entry through cracks varied lin-
early with indoor pressure and had an
exponential dependence on soil type,
which was represented by a surrogate of
water saturation fraction. These effects
were represented by simple, fitted para-
metric relationships that were combined
into a single equation to express indoor
radon as a function of sub-slab radon, soil
water content (a surrogate that includes
permeability effects), house width, floor
crack location and area fraction, indoor
pressure (driving soil gas entry), and house
and soil ventilation rates.
Empirical Basis and
Development
Empirical analyses of trends from prior
data sets characterized house air infiltra-
tion properties, concrete slab diffusion
properties, sub-slab ventilation effective-
ness, and ftoor crack areas. The data on
Florida house ventilation rates suggested
a nominal 0.25 air change per hour (ach)
passive infiltration rate with an
age-dependent increase of 0.007 ach per
year. The infiltration rates were expressed
in terms of the passive, ventilating indoor
air pressure, which was estimated from
the FRRP house data to be -0.7 Pa. This
value, combined with age-dependence,
corresponds to a ventilation-pressure re-
lationship of X,, = 0.31 (APV)06 + 0.007y,
where A,, is tne house air infiltration rate
(ach), APV is the ventilating indoor pres-
sure (Pa), and y is the age of the house
(years).
Radon diffusion measurements in
Florida concrete floor slabs suggested a
correlation with their water/cement ratio
as a surrogate for their radon diffusion
coefficient. The resulting trend had an
exponential-dependence on the water/
cement ratio of the concrete. Sub-slab
ventilation (SSV) effectiveness was ap-
proximated from reviews of prior perfor-
mance by an 80% effectiveness estimate
for active SSV systems and approxi-
mately 6% effectiveness for passive SSV
systems. Floor crack areas were recog-
nized as being difficult to characterize
by visual inspection, and to consist of
an approximate 0.2% leakage area plus
an additional 0.29% that could result
from a hollow stem wall.
Lumped-Parameter Model
The lumped-parameter model used the
combined fitting constants from the theo-
retical and empirical analyses to express
the net, soil-related indoor radon as a
function of 11 variables and 14 constants.
The resulting expression for the lumped
parameter model is
Cnet - Cin -Coul - hp^1(APv)" + 0.007yl
[(2x10-3 + 2,9x10-38) (4r+APe;3.:0:045068)
"70
+2.9x10-V1'4W
xh
where
C.n =
net indoor radon concentration
from sub-slab sources (pCi L1)
total indoor radon concentra-
tion (pCi L-1)
outdoor (background) radon
concentration (pCi L1)
sub-slab radon concentration
(pCi L1)
radon reduction factor from
sub-slab ventilation (fssv=0.2
for active system, 0.936 for
passive system, and 1 for
capped or no system)
h = height of indoor volume (m)
Apv _ ventilation-driving
indoor-outdoor pressure differ-
ence (Pa)
y = house age (years)
8 = 1 for hollow-block stem walls
and 0 for poured monolithic
stem walls
AP = indoor-outdoor pressure differ-
ence (Pa)
S = sub-slab soil water content
(fraction of saturation)
W = concrete slab water/cement
ratio
xcik = location of dominant crack
from perimeter of house (m)
xh - house minor dimension (width)
(m) = rh->/7i.
The factor C^, fsStf, and h directly multi-
ply the net indoor radon concentration pre-
dicted by the lumped-parameter model.
The ventilation-driving indoor pressure,
APV, also directly affects the indoor radon
level with the nominal age dependence.
The remaining terms account for particu-
lar mechanisms or variations in indoor ra-
don entry. The first term in the brackets is
a product of crack area fraction (2x10'3 +
2.9 x 10'38) and radon entry velocity. The
1/70 term approximates diffusive entry
through the crack and the APexp(-3-
0.045e6S) term approximates advective
entry through the crack. The second ma-
jor term in the brackets accounts for ra-
don diffusion through the slab and depends
on the water/cement ratio of the concrete.
The third term in the brackets slightly in-
creases the effect of radon diffusion
through the slab due to slab crack loca-
tion, and the last term in brackets slightly
increases the diffusive entry through the
slab for small structures.
For the reference house conditions used
previously (AP = -2.4Pa), the lumped-pa-
rameter model estimates that slightly over
half of the" radon entry'Dccurs by advec-
tion through floor cracks and openings for
a monolithic slab house, and about two-
thirds occurs by advection through the
openings for a slab-in-stem wall (SSW)
house. To estimate absolute indoor radon
concentrations, the sub-slab radon con-
centration can be represented by mea-
sured values or calculated from surrogates
such as radon flux or soil-gas radon con-
centrations.
Model Validation
Calculated indoor/sub-slab radon ratios
(Cno,/Csub) from the lumped-parameter
model were compared with measured data
from the FRRP test cells, with reference
data used to generate soil radon potential
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maps, and with measured data from FRRP
demonstration and evaluation houses. The
test-cell data were compared for each of
eight sets of conditions, involving both
floating-slab and SSW construction, and
indoor pressures from passive to -50 Pa.
The ratios of calculated/measured C^/C^
ratios varied by about 40-50%, but aver-
aged 0.97 for the floating-slab cell and
0.83 for the SSW cell.
Comparisons of calculated C^/C^ ra-
tios with soil radon potentials calculated
for maps for Alachua County, Florida, had
an overall relative standard deviation of
30%, with most of the differences occur-
ring with the dry, sandy-soil profiles. In
general, the comparison showed a posi-
tive bias for the lumped-parameter model
that was associated with inherent differ-
ences between the lumped-parameter
model and the radon mapping algorithm.
Since neither explicitly defines fill soil lay-
ers or detailed crack properties, these are
represented implicitly, as required for the
different purposes of characterizing soil
radon potentials for mapping and house
radon resistance for the lumped-param-
eter model.
For comparisons of calculated C^/C^
ratios with FRRP house data, information
was compiled from 20 houses studied by
Geomet Technologies, Inc., 13 houses
studied by Florida Solar Energy Center,
and 30 houses studied by Southern Re-
search Institute. The houses varied from
new to 36 years old, were almost entirely
slab-on-grade, and included a nearly equal
mix of SSW and monolithic slab-wall con-
struction. Many had SSV systems installed,
including both suction-pit and ventilation-
mat designs.
Net indoor radon data were obtained
by subtracting outdoor radon concentra-
tions from measured indoor values. The
outdoor concentrations were estimated
from a set of theoretical, 1-dimensional
diffusion-dispersion calculations to de-
velop a correlation based on sub-slab
radon levels and soil properties. The
theoretical calculations used a reference
set of atmospheric dispersion data and
boundary conditions for normal turbu-
lence conditions, and led to the correla-
tions Cout « 0^(9x10'5-8.7x10-5S). This
relationship corresponds to about 0.1
pCi L1 outdoors for a sub-slab radon
concentration of 2,000 pCi L1 in sandy
soil.
Measured C^C^ ratios for both SSW
and monolithic slab houses were highest
for houses with no SSV system, were
slightly lower for houses with a passive
SSV system, and were significantly lower
for houses with active SSV systems.
Houses with capped SSV systems were
erratic, being higher than those without
SSV systems for SSW houses and lower
than those without SSV systems for mono-
lithic slab houses. Calculated CM/Csub ra-
tios, using best-estimate input parameters,
were lower than measured ratios for all of
the SSW and monolithic slab houses by a
factor that averaged 0.5+0.3 among the
SSV categories. The only exception was
the capped-SSV case for monolithic-slab
houses, where a 3-fold higher calculated
value raised the average ratio to 1.1+1.2
for the SSV categories if it is included.
Alternative estimates of floor crack and
opening areas were explored as a pos-
sible explanation of the higher observed
indoor radon concentrations. Estimates of
the areas required to give agreement be-
tween the lumped-parameter model and
the measured data were excessive. Alter-
native estimates of concrete water/cement
ratios to explain the differences may be
, plausible, however. Average water/cement
ratios of 0.64 to 0.70 were estimated to
explain the respective passive-SSV and
active-SSV data for both SSW and mono-
lithic houses. For no-SSV houses, how-
ever, higher water/cement ratios of 0.77
were required. Another explanation of the
discrepancy may be the void volume of
the SSV systems, in which the sub-slab
radon concentrations typically were mea-
sured. These sub-slab volumes may have
lower concentrations than the soil pore
volumes in contact with most of the slab,
therefore giving a high estimate for the
measured C^/C.^, ratio.
Conclusions and
Recommendations
The lumped-parameter model combines
theoretical and empirical trends to form a
simple expression to estimate indoor ra-
don concentrations for Florida slab-on-
grade houses. The expression retains the
fundamental, parametric dependencies of
the more detailed models and data sets. It
agrees with FRRP radon test cell data
within averages of 3-17%, and with indoor
radon data from over 60 houses within
about a factor of two (houses being
higher). Its agreement with radon map-
ping calculations is within about 30%, ow-
ing to fundamental differences in the
purposes of the two algorithms being com-
pared.
The present analyses suggest several
parameters to be included in future new
house evaluation projects. These include
outdoor radon concentrations (for obtain-
ing net indoor levels), concrete slab diffu-
sion properties (water/cement ratio,
diffusion coefficient, or density), and sur-
face soil radon fluxes (as possible surro-
gates for sub-slab radon on undeveloped
land). The analyses also suggest poten-
tial improvements in the lumped-param-
eter model. These include improved,
alternative definitions of floor crack and
opening areas and their associated per-
meability, an improved correlation for pre-
dicting concrete diffusivity, and
representation of sub-slab void volumes
associated with SSV systems for their ef-
fects on sub-slab radon measurements.
Even without these changes, however, the
lumped-parameter model is useful for pre-
dicting indoor radon concentrations from
a minimum of readily obtainable param-
eters. It also is useful for interpreting and
coordinating data from the FRRP New
House Evaluation Project and for provid-
ing additional focus for that project.
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K. Nielson, V.C. Rogers, and R. Holt are with Rogers and Associates Engineering
Corp., Salt Lake City, UT 84110-0330.
David C. Sanchez is the EPA Project Officer (see below).
Tha complete report, entitled "Development of a Lumped-Parameter Model of
Indoor Radon Concentrations," (Order No. PB95-142048; Cost: $27.00, subject
to change) will be available only from
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at
Air and Energy Engineering Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
United States
Environmental Protection Agency
Center for Environmental Research Information
Cincinnati, OH 45268
Official Business
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